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This is the fifth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

We have almost finished off the test problem of showing that {H_t(x+iy) \neq 0} whenever {t = y = 0.4}. We have two useful approximations for {H_t}, which we have denoted {A^{eff}+B^{eff}} and {A^{eff}+B^{eff}-C^{eff}}, and a normalising quantity {B^{eff}_0} that is asymptotically equal to the above expressions; see the wiki page for definitions. In practice, the {A^{eff}+B^{eff}} approximation seems to be accurate within about one or two significant figures, whilst the {A^{eff}+B^{eff}-C^{eff}} approximation is accurate to about three or four. We have an effective upper bound

\displaystyle  |H_t - A^{eff} - B^{eff}| \leq E_1 + E_2 + E_3^*

where the expressions {E_1,E_2,E_3^*} are quite small in practice ({E_3^*} is typically about two orders of magnitude smaller than the main term {B^{eff}_0} once {x} is moderately large, and the error terms {E_1,E_2} are even smaller). See this page for details. In principle we could also obtain an effective upper bound for {|H_t - (A^{eff} + B^{eff} - C^{eff})|} (the {E_3^*} term would be replaced by something smaller).

The ratio {\frac{A^{eff}+B^{eff}}{B^{eff}_0}} takes the form of a difference {\sum_{n=1}^N \frac{b_n}{n^s} - e^{i\theta} \sum_{n=1}^N \frac{a_n}{n^s}} of two Dirichlet series, where {e^{i\theta}} is a phase whose value is explicit but perhaps not terribly important, and the coefficients {b_n, a_n} are explicit and relatively simple ({b_n} is {\exp( \frac{t}{4} \log^2 n)}, and {a_n} is approximately {(n/N)^y b_n}). To bound this away from zero, we have found it advantageous to mollify this difference by multiplying by an Euler product {\prod_{p \leq P} (1 - \frac{b_p}{p^s})} to cancel much of the initial oscillation; also one can take advantage of the fact that the {b_n} are real and the {a_n} are (approximately) real. See this page for details. The upshot is that we seem to be getting good lower bounds for the size of this difference of Dirichlet series starting from about {x \geq 5 \times 10^5} or so. The error terms {E_1,E_2,E_3^*} are already quite small by this stage, so we should soon be able to rigorously keep {H_t} from vanishing at this point. We also have a scheme for lower bounding the difference of Dirichlet series below this range, though it is not clear at present how far we can continue this before the error terms {E_1,E_2,E_3^*} become unmanageable. For very small {x} we may have to explore some faster ways to compute the expression {H_t}, which is still difficult to compute directly with high accuracy. One will also need to bound the somewhat unwieldy expressions {E_1,E_2} by something more manageable. For instance, right now these quantities depend on the continuous variable {x}; it would be preferable to have a quantity that depends only on the parameter {N = \lfloor \sqrt{ \frac{x}{4\pi} + \frac{t}{16} }\rfloor}, as this could be computed numerically for all {x} in the remaining range of interest quite quickly.

As before, any other mathematical discussion related to the project is also welcome here, for instance any summaries of previous discussion that was not covered in this post.

This is the fourth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, continuing Progress will be summarised at this Polymath wiki page.

We are getting closer to finishing off the following test problem: can one show that {H_t(x+iy) \neq 0} whenever {t = y = 0.4}, {x \geq 0}? This would morally show that {\Lambda \leq 0.48}. A wiki page for this problem has now been created here. We have obtained a number of approximations {A+B, A'+B', A^{eff}+B^{eff}, A^{toy}+B^{toy}} to {H_t} (see wiki page), though numeric evidence indicates that the approximations are all very close to each other. (Many of these approximations come with a correction term {C}, but thus far it seems that we may be able to avoid having to use this refinement to the approximations.) The effective approximation {A^{eff} + B^{eff}} also comes with an effective error bound

\displaystyle |H_t - A^{eff} - B^{eff}| \leq E_1 + E_2 + E_3

for some explicit (but somewhat messy) error terms {E_1,E_2,E_3}: see this wiki page for details. The original approximations {A+B, A'+B'} can be considered deprecated at this point in favour of the (slightly more complicated) approximation {A^{eff}+B^{eff}}; the approximation {A^{toy}+B^{toy}} is a simplified version of {A^{eff}+B^{eff}} which is not quite as accurate but might be useful for testing purposes.

It is convenient to normalise everything by an explicit non-zero factor {B^{eff}_0}. Asymptotically, {(A^{eff} + B^{eff}) / B^{eff}_0} converges to 1; numerically, it appears that its magnitude (and also its real part) stays roughly between 0.4 and 3 in the range {10^5 \leq x \leq 10^6}, and we seem to be able to keep it (or at least the toy counterpart {(A^{toy} + B^{toy}) / B^{toy}_0}) away from zero starting from about {x \geq 4 \times 10^6} (here it seems that there is a useful trick of multiplying by Euler-type factors like {1 - \frac{1}{2^{1-s}}} to cancel off some of the oscillation). Also, the bounds on the error {(H_t - A^{eff} - B^{eff}) / B^{eff}_0} seem to be of size about 0.1 or better in these ranges also. So we seem to be on track to be able to rigorously eliminate zeroes starting from about {x \geq 10^5} or so. We have not discussed too much what to do with the small values of {x}; at some point our effective error bounds will become unusable, and we may have to find some more faster ways to compute {H_t}.

In addition to this main direction of inquiry, there have been additional discussions on the dynamics of zeroes, and some numerical investigations of the behaviour Lehmer pairs under heat flow. Contributors are welcome to summarise any findings from these discussions from previous threads (or on any other related topic, e.g. improvements in the code) in the comments below.

Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and I have uploaded to the arXiv our paper “Long gaps in sieved sets“, submitted to J. Europ. Math. Soc..

This paper originated from the MSRI program in analytic number theory last year, and was centred around variants of the question of finding large gaps between primes. As discussed for instance in this previous post, it is now known that within the set of primes {{\mathcal P} = \{2,3,5,\dots\}}, one can find infinitely many adjacent elements {a,b} whose gap {b-a} obeys a lower bound of the form

\displaystyle b-a \gg \log a \frac{\log_2 a \log_4 a}{\log_3 a}

where {\log_k} denotes the {k}-fold iterated logarithm. This compares with the trivial bound of {b-a \gg \log a} that one can obtain from the prime number theorem and the pigeonhole principle. Several years ago, Pomerance posed the question of whether analogous improvements to the trivial bound can be obtained for such sets as

\displaystyle {\mathcal P}_2 = \{ n \in {\bf N}: n^2+1 \hbox{ prime} \}.

Here there is the obvious initial issue that this set is not even known to be infinite (this is the fourth Landau problem), but let us assume for the sake of discussion that this set is indeed infinite, so that we have an infinite number of gaps to speak of. Standard sieve theory techniques give upper bounds for the density of {{\mathcal P}_2} that is comparable (up to an absolute constant) to the prime number theorem bounds for {{\mathcal P}}, so again we can obtain a trivial bound of {b-a \gg \log a} for the gaps of {{\mathcal P}_2}. In this paper we improve this to

\displaystyle b-a \gg \log a \log^c_2 a

for an absolute constant {c>0}; this is not as strong as the corresponding bound for {{\mathcal P}}, but still improves over the trivial bound. In fact we can handle more general “sifted sets” than just {{\mathcal P}_2}. Recall from the sieve of Eratosthenes that the elements of {{\mathcal P}} in, say, the interval {[x/2, x]} can be obtained by removing from {[x/2, x]} one residue class modulo {p} for each prime up to {\sqrt{x}}, namely the class {0} mod {p}. In a similar vein, the elements of {{\mathcal P}_2} in {[x/2,x]} can be obtained by removing for each prime {p} up to {x} zero, one, or two residue classes modulo {p}, depending on whether {-1} is a quadratic residue modulo {p}. On the average, one residue class will be removed (this is a very basic case of the Chebotarev density theorem), so this sieving system is “one-dimensional on the average”. Roughly speaking, our arguments apply to any other set of numbers arising from a sieving system that is one-dimensional on average. (One can consider other dimensions also, but unfortunately our methods seem to give results that are worse than a trivial bound when the dimension is less than or greater than one.)

The standard “Erdős-Rankin” method for constructing long gaps between primes proceeds by trying to line up some residue classes modulo small primes {p} so that they collectively occupy a long interval. A key tool in doing so are the smooth number estimates of de Bruijn and others, which among other things assert that if one removes from an interval such as {[1,x]} all the residue classes {0} mod {p} for {p} between {x^{1/u}} and {x} for some fixed {u>1}, then the set of survivors has exceptionally small density (roughly of the order of {u^{-u}}, with the precise density given by the Dickman function), in marked contrast to the situation in which one randomly removes one residue class for each such prime {p}, in which the density is more like {1/u}. One generally exploits this phenomenon to sieve out almost all the elements of a long interval using some of the primes available, and then using the remaining primes to cover up the remaining elements that have not already been sifted out. In the more recent work on this problem, advanced combinatorial tools such as hypergraph covering lemmas are used for the latter task.

In the case of {{\mathcal P}_2}, there does not appear to be any analogue of smooth numbers, in the sense that there is no obvious way to arrange the residue classes so that they have significantly fewer survivors than a random arrangement. Instead we adopt the following semi-random strategy to cover an interval {[1,y]} by residue classes. Firstly, we randomly remove residue classes for primes {p} up to some intermediate threshold {z} (smaller than {y} by a logarithmic factor), leaving behind a preliminary sifted set {S_{[2,z]}}. Then, for each prime {p} between {z} and another intermediate threshold {x/2}, we remove a residue class mod {p} that maximises (or nearly maximises) its intersection with {S_{[2,z]}}. This ends up reducing the number of survivors to be significantly below what one would achieve if one selects residue classes randomly, particularly if one also uses the hypergraph covering lemma from our previous paper. Finally, we cover each the remaining survivors by a residue class from a remaining available prime.

This is the second “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, continuing this previous thread. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

We now have the following proposition (see this page for a proof sketch) that looks like it can give a numerically feasible approach to bound {\Lambda}:

Proposition 1 Suppose that one has parameters {t_0, T, \varepsilon > 0} obeying the following properties:

  • All the zeroes of {H_0(x+iy)=0} with {0 \leq x \leq T} are real.
  • There are no zeroes {H_t(x+iy)=0} with {0 \leq t \leq t_0} in the region {\{ x+iy: x \geq T; 1-2t \geq y^2 \geq \varepsilon^2 + (T-x)^2 \}}.
  • There are no zeroes {H_{t_0}(x+iy)=0} with {x > T} and {y \geq \varepsilon}.

Then one has {\Lambda \leq t_0 + \frac{1}{2} \varepsilon^2}.

The first hypothesis is already known for {T} up to about {10^{12}} (we should find out exactly what we can reach here). Preliminary calculations suggest that we can obtain the third item provided that {t_0, \varepsilon \gg \frac{1}{\log T}}. The second hypothesis requires good numerical calculation for {H_t}, to which we now turn.

The initial definition of {H_t} is given by the formula

\displaystyle  H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du


\displaystyle  \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp(-\pi n^2 e^{4u}).

This formula has proven numerically computable to acceptable error up until about the first hundred zeroes of {H_t}, but degrades after that, and so other exact or approximate formulae for {H_t} are needed. One possible exact formula that could be useful is

\displaystyle  H_t(z) = \frac{1}{2} (K_{t,\theta}(z) + \overline{K_{t,\theta}(\overline{z})})


\displaystyle  K_{t,\theta}(z) := \sum_{n=1}^\infty (2\pi^2 n^4 I_{t,\theta}(z-9i, \pi n^2) - 3\pi n^2I_{t,\theta}(z-5i, \pi n^2))


\displaystyle  I_{t,\theta}(b,\beta) := \int_{i\theta}^{i\theta+i\infty} \exp(tu^2 - \beta e^{4u} + ibu)\ du

and {-\pi/8 < \theta < \pi/8} can be chosen arbitrarily. We are still trying to see if this can be implemented numerically to give better accuracy than the previous formula.

It seems particularly promising to develop a generalisation of the Riemann-Siegel approximate functional equation for {H_0}. Preliminary computations suggest in particular that we have the approximation

\displaystyle  H_t(x+iy) \approx \frac{1}{4} (F_t(\frac{1+ix-y}{2}) + \overline{F_t(\frac{1+ix+y}{2})})


\displaystyle  F_t(s) := \pi^{-s/2} \Gamma(\frac{s+4}{2}) \sum_{n \leq \sqrt{\mathrm{Im}(s)/2\pi}} \frac{\exp( \frac{t}{16} \log^2 \frac{s+4}{2\pi n^2})}{n^s}.

Some very preliminary numerics suggest that this formula is reasonably accurate even for moderate values of {x}, though further numerical verification is needed. As a proof of concept, one could take this approximation as exact for the purposes of seeing what ranges of {T} one can feasibly compute with (and for extremely large values of {T}, we will presumably have to introduce some version of the Odlyzko-Schönhage algorithm. Of course, to obtain a rigorous result, we will eventually need a rigorous version of this formula with explicit error bounds. It may also be necessary to add more terms to the approximation to reduce the size of the error.

Sujit Nair has kindly summarised for me the current state of affairs with the numerics as follows:

  • We need a real milestone and work backward to set up intermediate goals. This will definitely help bring in focus!
  • So far, we have some utilities to compute zeroes of {H_t} with a nonlinear solver which uses roots of {H_0} as an initial condition. The solver is a wrapper around MINPACK’s implementation of Powell’s method. There is some room for optimization. For example, we aren’t providing the solver with the analytical Jacobian which speeds up the computation and increases accuracy.
  • We have some results in the output folder which contains the first 1000 roots of {H_t} for some small values of {t \in \{0.01, 0.1, 0.22\}}, etc. They need some more organization and visualization.

We have a decent initial start but we have some ways to go. Moving forward, here is my proposition for some areas of focus. We should expand and prioritize after some open discussion.

  1. Short term Optimize the existing framework and target to have the first million zeros of {H_t} (for a reasonable range of {t}) and the corresponding plots. With better engineering practice and discipline, I am confident we can get to a few tens of millions range. Some things which will help include parallelization, iterative approaches (using zeroes of {H_t} to compute zeroes of {H_{t + \delta t}}), etc.
  2. Medium term We need to explore better ways to represent the zeros and compute them. An analogy is the computation of Riemann zeroes up to height {T}. It is computed by computing the sign changes of {Z(t)} (page 119 of Edwards) and by exploiting the {\sqrt T} speed up with the Riemann-Siegel formulation (over Euler-Maclaurin). For larger values of {j}, I am not sure the root solver based approach is going to work to understand the gaps between zeroes.
  3. Long term We also need a better understanding of the errors involved in the computation — truncation, hardware/software, etc.

This is the first official “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

The proposal naturally splits into at least three separate (but loosely related) topics:

  • Numerical computation of the entire functions {H_t(z)}, with the ultimate aim of establishing zero-free regions of the form {\{ x+iy: 0 \leq x \leq T, y \geq \varepsilon \}} for various {T, \varepsilon > 0}.
  • Improved understanding of the dynamics of the zeroes {z_j(t)} of {H_t}.
  • Establishing the zero-free nature of {H_t(x+iy)} when {y \geq \varepsilon > 0} and {x} is sufficiently large depending on {t} and {\varepsilon}.

Below the fold, I will present each of these topics in turn, to initiate further discussion in each of them. (I thought about splitting this post into three to have three separate discussions, but given the current volume of comments, I think we should be able to manage for now having all the comments in a single post. If this changes then of course we can split up some of the discussion later.)

To begin with, let me present some formulae for computing {H_t} (inspired by similar computations in the Ki-Kim-Lee paper) which may be useful. The initial definition of {H_t} is

\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du


\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(- \pi n^2 e^{4u} )

is a variant of the Jacobi theta function. We observe that {\Phi} in fact extends analytically to the strip

\displaystyle \{ u \in {\bf C}: -\frac{\pi}{8} < \mathrm{Im} u < \frac{\pi}{8} \}, \ \ \ \ \ (1)


as {e^{4u}} has positive real part on this strip. One can use the Poisson summation formula to verify that {\Phi} is even, {\Phi(-u) = \Phi(u)} (see this previous post for details). This lets us obtain a number of other formulae for {H_t}. Most obviously, one can unfold the integral to obtain

\displaystyle H_t(z) = \frac{1}{2} \int_{-\infty}^\infty e^{tu^2} \Phi(u) e^{izu}\ du.

In my previous paper with Brad, we used this representation, combined with Fubini’s theorem to swap the sum and integral, to obtain a useful series representation for {H_t} in the {t0} case because expressions such as {e^{tu^2} e^{9u} \exp( -\pi n^2 e^{4u} ) e^{izu}} diverge as {u} approaches {-\infty}. Nevertheless we can still perform the following contour integration manipulation. Let {0 \leq \theta < \frac{\pi}{8}} be fixed. The function {\Phi} decays super-exponentially fast (much faster than {e^{tu^2}}, in particular) as {\mathrm{Re} u \rightarrow +\infty} with {-\infty \leq \mathrm{Im} u \leq \theta}; as {\Phi} is even, we also have this decay as {\mathrm{Re} u \rightarrow -\infty} with {-\infty \leq \mathrm{Im} u \leq \theta} (this is despite each of the summands in {\Phi} having much slower decay in this direction – there is considerable cancellation!). Hence by the Cauchy integral formula we have

\displaystyle H_t(z) = \frac{1}{2} \int_{i\theta-\infty}^{i\theta+\infty} e^{tu^2} \Phi(u) e^{izu}\ du.

Splitting the horizontal line from {i\theta-\infty} to {i\theta+\infty} at {i\theta} and using the even nature of {\Phi(u)}, we thus have

\displaystyle H_t(z) = \frac{1}{2} (\int_{i\theta}^{i\theta+\infty} e^{tu^2} \Phi(u) e^{izu}\ du + \int_{-i\theta}^{-i\theta+\infty} e^{tu^2} \Phi(u) e^{-izu}\ du).

Using the functional equation {\Phi(\overline{u}) = \overline{\Phi(u)}}, we thus have the representation

\displaystyle H_t(z) = \frac{1}{2} ( K_{t,\theta}(z) + \overline{K_{t,\theta}(\overline{z})} ) \ \ \ \ \ (2)



\displaystyle K_{t,\theta}(z) := \int_{i\theta}^{i \theta+\infty} e^{tu^2} \Phi(u) e^{izu}\ du

\displaystyle = \sum_{n=1}^\infty 2 \pi^2 n^4 I_{t, \theta}( z - 9i, \pi n^2 ) - 3 \pi n^2 I_{t,\theta}( z - 5i, \pi n^2 )

where {I_{t,\theta}(b,\beta)} is the oscillatory integral

\displaystyle I_{t,\theta}(b,\beta) := \int_{i\theta}^{i\theta+\infty} \exp( tu^2 - \beta e^{4u} + i b u )\ du. \ \ \ \ \ (3)


The formula (2) is valid for any {0 \leq \theta < \frac{\pi}{8}}. Naively one would think that it would be simplest to take {\theta=0}; however, when {z=x+iy} and {x} is large (with {y} bounded), it seems asymptotically better to take {\theta} closer to {\pi/8}, in particular something like {\theta = \frac{\pi}{8} - \frac{1}{4x}} seems to be a reasonably good choice. This is because the integrand in (3) becomes significantly less oscillatory and also much lower in amplitude; the {\exp(ibu)} term in (3) now generates a factor roughly comparable to {\exp( - \pi x/8 )} (which, as we will see below, is the main term in the decay asymptotics for {H_t(x+iy)}), while the {\exp( - \beta e^{4u} )} term still exhibits a reasonable amount of decay as {u \rightarrow \infty}. We will use the representation (2) in the asymptotic analysis of {H_t} below, but it may also be a useful representation to use for numerical purposes.

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Building on the interest expressed in the comments to this previous post, I am now formally proposing to initiate a “Polymath project” on the topic of obtaining new upper bounds on the de Bruijn-Newman constant {\Lambda}. The purpose of this post is to describe the proposal and discuss the scope and parameters of the project.

De Bruijn introduced a family {H_t: {\bf C} \rightarrow {\bf C}} of entire functions for each real number {t}, defined by the formula

\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du

where {\Phi} is the super-exponentially decaying function

\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(-\pi n^2 e^{4u}).

As discussed in this previous post, the Riemann hypothesis is equivalent to the assertion that all the zeroes of {H_0} are real.

De Bruijn and Newman showed that there existed a real constant {\Lambda} – the de Bruijn-Newman constant – such that {H_t} has all zeroes real whenever {t \geq \Lambda}, and at least one non-real zero when {t < \Lambda}. In particular, the Riemann hypothesis is equivalent to the upper bound {\Lambda \leq 0}. In the opposite direction, several lower bounds on {\Lambda} have been obtained over the years, most recently in my paper with Brad Rodgers where we showed that {\Lambda \geq 0}, a conjecture of Newman.

As for upper bounds, de Bruijn showed back in 1950 that {\Lambda \leq 1/2}. The only progress since then has been the work of Ki, Kim and Lee in 2009, who improved this slightly to {\Lambda < 1/2}. The primary proposed aim of this Polymath project is to obtain further explicit improvements to the upper bound of {\Lambda}. Of course, if we could lower the upper bound all the way to zero, this would solve the Riemann hypothesis, but I do not view this as a realistic outcome of this project; rather, the upper bounds that one could plausibly obtain by known methods and numerics would be comparable in achievement to the various numerical verifications of the Riemann hypothesis that exist in the literature (e.g., that the first {N} non-trivial zeroes of the zeta function lie on the critical line, for various large explicit values of {N}).

In addition to the primary goal, one could envisage some related secondary goals of the project, such as a better understanding (both analytic and numerical) of the functions {H_t} (or of similar functions), and of the dynamics of the zeroes of these functions. Perhaps further potential goals could emerge in the discussion to this post.

I think there is a plausible plan of attack on this project that proceeds as follows. Firstly, there are results going back to the original work of de Bruijn that demonstrate that the zeroes of {H_t} become attracted to the real line as {t} increases; in particular, if one defines {\sigma_{max}(t)} to be the supremum of the imaginary parts of all the zeroes of {H_t}, then it is known that this quantity obeys the differential inequality

\displaystyle \frac{d}{dt} \sigma_{max}(t) \leq - \frac{1}{\sigma_{max}(t)} \ \ \ \ \ (1)


whenever {\sigma_{max}(t)} is positive; furthermore, once {\sigma_{max}(t) = 0} for some {t}, then {\sigma_{max}(t') = 0} for all {t' > t}. I hope to explain this in a future post (it is basically due to the attraction that a zero off the real axis has to its complex conjugate). As a corollary of this inequality, we have the upper bound

\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2 \ \ \ \ \ (2)


for any real number {t}. For instance, because all the non-trivial zeroes of the Riemann zeta function lie in the critical strip {\{ s: 0 \leq \mathrm{Re} s \leq 1 \}}, one has {\sigma_{max}(0) \leq 1}, which when inserted into (2) gives {\Lambda \leq 1/2}. The inequality (1) also gives {\sigma_{max}(t) \leq \sqrt{1-2t}} for all {0 \leq t \leq 1/2}. If we could find some explicit {t} between {0} and {1/2} where we can improve this upper bound on {\sigma_{max}(t)} by an explicit constant, this would lead to a new upper bound on {\Lambda}.

Secondly, the work of Ki, Kim and Lee (based on an analysis of the various terms appearing in the expression for {H_t}) shows that for any positive {t}, all but finitely many of the zeroes of {H_t} are real (in contrast with the {t=0} situation, where it is still an open question as to whether the proportion of non-trivial zeroes of the zeta function on the critical line is asymptotically equal to {1}). As a key step in this analysis, Ki, Kim, and Lee show that for any {t>0} and {\varepsilon>0}, there exists a {T>0} such that all the zeroes of {H_t} with real part at least {T}, have imaginary part at most {\varepsilon}. Ki, Kim and Lee do not explicitly compute how {T} depends on {t} and {\varepsilon}, but it looks like this bound could be made effective.

If so, this suggests a possible strategy to get a new upper bound on {\Lambda}:

  • Select a good choice of parameters {t, \varepsilon > 0}.
  • By refining the Ki-Kim-Lee analysis, find an explicit {T} such that all zeroes of {H_t} with real part at least {T} have imaginary part at most {\varepsilon}.
  • By a numerical computation (e.g. using the argument principle), also verify that zeroes of {H_t} with real part between {0} and {T} have imaginary part at most {\varepsilon}.
  • Combining these facts, we obtain that {\sigma_{max}(t) \leq \varepsilon}; hopefully, one can insert this into (2) and get a new upper bound for {\Lambda}.

Of course, there may also be alternate strategies to upper bound {\Lambda}, and I would imagine this would also be a legitimate topic of discussion for this project.

One appealing thing about the above strategy for the purposes of a polymath project is that it naturally splits the project into several interacting but reasonably independent parts: an analytic part in which one tries to refine the Ki-Kim-Lee analysis (based on explicitly upper and lower bounding various terms in a certain series expansion for {H_t} – I may detail this later in a subsequent post); a numerical part in which one controls the zeroes of {H_t} in a certain finite range; and perhaps also a dynamical part where one sees if there is any way to improve the inequality (2). For instance, the numerical “team” might, over time, be able to produce zero-free regions for {H_t} with an increasingly large value of {T}, while in parallel the analytic “team” might produce increasingly smaller values of {T} beyond which they can control zeroes, and eventually the two bounds would meet up and we obtain a new bound on {\Lambda}. This factoring of the problem into smaller parts was also a feature of the successful Polymath8 project on bounded gaps between primes.

The project also resembles Polymath8 in another aspect: that there is an obvious way to numerically measure progress, by seeing how the upper bound for {\Lambda} decreases over time (and presumably there will also be another metric of progress regarding how well we can control {T} in terms of {t} and {\varepsilon}). However, in Polymath8 the final measure of progress (the upper bound {H} on gaps between primes) was a natural number, and thus could not decrease indefinitely. Here, the bound will be a real number, and there is a possibility that one may end up having an infinite descent in which progress slows down over time, with refinements to increasingly less significant digits of the bound as the project progresses. Because of this, I think it makes sense to follow recent Polymath projects and place an expiration date for the project, for instance one year after the launch date, in which we will agree to end the project and (if the project was successful enough) write up the results, unless there is consensus at that time to extend the project. (In retrospect, we should probably have imposed similar sunset dates on older Polymath projects, some of which have now been inactive for years, but that is perhaps a discussion for another time.)

Some Polymath projects have been known for a breakneck pace, making it hard for some participants to keep up. It’s hard to control these things, but I am envisaging a relatively leisurely project here, perhaps taking the full year mentioned above. It may well be that as the project matures we will largely be waiting for the results of lengthy numerical calculations to come in, for instance. Of course, as with previous projects, we would maintain some wiki pages (and possibly some other resources, such as a code repository) to keep track of progress and also to summarise what we have learned so far. For instance, as was done with some previous Polymath projects, we could begin with some “online reading seminars” where we go through some relevant piece of literature (most obviously the Ki-Kim-Lee paper, but there may be other resources that become relevant, e.g. one could imagine the literature on numerical verification of RH to be of value).

One could also imagine some incidental outcomes of this project, such as a more efficient way to numerically establish zero free regions for various analytic functions of interest; in particular, the project may well end up focusing on some other aspect of mathematics than the specific questions posed here.

Anyway, I would be interested to hear in the comments below from others who might be interested in participating, or at least observing, this project, particularly if they have suggestions regarding the scope and direction of the project, and on organisational structure (e.g. if one should start with reading seminars, or some initial numerical exploration of the functions {H_t}, etc..) One could also begin some preliminary discussion of the actual mathematics of the project itself, though (in line with the leisurely pace I was hoping for), I expect that the main burst of mathematical activity would happen later, once the project is formally launched (with wiki page resources, blog posts dedicated to specific aspects of the project, etc.).


In this post we assume the Riemann hypothesis and the simplicity of zeroes, thus the zeroes of {\zeta} in the critical strip take the form {\frac{1}{2} \pm i \gamma_j} for some real number ordinates {0 < \gamma_1 < \gamma_2 < \dots}. From the Riemann-von Mangoldt formula, one has the asymptotic

\displaystyle  \gamma_n = (1+o(1)) \frac{2\pi}{\log n} n

as {n \rightarrow \infty}; in particular, the spacing {\gamma_{n+1} - \gamma_n} should behave like {\frac{2\pi}{\log n}} on the average. However, it can happen that some gaps are unusually small compared to other nearby gaps. For the sake of concreteness, let us define a Lehmer pair to be a pair of adjacent ordinates {\gamma_n, \gamma_{n+1}} such that

\displaystyle  \frac{1}{(\gamma_{n+1} - \gamma_n)^2} \geq 1.3 \sum_{m \neq n,n+1} \frac{1}{(\gamma_m - \gamma_n)^2} + \frac{1}{(\gamma_m - \gamma_{n+1})^2}. \ \ \ \ \ (1)

The specific value of constant {1.3} is not particularly important here; anything larger than {\frac{5}{4}} would suffice. An example of such a pair would be the classical pair

\displaystyle  \gamma_{6709} = 7005.062866\dots

\displaystyle  \gamma_{6710} = 7005.100564\dots

discovered by Lehmer. It follows easily from the main results of Csordas, Smith, and Varga that if an infinite number of Lehmer pairs (in the above sense) existed, then the de Bruijn-Newman constant {\Lambda} is non-negative. This implication is now redundant in view of the unconditional results of this recent paper of myself and Rodgers; however, the question of whether an infinite number of Lehmer pairs exist remain open.

In this post, I sketch an argument that Brad and I came up with (as initially suggested by Odlyzko) the GUE hypothesis implies the existence of infinitely many Lehmer pairs. We argue probabilistically: pick a sufficiently large number {T}, pick {n} at random from {T \log T} to {2 T \log T} (so that the average gap size is close to {\frac{2\pi}{\log T}}), and prove that the Lehmer pair condition (1) occurs with positive probability.

Introduce the renormalised ordinates {x_n := \frac{\log T}{2\pi} \gamma_n} for {T \log T \leq n \leq 2 T \log T}, and let {\varepsilon > 0} be a small absolute constant (independent of {T}). It will then suffice to show that

\displaystyle  \frac{1}{(x_{n+1} - x_n)^2} \geq

\displaystyle 1.3 \sum_{m \in [T \log T, 2T \log T]: m \neq n,n+1} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2}

\displaystyle  + \frac{1}{6\varepsilon^2}

(say) with probability {\gg \varepsilon^4 - o(1)}, since the contribution of those {m} outside of {[T \log T, 2T \log T]} can be absorbed by the {\frac{1}{\varepsilon^2}} factor with probability {o(1)}.

As one consequence of the GUE hypothesis, we have {x_{n+1} - x_n \leq \varepsilon^2} with probability {O(\varepsilon^6)}. Thus, if {E := \{ m \in [T \log T, 2T \log T]: x_{m+1} - x_m \leq \varepsilon^2 \}}, then {E} has density {O( \varepsilon^6 )}. Applying the Hardy-Littlewood maximal inequality, we see that with probability {O(\varepsilon^6)}, we have

\displaystyle  \sup_{h \geq 1} | \# E \cap [n+h, n-h] | \leq \frac{1}{10}

which implies in particular that

\displaystyle  |x_m - x_n|, |x_{m+1} - x_n| \gg \varepsilon^2 |m-n|

for all {m \in [T \log T, 2 T \log T] \backslash \{ n, n+1\}}. This implies in particular that

\displaystyle  \sum_{m \in [T \log T, 2T \log T]: |m-n| \geq \varepsilon^{-3}} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2} \ll \varepsilon^{-1}

and so it will suffice to show that

\displaystyle  \frac{1}{(x_{n+1} - x_n)^2}

\displaystyle \geq 1.3 \sum_{m \in [T \log T, 2T \log T]: m \neq n,n+1; |m-n| < \varepsilon^{-3}} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2} + \frac{1}{5\varepsilon^2}

(say) with probability {\gg \varepsilon^4 - o(1)}.

By the GUE hypothesis (and the fact that {\varepsilon} is independent of {T}), it suffices to show that a Dyson sine process {(x_n)_{n \in {\bf Z}}}, normalised so that {x_0} is the first positive point in the process, obeys the inequality

\displaystyle  \frac{1}{(x_{1} - x_0)^2} \geq 1.3 \sum_{|m| < \varepsilon^{-3}: m \neq 0,1} \frac{1}{(x_m - x_0)^2} + \frac{1}{(x_m - x_1)^2} \ \ \ \ \ (2)

with probability {\gg \varepsilon^4}. However, if we let {A > 0} be a moderately large constant (and assume {\varepsilon} small depending on {A}), one can show using {k}-point correlation functions for the Dyson sine process (and the fact that the Dyson kernel {K(x,y) = \sin(\pi(x-y))/\pi(x-y)} equals {1} to second order at the origin) that

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} \gg \varepsilon^4

\displaystyle  {\bf E} N_{[-\varepsilon,0]} \binom{N_{[0,\varepsilon]}}{2} \ll \varepsilon^7

\displaystyle  {\bf E} \binom{N_{[-\varepsilon,0]}}{2} N_{[0,\varepsilon]} \ll \varepsilon^7

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[\varepsilon,A^{-1}]} \ll A^{-3} \varepsilon^4

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[-A^{-1}, -\varepsilon]} \ll A^{-3} \varepsilon^4

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[-k, k]}^2 \ll k^2 \varepsilon^4 \ \ \ \ \ (3)

for any natural number {k}, where {N_{I}} denotes the number of elements of the process in {I}. For instance, the expression {{\bf E} N_{[-\varepsilon,0]} \binom{N_{[0,\varepsilon]}}{2} } can be written in terms of the three-point correlation function {\rho_3(x_1,x_2,x_3) = \mathrm{det}(K(x_i,x_j))_{1 \leq i,j \leq 3}} as

\displaystyle  \int_{-\varepsilon \leq x_1 \leq 0 \leq x_2 \leq x_3 \leq \varepsilon} \rho_3( x_1, x_2, x_3 )\ dx_1 dx_2 dx_3

which can easily be estimated to be {O(\varepsilon^7)} (since {\rho_3 = O(\varepsilon^4)} in this region), and similarly for the other estimates claimed above.

Since for natural numbers {a,b}, the quantity {ab - 2 a \binom{b}{2} - 2 b \binom{a}{2} = ab (5-2a-2b)} is only positive when {a=b=1}, we see from the first three estimates that the event {E} that {N_{[-\varepsilon,0]} = N_{[0,\varepsilon]} = 1} occurs with probability {\gg \varepsilon^4}. In particular, by Markov’s inequality we have the conditional probabilities

\displaystyle  {\bf P} ( N_{[\varepsilon,A^{-1}]} \geq 1 | E ) \ll A^{-3}

\displaystyle  {\bf P} ( N_{[-A^{-1}, -\varepsilon]} \geq 1 | E ) \ll A^{-3}

\displaystyle  {\bf P} ( N_{[-k, k]} \geq A k^{5/3} | E ) \ll A^{-4} k^{-4/3}

and thus, if {A} is large enough, and {\varepsilon} small enough, it will be true with probability {\gg \varepsilon^4} that

\displaystyle  N_{[-\varepsilon,0]}, N_{[0,\varepsilon]} = 1


\displaystyle  N_{[A^{-1}, \varepsilon]} = N_{[\varepsilon, A^{-1}]} = 0

and simultaneously that

\displaystyle  N_{[-k,k]} \leq A k^{5/3}

for all natural numbers {k}. This implies in particular that

\displaystyle  x_1 - x_0 \leq 2\varepsilon


\displaystyle  |x_m - x_0|, |x_m - x_1| \gg_A |m|^{3/5}

for all {m \neq 0,1}, which gives (2) for {\varepsilon} small enough.

Remark 1 The above argument needed the GUE hypothesis for correlations up to fourth order (in order to establish (3)). It might be possible to reduce the number of correlations needed, but I do not see how to obtain the claim just using pair correlations only.

Brad Rodgers and I have uploaded to the arXiv our paper “The De Bruijn-Newman constant is non-negative“. This paper affirms a conjecture of Newman regarding to the extent to which the Riemann hypothesis, if true, is only “barely so”. To describe the conjecture, let us begin with the Riemann xi function

\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(\frac{s}{2}) \zeta(s)

where {\Gamma(s) := \int_0^\infty e^{-t} t^{s-1}\ dt} is the Gamma function and {\zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s}} is the Riemann zeta function. Initially, this function is only defined for {\mathrm{Re} s > 1}, but, as was already known to Riemann, we can manipulate it into a form that extends to the entire complex plane as follows. Firstly, in view of the standard identity {s \Gamma(s) = \Gamma(s+1)}, we can write

\displaystyle \frac{s(s-1)}{2} \Gamma(\frac{s}{2}) = 2 \Gamma(\frac{s+4}{2}) - 3 \Gamma( \frac{s+2}{2} )

and hence

\displaystyle \xi(s) = \sum_{n=1}^\infty 2 \pi^{-s/2} n^{-s} \int_0^\infty e^{-t} t^{\frac{s+4}{2}-1}\ dt - 3 \pi^{-s/2} n^{-s} \int_0^\infty e^{-t} t^{\frac{s+2}{2}-1}\ dt.

By a rescaling, one may write

\displaystyle \int_0^\infty e^{-t} t^{\frac{s+4}{2}-1}\ dt = (\pi n^2)^{\frac{s+4}{2}} \int_0^\infty e^{-\pi n^2 t} t^{\frac{s+4}{2}-1}\ dt

and similarly

\displaystyle \int_0^\infty e^{-t} t^{\frac{s+2}{2}-1}\ dt = (\pi n^2)^{\frac{s+2}{2}} \int_0^\infty e^{-\pi n^2 t} t^{\frac{s+2}{2}-1}\ dt

and thus (after applying Fubini’s theorem)

\displaystyle \xi(s) = \int_0^\infty \sum_{n=1}^\infty 2 \pi^2 n^4 e^{-\pi n^2 t} t^{\frac{s+4}{2}-1} - 3 \pi n^2 e^{-\pi n^2 t} t^{\frac{s+2}{2}-1}\ dt.

We’ll make the change of variables {t = e^{4u}} to obtain

\displaystyle \xi(s) = 4 \int_{\bf R} \sum_{n=1}^\infty (2 \pi^2 n^4 e^{8u} - 3 \pi n^2 e^{4u}) \exp( 2su - \pi n^2 e^{4u} )\ du.

If we introduce the mild renormalisation

\displaystyle H_0(z) := \frac{1}{8} \xi( \frac{1}{2} + \frac{iz}{2} )

of {\xi}, we then conclude (at least for {\mathrm{Im} z > 1}) that

\displaystyle H_0(z) = \frac{1}{2} \int_{\bf R} \Phi(u)\exp(izu)\ du \ \ \ \ \ (1)


where {\Phi: {\bf R} \rightarrow {\bf C}} is the function

\displaystyle \Phi(u) := \sum_{n=1}^\infty (2 \pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp( - \pi n^2 e^{4u} ), \ \ \ \ \ (2)


which one can verify to be rapidly decreasing both as {u \rightarrow +\infty} and as {u \rightarrow -\infty}, with the decrease as {u \rightarrow +\infty} faster than any exponential. In particular {H_0} extends holomorphically to the upper half plane.

If we normalize the Fourier transform {{\mathcal F} f(\xi)} of a (Schwartz) function {f(x)} as {{\mathcal F} f(\xi) := \int_{\bf R} f(x) e^{-2\pi i \xi x}\ dx}, it is well known that the Gaussian {x \mapsto e^{-\pi x^2}} is its own Fourier transform. The creation operator {2\pi x - \frac{d}{dx}} interacts with the Fourier transform by the identity

\displaystyle {\mathcal F} (( 2\pi x - \frac{d}{dx} ) f) (\xi) = -i (2 \pi \xi - \frac{d}{d\xi} ) {\mathcal F} f(\xi).

Since {(-i)^4 = 1}, this implies that the function

\displaystyle x \mapsto (2\pi x - \frac{d}{dx})^4 e^{-\pi x^2} = 128 \pi^2 (2 \pi^2 x^4 - 3 \pi x^2) e^{-\pi x^2} + 48 \pi^2 e^{-\pi x^2}

is its own Fourier transform. (One can view the polynomial {128 \pi^2 (2\pi^2 x^4 - 3 \pi x^2) + 48 \pi^2} as a renormalised version of the fourth Hermite polynomial.) Taking a suitable linear combination of this with {x \mapsto e^{-\pi x^2}}, we conclude that

\displaystyle x \mapsto (2 \pi^2 x^4 - 3 \pi x^2) e^{-\pi x^2}

is also its own Fourier transform. Rescaling {x} by {e^{2u}} and then multiplying by {e^u}, we conclude that the Fourier transform of

\displaystyle x \mapsto (2 \pi^2 x^4 e^{9u} - 3 \pi x^2 e^{5u}) \exp( - \pi x^2 e^{4u} )


\displaystyle x \mapsto (2 \pi^2 x^4 e^{-9u} - 3 \pi x^2 e^{-5u}) \exp( - \pi x^2 e^{-4u} ),

and hence by the Poisson summation formula (using symmetry and vanishing at {n=0} to unfold the {n} summation in (2) to the integers rather than the natural numbers) we obtain the functional equation

\displaystyle \Phi(-u) = \Phi(u),

which implies that {\Phi} and {H_0} are even functions (in particular, {H_0} now extends to an entire function). From this symmetry we can also rewrite (1) as

\displaystyle H_0(z) = \int_0^\infty \Phi(u) \cos(zu)\ du,

which now gives a convergent expression for the entire function {H_0(z)} for all complex {z}. As {\Phi} is even and real-valued on {{\bf R}}, {H_0(z)} is even and also obeys the functional equation {H_0(\overline{z}) = \overline{H_0(z)}}, which is equivalent to the usual functional equation for the Riemann zeta function. The Riemann hypothesis is equivalent to the claim that all the zeroes of {H_0} are real.

De Bruijn introduced the family {H_t: {\bf C} \rightarrow {\bf C}} of deformations of {H_0: {\bf C} \rightarrow {\bf C}}, defined for all {t \in {\bf R}} and {z \in {\bf C}} by the formula

\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du.

From a PDE perspective, one can view {H_t} as the evolution of {H_0} under the backwards heat equation {\partial_t H_t(z) = - \partial_{zz} H_t(z)}. As with {H_0}, the {H_t} are all even entire functions that obey the functional equation {H_t(\overline{z}) = \overline{H_t(z)}}, and one can ask an analogue of the Riemann hypothesis for each such {H_t}, namely whether all the zeroes of {H_t} are real. De Bruijn showed that these hypotheses were monotone in {t}: if {H_t} had all real zeroes for some {t}, then {H_{t'}} would also have all zeroes real for any {t' \geq t}. Newman later sharpened this claim by showing the existence of a finite number {\Lambda \leq 1/2}, now known as the de Bruijn-Newman constant, with the property that {H_t} had all zeroes real if and only if {t \geq \Lambda}. Thus, the Riemann hypothesis is equivalent to the inequality {\Lambda \leq 0}. Newman then conjectured the complementary bound {\Lambda \geq 0}; in his words, this conjecture asserted that if the Riemann hypothesis is true, then it is only “barely so”, in that the reality of all the zeroes is destroyed by applying heat flow for even an arbitrarily small amount of time. Over time, a significant amount of evidence was established in favour of this conjecture; most recently, in 2011, Saouter, Gourdon, and Demichel showed that {\Lambda \geq -1.15 \times 10^{-11}}.

In this paper we finish off the proof of Newman’s conjecture, that is we show that {\Lambda \geq 0}. The proof is by contradiction, assuming that {\Lambda < 0} (which among other things, implies the truth of the Riemann hypothesis), and using the properties of backwards heat evolution to reach a contradiction.

Very roughly, the argument proceeds as follows. As observed by Csordas, Smith, and Varga (and also discussed in this previous blog post, the backwards heat evolution of the {H_t} introduces a nice ODE dynamics on the zeroes {x_j(t)} of {H_t}, namely that they solve the ODE

\displaystyle \frac{d}{dt} x_j(t) = -2 \sum_{j \neq k} \frac{1}{x_k(t) - x_j(t)} \ \ \ \ \ (3)


for all {j} (one has to interpret the sum in a principal value sense as it is not absolutely convergent, but let us ignore this technicality for the current discussion). Intuitively, this ODE is asserting that the zeroes {x_j(t)} repel each other, somewhat like positively charged particles (but note that the dynamics is first-order, as opposed to the second-order laws of Newtonian mechanics). Formally, a steady state (or equilibrium) of this dynamics is reached when the {x_k(t)} are arranged in an arithmetic progression. (Note for instance that for any positive {u}, the functions {z \mapsto e^{tu^2} \cos(uz)} obey the same backwards heat equation as {H_t}, and their zeroes are on a fixed arithmetic progression {\{ \frac{2\pi (k+\tfrac{1}{2})}{u}: k \in {\bf Z} \}}.) The strategy is to then show that the dynamics from time {-\Lambda} to time {0} creates a convergence to local equilibrium, in which the zeroes {x_k(t)} locally resemble an arithmetic progression at time {t=0}. This will be in contradiction with known results on pair correlation of zeroes (or on related statistics, such as the fluctuations on gaps between zeroes), such as the results of Montgomery (actually for technical reasons it is slightly more convenient for us to use related results of Conrey, Ghosh, Goldston, Gonek, and Heath-Brown). Another way of thinking about this is that even very slight deviations from local equilibrium (such as a small number of gaps that are slightly smaller than the average spacing) will almost immediately lead to zeroes colliding with each other and leaving the real line as one evolves backwards in time (i.e., under the forward heat flow). This is a refinement of the strategy used in previous lower bounds on {\Lambda}, in which “Lehmer pairs” (pairs of zeroes of the zeta function that were unusually close to each other) were used to limit the extent to which the evolution continued backwards in time while keeping all zeroes real.

How does one obtain this convergence to local equilibrium? We proceed by broad analogy with the “local relaxation flow” method of Erdos, Schlein, and Yau in random matrix theory, in which one combines some initial control on zeroes (which, in the case of the Erdos-Schlein-Yau method, is referred to with terms such as “local semicircular law”) with convexity properties of a relevant Hamiltonian that can be used to force the zeroes towards equilibrium.

We first discuss the initial control on zeroes. For {H_0}, we have the classical Riemann-von Mangoldt formula, which asserts that the number of zeroes in the interval {[0,T]} is {\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + O(\log T)} as {T \rightarrow \infty}. (We have a factor of {4\pi} here instead of the more familiar {2\pi} due to the way {H_0} is normalised.) This implies for instance that for a fixed {\alpha}, the number of zeroes in the interval {[T, T+\alpha]} is {\frac{\alpha}{4\pi} \log T + O(\log T)}. Actually, because we get to assume the Riemann hypothesis, we can sharpen this to {\frac{\alpha}{4\pi} \log T + o(\log T)}, a result of Littlewood (see this previous blog post for a proof). Ideally, we would like to obtain similar control for the other {H_t}, {\Lambda \leq t < 0}, as well. Unfortunately we were only able to obtain the weaker claims that the number of zeroes of {H_t} in {[0,T]} is {\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + O(\log^2 T)}, and that the number of zeroes in {[T, T+\alpha \log T]} is {\frac{\alpha}{4 \pi} \log^2 T + o(\log^2 T)}, that is to say we only get good control on the distribution of zeroes at scales {\gg \log T} rather than at scales {\gg 1}. Ultimately this is because we were only able to get control (and in particular, lower bounds) on {|H_t(x-iy)|} with high precision when {y \gg \log x} (whereas {|H_0(x-iy)|} has good estimates as soon as {y} is larger than (say) {2}). This control is obtained by the expressing {H_t(x-iy)} in terms of some contour integrals and using the method of steepest descent (actually it is slightly simpler to rely instead on the Stirling approximation for the Gamma function, which can be proven in turn by steepest descent methods). Fortunately, it turns out that this weaker control is still (barely) enough for the rest of our argument to go through.

Once one has the initial control on zeroes, we now need to force convergence to local equilibrium by exploiting convexity of a Hamiltonian. Here, the relevant Hamiltonian is

\displaystyle H(t) := \sum_{j,k: j \neq k} \log \frac{1}{|x_j(t) - x_k(t)|},

ignoring for now the rather important technical issue that this sum is not actually absolutely convergent. (Because of this, we will need to truncate and renormalise the Hamiltonian in a number of ways which we will not detail here.) The ODE (3) is formally the gradient flow for this Hamiltonian. Furthermore, this Hamiltonian is a convex function of the {x_j} (because {t \mapsto \log \frac{1}{t}} is a convex function on {(0,+\infty)}). We therefore expect the Hamiltonian to be a decreasing function of time, and that the derivative should be an increasing function of time. As time passes, the derivative of the Hamiltonian would then be expected to converge to zero, which should imply convergence to local equilibrium.

Formally, the derivative of the above Hamiltonian is

\displaystyle \partial_t H(t) = -4 E(t), \ \ \ \ \ (4)


where {E(t)} is the “energy”

\displaystyle E(t) := \sum_{j,k: j \neq k} \frac{1}{|x_j(t) - x_k(t)|^2}.

Again, there is the important technical issue that this quantity is infinite; but it turns out that if we renormalise the Hamiltonian appropriately, then the energy will also become suitably renormalised, and in particular will vanish when the {x_j} are arranged in an arithmetic progression, and be positive otherwise. One can also formally calculate the derivative of {E(t)} to be a somewhat complicated but manifestly non-negative quantity (a sum of squares); see this previous blog post for analogous computations in the case of heat flow on polynomials. After flowing from time {\Lambda} to time {0}, and using some crude initial bounds on {H(t)} and {E(t)} in this region (coming from the Riemann-von Mangoldt type formulae mentioned above and some further manipulations), we can eventually show that the (renormalisation of the) energy {E(0)} at time zero is small, which forces the {x_j} to locally resemble an arithmetic progression, which gives the required convergence to local equilibrium.

There are a number of technicalities involved in making the above sketch of argument rigorous (for instance, justifying interchanges of derivatives and infinite sums turns out to be a little bit delicate). I will highlight here one particular technical point. One of the ways in which we make expressions such as the energy {E(t)} finite is to truncate the indices {j,k} to an interval {I} to create a truncated energy {E_I(t)}. In typical situations, we would then expect {E_I(t)} to be decreasing, which will greatly help in bounding {E_I(0)} (in particular it would allow one to control {E_I(0)} by time-averaged quantities such as {\int_{\Lambda/2}^0 E_I(t)\ dt}, which can in turn be controlled using variants of (4)). However, there are boundary effects at both ends of {I} that could in principle add a large amount of energy into {E_I}, which is bad news as it could conceivably make {E_I(0)} undesirably large even if integrated energies such as {\int_{\Lambda/2}^0 E_I(t)\ dt} remain adequately controlled. As it turns out, such boundary effects are negligible as long as there is a large gap between adjacent zeroes at boundary of {I} – it is only narrow gaps that can rapidly transmit energy across the boundary of {I}. Now, narrow gaps can certainly exist (indeed, the GUE hypothesis predicts these happen a positive fraction of the time); but the pigeonhole principle (together with the Riemann-von Mangoldt formula) can allow us to pick the endpoints of the interval {I} so that no narrow gaps appear at the boundary of {I} for any given time {t}. However, there was a technical problem: this argument did not allow one to find a single interval {I} that avoided gaps for all times {\Lambda/2 \leq t \leq 0} simultaneously – the pigeonhole principle could produce a different interval {I} for each time {t}! Since the number of times was uncountable, this was a serious issue. (In physical terms, the problem was that there might be very fast “longitudinal waves” in the dynamics that, at each time, cause some gaps between zeroes to be highly compressed, but the specific gap that was narrow changed very rapidly with time. Such waves could, in principle, import a huge amount of energy into {E_I} by time {0}.) To resolve this, we borrowed a PDE trick of Bourgain’s, in which the pigeonhole principle was coupled with local conservation laws. More specifically, we use the phenomenon that very narrow gaps {g_i = x_{i+1}-x_i} take a nontrivial amount of time to expand back to a reasonable size (this can be seen by comparing the evolution of this gap with solutions of the scalar ODE {\partial_t g = \frac{4}{g^2}}, which represents the fastest at which a gap such as {g_i} can expand). Thus, if a gap {g_i} is reasonably large at some time {t_0}, it will also stay reasonably large at slightly earlier times {t \in [t_0-\delta, t_0]} for some moderately small {\delta>0}. This lets one locate an interval {I} that has manageable boundary effects during the times in {[t_0-\delta, t_0]}, so in particular {E_I} is basically non-increasing in this time interval. Unfortunately, this interval is a little bit too short to cover all of {[\Lambda/2,0]}; however it turns out that one can iterate the above construction and find a nested sequence of intervals {I_k}, with each {E_{I_k}} non-increasing in a different time interval {[t_k - \delta, t_k]}, and with all of the time intervals covering {[\Lambda/2,0]}. This turns out to be enough (together with the obvious fact that {E_I} is monotone in {I}) to still control {E_I(0)} for some reasonably sized interval {I}, as required for the rest of the arguments.

ADDED LATER: the following analogy (involving functions with just two zeroes, rather than an infinite number of zeroes) may help clarify the relation between this result and the Riemann hypothesis (and in particular why this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis). Suppose one had a quadratic polynomial {P} of the form {P(z) = z^2 + \Lambda}, where {\Lambda} was an unknown real constant. Suppose that one was for some reason interested in the analogue of the “Riemann hypothesis” for {P}, namely that all the zeroes of {P} are real. A priori, there are three scenarios:

  • (Riemann hypothesis false) {\Lambda > 0}, and {P} has zeroes {\pm i |\Lambda|^{1/2}} off the real axis.
  • (Riemann hypothesis true, but barely so) {\Lambda = 0}, and both zeroes of {P} are on the real axis; however, any slight perturbation of {\Lambda} in the positive direction would move zeroes off the real axis.
  • (Riemann hypothesis true, with room to spare) {\Lambda < 0}, and both zeroes of {P} are on the real axis. Furthermore, any slight perturbation of {P} will also have both zeroes on the real axis.

The analogue of our result in this case is that {\Lambda \geq 0}, thus ruling out the third of the three scenarios here. In this simple example in which only two zeroes are involved, one can think of the inequality {\Lambda \geq 0} as asserting that if the zeroes of {P} are real, then they must be repeated. In our result (in which there are an infinity of zeroes, that become increasingly dense near infinity), and in view of the convergence to local equilibrium properties of (3), the analogous assertion is that if the zeroes of {H_0} are real, then they do not behave locally as if they were in arithmetic progression.

Kaisa Matomaki, Maksym Radziwill, and I have uploaded to the arXiv our paper “Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges“. This is a sequel of sorts to our previous paper on divisor correlations, though the proof techniques in this paper are rather different. As with the previous paper, our interest is in correlations such as

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) \ \ \ \ \ (1)

for medium-sized {h} and large {X}, where {k \geq l \geq 1} are natural numbers and {d_k(n) = \sum_{n = m_1 \dots m_k} 1} is the {k^{th}} divisor function (actually our methods can also treat a generalisation in which {k} is non-integer, but for simplicity let us stick with the integer case for this discussion). Our methods also allow for one of the divisor function factors to be replaced with a von Mangoldt function, but (in contrast to the previous paper) we cannot treat the case when both factors are von Mangoldt.

As discussed in this previous post, one heuristically expects an asymptotic of the form

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) = P_{k,l,h}( \log X ) X + O( X^{1/2+\varepsilon})

for any fixed {\varepsilon>0}, where {P_{k,l,h}} is a certain explicit (but rather complicated) polynomial of degree {k+l-1}. Such asymptotics are known when {l \leq 2}, but remain open for {k \geq l \geq 3}. In the previous paper, we were able to obtain a weaker bound of the form

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) = P_{k,l,h}( \log X ) X + O_A( X \log^{-A} X)

for {1-O_A(\log^{-A} X)} of the shifts {-H \leq h \leq H}, whenever the shift range {H} lies between {X^{8/33+\varepsilon}} and {X^{1-\varepsilon}}. But the methods become increasingly hard to use as {H} gets smaller. In this paper, we use a rather different method to obtain the even weaker bound

\displaystyle  \sum_{n \leq X} d_k(n) d_l(n+h) = (1+o(1)) P_{k,l,h}( \log X ) X

for {1-o(1)} of the shifts {-H \leq h \leq H}, where {H} can now be as short as {H = \log^{10^4 k \log k} X}. The constant {10^4} can be improved, but there are serious obstacles to using our method to go below {\log^{k \log k} X} (as the exceptionally large values of {d_k} then begin to dominate). This can be viewed as an analogue to our previous paper on correlations of bounded multiplicative functions on average, in which the functions {d_k,d_l} are now unbounded, and indeed our proof strategy is based in large part on that paper (but with many significant new technical complications).

We now discuss some of the ingredients of the proof. Unsurprisingly, the first step is the circle method, expressing (1) in terms of exponential sums such as

\displaystyle  S(\alpha) := \sum_{n \leq X} d_k(n) e(\alpha).

Actually, it is convenient to first prune {d_k} slightly by zeroing out this function on “atypical” numbers {n} that have an unusually small or large number of factors in a certain sense, but let us ignore this technicality for this discussion. The contribution of {S(\alpha)} for “major arc” {\alpha} can be treated by standard techniques (and is the source of the main term {P_{k,l,h}(\log X) X}; the main difficulty comes from treating the contribution of “minor arc” {\alpha}.

In our previous paper on bounded multiplicative functions, we used Plancherel’s theorem to estimate the global {L^2} norm {\int_{{\bf R}/{\bf Z}} |S(\alpha)|^2\ d\alpha}, and then also used the Katai-Bourgain-Sarnak-Ziegler orthogonality criterion to control local {L^2} norms {\int_I |S(\alpha)|^2\ d\alpha}, where {I} was a minor arc interval of length about {1/H}, and these two estimates together were sufficient to get a good bound on correlations by an application of Hölder’s inequality. For {d_k}, it is more convenient to use Dirichlet series methods (and Ramaré-type factorisations of such Dirichlet series) to control local {L^2} norms on minor arcs, in the spirit of the proof of the Matomaki-Radziwill theorem; a key point is to develop “log-free” mean value theorems for Dirichlet series associated to functions such as {d_k}, so as not to wipe out the (rather small) savings one will get over the trivial bound from this method. On the other hand, the global {L^2} bound will definitely be unusable, because the {\ell^2} sum {\sum_{n \leq X} d_k(n)^2} has too many unwanted factors of {\log X}. Fortunately, we can substitute this global {L^2} bound with a “large values” bound that controls expressions such as

\displaystyle  \sum_{i=1}^J \int_{I_i} |S(\alpha)|^2\ d\alpha

for a moderate number of disjoint intervals {I_1,\dots,I_J}, with a bound that is slightly better (for {J} a medium-sized power of {\log X}) than what one would have obtained by bounding each integral {\int_{I_i} |S(\alpha)|^2\ d\alpha} separately. (One needs to save more than {J^{1/2}} for the argument to work; we end up saving a factor of about {J^{3/4}}.) This large values estimate is probably the most novel contribution of the paper. After taking the Fourier transform, matters basically reduce to getting a good estimate for

\displaystyle  \sum_{i=1}^J (\int_X^{2X} |\sum_{x \leq n \leq x+H} d_k(n) e(\alpha_i n)|^2\ dx)^{1/2},

where {\alpha_i} is the midpoint of {I_i}; thus we need some upper bound on the large local Fourier coefficients of {d_k}. These coefficients are difficult to calculate directly, but, in the spirit of a paper of Ben Green and myself, we can try to replace {d_k} by a more tractable and “pseudorandom” majorant {\tilde d_k} for which the local Fourier coefficients are computable (on average). After a standard duality argument, one ends up having to control expressions such as

\displaystyle  |\sum_{x \leq n \leq x+H} \tilde d_k(n) e((\alpha_i -\alpha_{i'}) n)|

after various averaging in the {x, i,i'} parameters. These local Fourier coefficients of {\tilde d_k} turn out to be small on average unless {\alpha_i -\alpha_{i'}} is “major arc”. One then is left with a mostly combinatorial problem of trying to bound how often this major arc scenario occurs. This is very close to a computation in the previously mentioned paper of Ben and myself; there is a technical wrinkle in that the {\alpha_i} are not as well separated as they were in my paper with Ben, but it turns out that one can modify the arguments in that paper to still obtain a satisfactory estimate in this case (after first grouping nearby frequencies {\alpha_i} together, and modifying the duality argument accordingly).

A basic object of study in multiplicative number theory are the arithmetic functions: functions {f: {\bf N} \rightarrow {\bf C}} from the natural numbers to the complex numbers. Some fundamental examples of such functions include

Given an arithmetic function {f}, we are often interested in statistics such as the summatory function

\displaystyle \sum_{n \leq x} f(n), \ \ \ \ \ (1)


the logarithmically (or harmonically) weighted summatory function

\displaystyle \sum_{n \leq x} \frac{f(n)}{n}, \ \ \ \ \ (2)


or the Dirichlet series

\displaystyle {\mathcal D}[f](s) := \sum_n \frac{f(n)}{n^s}.

In the latter case, one typically has to first restrict {s} to those complex numbers whose real part is large enough in order to ensure the series on the right converges; but in many important cases, one can then extend the Dirichlet series to almost all of the complex plane by analytic continuation. One is also interested in correlations involving additive shifts, such as {\sum_{n \leq x} f(n) f(n+h)}, but these are significantly more difficult to study and cannot be easily estimated by the methods of classical multiplicative number theory.

A key operation on arithmetic functions is that of Dirichlet convolution, which when given two arithmetic functions {f,g: {\bf N} \rightarrow {\bf C}}, forms a new arithmetic function {f*g: {\bf N} \rightarrow {\bf C}}, defined by the formula

\displaystyle f*g(n) := \sum_{d|n} f(d) g(\frac{n}{d}).

Thus for instance {1*1 = d_2}, {1 * \Lambda = L}, {1 * \mu = \delta}, and {\delta * f = f} for any arithmetic function {f}. Dirichlet convolution and Dirichlet series are related by the fundamental formula

\displaystyle {\mathcal D}[f * g](s) = {\mathcal D}[f](s) {\mathcal D}[g](s), \ \ \ \ \ (3)


at least when the real part of {s} is large enough that all sums involved become absolutely convergent (but in practice one can use analytic continuation to extend this identity to most of the complex plane). There is also the identity

\displaystyle {\mathcal D}[Lf](s) = - \frac{d}{ds} {\mathcal D}[f](s), \ \ \ \ \ (4)


at least when the real part of {s} is large enough to justify interchange of differentiation and summation. As a consequence, many Dirichlet series can be expressed in terms of the Riemann zeta function {\zeta = {\mathcal D}[1]}, thus for instance

\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s)

\displaystyle {\mathcal D}[L](s) = - \zeta'(s)

\displaystyle {\mathcal D}[\delta](s) = 1

\displaystyle {\mathcal D}[\mu](s) = \frac{1}{\zeta(s)}

\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)}.

Much of the difficulty of multiplicative number theory can be traced back to the discrete nature of the natural numbers {{\bf N}}, which form a rather complicated abelian semigroup with respect to multiplication (in particular the set of generators is the set of prime numbers). One can obtain a simpler analogue of the subject by working instead with the half-infinite interval {{\bf N}_\infty := [1,+\infty)}, which is a much simpler abelian semigroup under multiplication (being a one-dimensional Lie semigroup). (I will think of this as a sort of “completion” of {{\bf N}} at the infinite place {\infty}, hence the terminology.) Accordingly, let us define a continuous arithmetic function to be a locally integrable function {f: {\bf N}_\infty \rightarrow {\bf C}}. The analogue of the summatory function (1) is then an integral

\displaystyle \int_1^x f(t)\ dt,

and similarly the analogue of (2) is

\displaystyle \int_1^x \frac{f(t)}{t}\ dt.

The analogue of the Dirichlet series is the Mellin-type transform

\displaystyle {\mathcal D}_\infty[f](s) := \int_1^\infty \frac{f(t)}{t^s}\ dt,

which will be well-defined at least if the real part of {s} is large enough and if the continuous arithmetic function {f: {\bf N}_\infty \rightarrow {\bf C}} does not grow too quickly, and hopefully will also be defined elsewhere in the complex plane by analytic continuation.

For instance, the continuous analogue of the discrete constant function {1: {\bf N} \rightarrow {\bf C}} would be the constant function {1_\infty: {\bf N}_\infty \rightarrow {\bf C}}, which maps any {t \in [1,+\infty)} to {1}, and which we will denote by {1_\infty} in order to keep it distinct from {1}. The two functions {1_\infty} and {1} have approximately similar statistics; for instance one has

\displaystyle \sum_{n \leq x} 1 = \lfloor x \rfloor \approx x-1 = \int_1^x 1\ dt


\displaystyle \sum_{n \leq x} \frac{1}{n} = H_{\lfloor x \rfloor} \approx \log x = \int_1^x \frac{1}{t}\ dt

where {H_n} is the {n^{th}} harmonic number, and we are deliberately vague as to what the symbol {\approx} means. Continuing this analogy, we would expect

\displaystyle {\mathcal D}[1](s) = \zeta(s) \approx \frac{1}{s-1} = {\mathcal D}_\infty[1_\infty](s)

which reflects the fact that {\zeta} has a simple pole at {s=1} with residue {1}, and no other poles. Note that the identity {{\mathcal D}_\infty[1_\infty](s) = \frac{1}{s-1}} is initially only valid in the region {\mathrm{Re} s > 1}, but clearly the right-hand side can be continued analytically to the entire complex plane except for the pole at {1}, and so one can define {{\mathcal D}_\infty[1_\infty]} in this region also.

In a similar vein, the logarithm function {L: {\bf N} \rightarrow {\bf C}} is approximately similar to the logarithm function {L_\infty: {\bf N}_\infty \rightarrow {\bf C}}, giving for instance the crude form

\displaystyle \sum_{n \leq x} L(n) = \log \lfloor x \rfloor! \approx x \log x - x = \int_1^\infty L_\infty(t)\ dt

of Stirling’s formula, or the Dirichlet series approximation

\displaystyle {\mathcal D}[L](s) = -\zeta'(s) \approx \frac{1}{(s-1)^2} = {\mathcal D}_\infty[L_\infty](s).

The continuous analogue of Dirichlet convolution is multiplicative convolution using the multiplicative Haar measure {\frac{dt}{t}}: given two continuous arithmetic functions {f_\infty, g_\infty: {\bf N}_\infty \rightarrow {\bf C}}, one can define their convolution {f_\infty *_\infty g_\infty: {\bf N}_\infty \rightarrow {\bf C}} by the formula

\displaystyle f_\infty *_\infty g_\infty(t) := \int_1^t f_\infty(s) g_\infty(\frac{t}{s}) \frac{ds}{s}.

Thus for instance {1_\infty * 1_\infty = L_\infty}. A short computation using Fubini’s theorem shows the analogue

\displaystyle D_\infty[f_\infty *_\infty g_\infty](s) = D_\infty[f_\infty](s) D_\infty[g_\infty](s)

of (3) whenever the real part of {s} is large enough that Fubini’s theorem can be justified; similarly, differentiation under the integral sign shows that

\displaystyle D_\infty[L_\infty f_\infty](s) = -\frac{d}{ds} D_\infty[f_\infty](s) \ \ \ \ \ (5)


again assuming that the real part of {s} is large enough that differentiation under the integral sign (or some other tool like this, such as the Cauchy integral formula for derivatives) can be justified.

Direct calculation shows that for any complex number {\rho}, one has

\displaystyle \frac{1}{s-\rho} = D_\infty[ t \mapsto t^{\rho-1} ](s)

(at least for the real part of {s} large enough), and hence by several applications of (5)

\displaystyle \frac{1}{(s-\rho)^k} = D_\infty[ t \mapsto \frac{1}{(k-1)!} t^{\rho-1} \log^{k-1} t ](s)

for any natural number {k}. This can lead to the following heuristic: if a Dirichlet series {D[f](s)} behaves like a linear combination of poles {\frac{1}{(s-\rho)^k}}, in that

\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{(s-\rho)^{k_\rho}}

for some set {\rho} of poles and some coefficients {c_\rho} and natural numbers {k_\rho} (where we again are vague as to what {\approx} means, and how to interpret the sum {\sum_\rho} if the set of poles is infinite), then one should expect the arithmetic function {f} to behave like the continuous arithmetic function

\displaystyle t \mapsto \sum_\rho \frac{c_\rho}{(k_\rho-1)!} t^{\rho-1} \log^{k_\rho-1} t.

In particular, if we only have simple poles,

\displaystyle D[f](s) \approx \sum_\rho \frac{c_\rho}{s-\rho}

then we expect to have {f} behave like continuous arithmetic function

\displaystyle t \mapsto \sum_\rho c_\rho t^{\rho-1}.

Integrating this from {1} to {x}, this heuristically suggests an approximation

\displaystyle \sum_{n \leq x} f(n) \approx \sum_\rho c_\rho \frac{x^\rho-1}{\rho}

for the summatory function, and similarly

\displaystyle \sum_{n \leq x} \frac{f(n)}{n} \approx \sum_\rho c_\rho \frac{x^{\rho-1}-1}{\rho-1},

with the convention that {\frac{x^\rho-1}{\rho}} is {\log x} when {\rho=0}, and similarly {\frac{x^{\rho-1}-1}{\rho-1}} is {\log x} when {\rho=1}. One can make these sorts of approximations more rigorous by means of Perron’s formula (or one of its variants) combined with the residue theorem, provided that one has good enough control on the relevant Dirichlet series, but we will not pursue these rigorous calculations here. (But see for instance this previous blog post for some examples.)

For instance, using the more refined approximation

\displaystyle \zeta(s) \approx \frac{1}{s-1} + \gamma

to the zeta function near {s=1}, we have

\displaystyle {\mathcal D}[d_2](s) = \zeta^2(s) \approx \frac{1}{(s-1)^2} + \frac{2 \gamma}{s-1}

we would expect that

\displaystyle d_2 \approx L_\infty + 2 \gamma

and thus for instance

\displaystyle \sum_{n \leq x} d_2(n) \approx x \log x - x + 2 \gamma x

which matches what one actually gets from the Dirichlet hyperbola method (see e.g. equation (44) of this previous post).

Or, noting that {\zeta(s)} has a simple pole at {s=1} and assuming simple zeroes elsewhere, the log derivative {-\zeta'(s)/\zeta(s)} will have simple poles of residue {+1} at {s=1} and {-1} at all the zeroes, leading to the heuristic

\displaystyle {\mathcal D}[\Lambda](s) = -\frac{\zeta'(s)}{\zeta(s)} \approx \frac{1}{s-1} - \sum_\rho \frac{1}{s-\rho}

suggesting that {\Lambda} should behave like the continuous arithmetic function

\displaystyle t \mapsto 1 - \sum_\rho t^{\rho-1}

leading for instance to the summatory approximation

\displaystyle \sum_{n \leq x} \Lambda(n) \approx x - \sum_\rho \frac{x^\rho-1}{\rho}

which is a heuristic form of the Riemann-von Mangoldt explicit formula (see Exercise 45 of these notes for a rigorous version of this formula).

Exercise 1 Go through some of the other explicit formulae listed at this Wikipedia page and give heuristic justifications for them (up to some lower order terms) by similar calculations to those given above.

Given the “adelic” perspective on number theory, I wonder if there are also {p}-adic analogues of arithmetic functions to which a similar set of heuristics can be applied, perhaps to study sums such as {\sum_{n \leq x: n = a \hbox{ mod } p^j} f(n)}. A key problem here is that there does not seem to be any good interpretation of the expression {\frac{1}{t^s}} when {s} is complex and {t} is a {p}-adic number, so it is not clear that one can analyse a Dirichlet series {p}-adically. For similar reasons, we don’t have a canonical way to define {\chi(t)} for a Dirichlet character {\chi} (unless its conductor happens to be a power of {p}), so there doesn’t seem to be much to say in the {q}-aspect either.