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This is the seventh “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.

The most recent news is that we appear to have completed the verification that {H_t(x+iy)} is free of zeroes when {t=0.4} and {y \geq 0.4}, which implies that {\Lambda \leq 0.48}. For very large {x} (for instance when the quantity {N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor} is at least {300}) this can be done analytically; for medium values of {x} (say when {N} is between {11} and {300}) this can be done by numerically evaluating a fast approximation {A^{eff} + B^{eff}} to {H_t} and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of {x}, in part due to some new, and significantly faster, numerical ways to evaluate {H_t} in this range.

One obvious thing to do now is to experiment with lowering the parameters {t} and {y} and see what happens. However there are two other potential ways to bound {\Lambda} which may also be numerically feasible. One approach is based on trying to exclude zeroes of {H_t(x+iy)=0} in a region of the form {0 \leq t \leq t_0}, {X \leq x \leq X+1} and {y \geq y_0} for some moderately large {X} (this acts as a “barrier” to prevent zeroes from flowing into the region {\{ 0 \leq x \leq X, y \geq y_0 \}} at time {t_0}, assuming that they were not already there at time {0}). This require significantly less numerical verification in the {x} aspect, but more numerical verification in the {t} aspect, so it is not yet clear whether this is a net win.

Another, rather different approach, is to study the evolution of statistics such as {S(t) = \sum_{H_t(x+iy)=0: x,y>0} y e^{-x/X}} over time. One has fairly good control on such quantities at time zero, and their time derivative looks somewhat manageable, so one may be able to still have good control on this quantity at later times {t_0>0}. However for this approach to work, one needs an effective version of the Riemann-von Mangoldt formula for {H_t}, which at present is only available asymptotically (or at time {t=0}). This approach may be able to avoid almost all numerical computation, except for numerical verification of the Riemann hypothesis, for which we can appeal to existing literature.

Participants are also welcome to add any further summaries of the situation in the comments below.

This is the sixth “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.

The last two threads have been focused primarily on the test problem of showing that {H_t(x+iy) \neq 0} whenever {t = y = 0.4}. We have been able to prove this for most regimes of {x}, or equivalently for most regimes of the natural number parameter {N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor}. In many of these regimes, a certain explicit approximation {A^{eff}+B^{eff}} to {H_t} was used, together with a non-zero normalising factor {B^{eff}_0}; see the wiki for definitions. The explicit upper bound

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

has been proven for certain explicit expressions {E_1, E_2, E_3} (see here) depending on {x}. In particular, if {x} satisfies the inequality

\displaystyle  |\frac{A^{eff}+B^{eff}}{B^{eff}_0}| > \frac{E_1}{|B^{eff}_0|} + \frac{E_2}{|B^{eff}_0|} + \frac{E_3}{|B^{eff}_0|}

then {H_t(x+iy)} is non-vanishing thanks to the triangle inequality. (In principle we have an even more accurate approximation {A^{eff}+B^{eff}-C^{eff}} available, but it is looking like we will not need it for this test problem at least.)

We have explicit upper bounds on {\frac{E_1}{|B^{eff}_0|}}, {\frac{E_2}{|B^{eff}_0|}}, {\frac{E_3}{|B^{eff}_0|}}; see this wiki page for details. They are tabulated in the range {3 \leq N \leq 2000} here. For {N \geq 2000}, the upper bound {\frac{E_3^*}{|B^{eff}_0|}} for {\frac{E_3}{|B^{eff}_0|}} is monotone decreasing, and is in particular bounded by {1.53 \times 10^{-5}}, while {\frac{E_2}{|B^{eff}_0|}} and {\frac{E_1}{|B^{eff}_0|}} are known to be bounded by {2.9 \times 10^{-7}} and {2.8 \times 10^{-8}} respectively (see here).

Meanwhile, the quantity {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|} can be lower bounded by

\displaystyle  |\sum_{n=1}^N \frac{b_n}{n^s}| - |\sum_{n=1}^N \frac{a_n}{n^s}|

for certain explicit coefficients {a_n,b_n} and an explicit complex number {s = \sigma + i\tau}. Using the triangle inequality to lower bound this by

\displaystyle  |b_1| - \sum_{n=2}^N \frac{|b_n|}{n^\sigma} - \sum_{n=1}^N \frac{|a_n|}{n^\sigma}

we can obtain a lower bound of {0.18} for {N \geq 2000}, which settles the test problem in this regime. One can get more efficient lower bounds by multiplying both Dirichlet series by a suitable Euler product mollifier; we have found {\prod_{p \leq P} (1 - \frac{b_p}{p^s})} for {P=2,3,5,7} to be good choices to get a variety of further lower bounds depending only on {N}, see this table and this wiki page. Comparing this against our tabulated upper bounds for the error terms we can handle the range {300 \leq N \leq 2000}.

In the range {11 \leq N \leq 300}, we have been able to obtain a suitable lower bound {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}| \geq c} (where {c} exceeds the upper bound for {\frac{E_1}{|B^{eff}_0|} + \frac{E_2}{|B^{eff}_0|} + \frac{E_3}{|B^{eff}_0|}}) by numerically evaluating {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|} at a mesh of points for each choice of {N}, with the mesh spacing being adaptive and determined by {c} and an upper bound for the derivative of {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|}; the data is available here.

This leaves the final range {N \leq 10} (roughly corresponding to {x \leq 1600}). Here we can numerically evaluate {H_t(x+iy)} to high accuracy at a fine mesh (see the data here), but to fill in the mesh we need good upper bounds on {H'_t(x+iy)}. It seems that we can get reasonable estimates using some contour shifting from the original definition of {H_t} (see here). We are close to finishing off this remaining region and thus solving the toy problem.

Beyond this, we need to figure out how to show that {H_t(x+iy) \neq 0} for {y > 0.4} as well. General theory lets one do this for {y \geq \sqrt{1-2t} = 0.447\dots}, leaving the region {0.4 < y < 0.448}. The analytic theory that handles {N \geq 2000} and {300 \leq N \leq 2000} should also handle this region; for {N \leq 300} presumably the argument principle will become relevant.

The full argument also needs to be streamlined and organised; right now it sprawls over many wiki pages and github code files. (A very preliminary writeup attempt has begun here). We should also see if there is much hope of extending the methods to push much beyond the bound of {\Lambda \leq 0.48} that we would get from the above calculations. This would also be a good time to start discussing whether to move to the writing phase of the project, or whether there are still fruitful research directions for the project to explore.

Participants are also welcome to add any further summaries of the situation in the comments below.

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 https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/. 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.

This is the third “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 are making progress on the following test problem: can one show that {H_t(x+iy) \neq 0} whenever {t = 0.4}, {x \geq 0}, and {y \geq 0.4}? This would imply that

\displaystyle \Lambda \leq 0.4 + \frac{1}{2} (0.4)^2 = 0.48

which would be the first quantitative improvement over the de Bruijn bound of {\Lambda \leq 1/2} (or the Ki-Kim-Lee refinement of {\Lambda < 1/2}). Of course we can try to lower the two parameters of {0.4} later on in the project, but this seems as good a place to start as any. One could also potentially try to use finer analysis of dynamics of zeroes to improve the bound {\Lambda \leq 0.48} further, but this seems to be a less urgent task.

Probably the hardest case is {y=0.4}, as there is a good chance that one can then recover the {y>0.4} case by a suitable use of the argument principle. Here we appear to have a workable Riemann-Siegel type formula that gives a tractable approximation for {H_t}. To describe this formula, first note that in the {t=0} case we have

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

and the Riemann-Siegel formula gives

\displaystyle \xi(s) = \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \sum_{n=1}^N \frac{1}{n^s}

\displaystyle + \frac{s(s-1)}{2} \pi^{-(1-s)/2} \Gamma((1-s)/2) \sum_{m=1}^M \frac{1}{m^{1-s}}

\displaystyle + \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \frac{e^{-i\pi s} \Gamma(1-s)}{2\pi i} \int_{C_M} \frac{w^{s-1} e^{-Nw}}{e^w-1}\ dw

for any natural numbers {N,M}, where {C_M} is a contour from {+\infty} to {+\infty} that winds once anticlockwise around the zeroes {e^{2\pi im}, |m| \leq M} of {e^w-1} but does not wind around any other zeroes. A good choice of {N,M} to use here is

\displaystyle N=M=\lfloor \sqrt{\mathrm{Im}(s)/2\pi}\rfloor = \lfloor \sqrt{\mathrm{Re}(z)/4\pi} \rfloor. \ \ \ \ \ (1)

 

In this case, a classical steepest descent computation (see wiki) yields the approximation

\displaystyle \int_{C_M} \frac{w^{s-1} e^{-Nw}}{e^w-1}\ dw \approx - (2\pi i M)^{s-1} \Psi( \frac{s}{2\pi i M} - N )

where

\displaystyle \Psi(\alpha) := 2\pi \frac{\cos \pi(\frac{1}{2}\alpha^2 - \alpha - \pi/8)}{\cos(\pi \alpha)} \exp( \frac{i\pi}{2} \alpha^2 - \frac{5\pi i}{8} ).

Thus we have

\displaystyle H_0(z) \approx A^{(0)} + B^{(0)} - C^{(0)}

where

\displaystyle A^{(0)} := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \sum_{n=1}^N \frac{1}{n^s}

\displaystyle B^{(0)} := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-(1-s)/2} \Gamma((1-s)/2) \sum_{m=1}^M \frac{1}{m^{1-s}}

\displaystyle C^{(0)} := \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \frac{e^{-i\pi s} \Gamma(1-s)}{2\pi i} (2\pi i M)^{s-1} \Psi( \frac{s}{2\pi i M} - N )

with {s := \frac{1+iz}{2}} and {N,M} given by (1).

Heuristically, we have derived (see wiki) the more general approximation

\displaystyle H_t(z) \approx A + B - C

for {t>0} (and in particular for {t=0.4}), where

\displaystyle A := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \sum_{n=1}^N \frac{\exp(\frac{t}{16} \log^2 \frac{s+4}{2\pi n^2} )}{n^s}

\displaystyle B := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-(1-s)/2} \Gamma((1-s)/2) \sum_{m=1}^M \frac{\exp(\frac{t}{16} \log^2 \frac{5-s}{2\pi m^2} )}{m^{1-s}}

\displaystyle C := \exp(-\frac{t \pi^2}{64}) C^{(0)}.

In practice it seems that the {C} term is negligible once the real part {x} of {z} is moderately large, so one also has the approximation

\displaystyle H_t(z) \approx A + B.

For large {x}, and for fixed {t,y>0}, e.g. {t=y=0.4}, the sums {A,B} converge fairly quickly (in fact the situation seems to be significantly better here than the much more intensively studied {t=0} case), and we expect the first term

\displaystyle B_0 := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-(1-s)/2} \Gamma((1-s)/2) \exp( \frac{t}{16} \log^2 \frac{5-s}{2\pi} )

of the {B} series to dominate. Indeed, analytically we know that {\frac{A+B-C}{B_0} \rightarrow 1} (or {\frac{A+B}{B_0} \rightarrow 1}) as {x \rightarrow \infty} (holding {y} fixed), and it should also be provable that {\frac{H_t}{B_0} \rightarrow 1} as well. Numerically with {t=y=0.4}, it seems in fact that {\frac{A+B-C}{B_0}} (or {\frac{A+B}{B_0}}) stay within a distance of about {1/2} of {1} once {x} is moderately large (e.g. {x \geq 2 \times 10^5}). This raises the hope that one can solve the toy problem of showing {H_t(x+iy) \neq 0} for {t=y=0.4} by numerically controlling {H_t(x+iy) / B_0} for small {x} (e.g. {x \leq 2 \times 10^5}), numerically controlling {(A+B)/B_0} and analytically bounding the error {(H_t - A - B)/B_0} for medium {x} (e.g. {2 \times 10^5 \leq x \leq 10^7}), and analytically bounding both {(A+B)/B_0} and {(H_t-A-B)/B_0} for large {x} (e.g. {x \geq 10^7}). (These numbers {2 \times 10^5} and {10^7} are arbitrarily chosen here and may end up being optimised to something else as the computations become clearer.)

Thus, we now have four largely independent tasks (for suitable ranges of “small”, “medium”, and “large” {x}):

  1. Numerically computing {H_t(x+iy) / B_0} for small {x} (with enough accuracy to verify that there are no zeroes)
  2. Numerically computing {(A+B)/B_0} for medium {x} (with enough accuracy to keep it away from zero)
  3. Analytically bounding {(A+B)/B_0} for large {x} (with enough accuracy to keep it away from zero); and
  4. Analytically bounding {(H_t - A - B)/B_0} for medium and large {x} (with a bound that is better than the bound away from zero in the previous two tasks).

Note that tasks 2 and 3 do not directly require any further understanding of the function {H_t}.

Below we will give a progress report on the numeric and analytic sides of these tasks.

— 1. Numerics report (contributed by Sujit Nair) —

There is some progress on the code side but not at the pace I was hoping. Here are a few things which happened (rather, mistakes which were taken care of).

  1. We got rid of code which wasn’t being used. For example, @dhjpolymath computed {H_t} based on an old version but only realized it after the fact.
  2. We implemented tests to catch human/numerical bugs before a computation starts. Again, we lost some numerical cycles but moving forward these can be avoided.
  3. David got set up on GitHub and he is able to compare his output (in C) with the Python code. That is helping a lot.

Two areas which were worked on were

  1. Computing {H_t} and zeroes for {t} around {0.4}
  2. Computing quantities like {(A+B-C)/B_0}, {(A+B)/B_0}, {C/B_0}, etc. with the goal of understanding the zero free regions.

Some observations for {t=0.4}, {y=0.4}, {x \in ( 10^4, 10^7)} include:

  • {(A+B) / B_0} does seem to avoid the negative real axis
  • {|(A+B) / B0| > 0.4} (based on the oscillations and trends in the plots)
  • {|C/B_0|} seems to be settling around {10^{-4}} range.

See the figure below. The top plot is on the complex plane and the bottom plot is the absolute value. The code to play with this is here.

— 2. Analysis report —

The Riemann-Siegel formula and some manipulations (see wiki) give {H_0 = A^{(0)} + B^{(0)} - \tilde C^{(0)}}, where

\displaystyle A^{(0)} = \frac{2}{8} \sum_{n=1}^N \int_C \exp( \frac{s+4}{2} u - e^u - \frac{s}{2} \log(\pi n^2) )\ du

\displaystyle - \frac{3}{8} \sum_{n=1}^N \int_C \exp( \frac{s+2}{2} u - e^u - \frac{s}{2} \log(\pi n^2) )\ du

\displaystyle B^{(0)} = \frac{2}{8} \sum_{m=1}^M \int_{\overline{C}} \exp( \frac{5-s}{2} u - e^u - \frac{1-s}{2} \log(\pi m^2) )\ du

\displaystyle - \frac{3}{8} \sum_{m=1}^M \int_C \exp( \frac{3-s}{2} u - e^u - \frac{1-s}{2} \log(\pi m^2) )\ du

\displaystyle \tilde C^{(0)} := -\frac{2}{8} \sum_{n=0}^\infty \frac{e^{-i\pi s/2} e^{i\pi s n}}{2^s \pi^{1/2}} \int_{\overline{C}} \int_{C_M} \frac{w^{s-1} e^{-Nw}}{e^w-1} \exp( \frac{5-s}{2} u - e^u)\ du dw

\displaystyle +\frac{3}{8} \sum_{n=0}^\infty \frac{e^{-i\pi s/2} e^{i\pi s n}}{2^s \pi^{1/2}} \int_{\overline{C}} \int_{C_M} \frac{w^{s-1} e^{-Nw}}{e^w-1} \exp( \frac{3-s}{2} u - e^u)\ du dw

where {C} is a contour that goes from {+i\infty} to {+\infty} staying a bounded distance away from the upper imaginary and right real axes, and {\overline{C}} is the complex conjugate of {C}. (In each of these sums, it is the first term that should dominate, with the second one being about {O(1/x)} as large.) One can then evolve by the heat flow to obtain {H_t = \tilde A + \tilde B - \tilde C}, where

\displaystyle \tilde A := \frac{2}{8} \sum_{n=1}^N \int_C \exp( \frac{s+4}{2} u - e^u - \frac{s}{2} \log(\pi n^2) + \frac{t}{16} (u - \log(\pi n^2))^2)\ du

\displaystyle - \frac{3}{8} \sum_{n=1}^N \int_C \exp( \frac{s+2}{2} u - e^u - \frac{s}{2} \log(\pi n^2) + \frac{t}{16} (u - \log(\pi n^2))^2)\ du

\displaystyle \tilde B := \frac{2}{8} \sum_{m=1}^M \int_{\overline{C}} \exp( \frac{5-s}{2} u - e^u - \frac{1-s}{2} \log(\pi m^2) + \frac{t}{16} (u - \log(\pi m^2))^2)\ du

\displaystyle - \frac{3}{8} \sum_{m=1}^M \int_C \exp( \frac{3-s}{2} u - e^u - \frac{1-s}{2} \log(\pi m^2) + \frac{t}{16} (u - \log(\pi m^2))^2)\ du

\displaystyle \tilde C := -\frac{2}{8} \sum_{n=0}^\infty \frac{e^{-i\pi s/2} e^{i\pi s n}}{2^s \pi^{1/2}} \int_{\overline{C}} \int_{C_M}

\displaystyle \frac{w^{s-1} e^{-Nw}}{e^w-1} \exp( \frac{5-s}{2} u - e^u + \frac{t}{4} (i \pi(n-1/2) + \log \frac{w}{2\sqrt{\pi}} - \frac{u}{2})^2) \ du dw

\displaystyle +\frac{3}{8} \sum_{n=0}^\infty \frac{e^{-i\pi s/2} e^{i\pi s n}}{2^s \pi^{1/2}} \int_{\overline{C}} \int_{C_M}

\displaystyle \frac{w^{s-1} e^{-Nw}}{e^w-1} \exp( \frac{3-s}{2} u - e^u + \frac{t}{4} (i \pi(n-1/2) + \log \frac{w}{2\sqrt{\pi}} - \frac{u}{2})^2)\ du dw.

Steepest descent heuristics then predict that {\tilde A \approx A}, {\tilde B \approx B}, and {\tilde C \approx C}. For the purposes of this project, we will need effective error estimates here, with explicit error terms.

A start has been made towards this goal at this wiki page. Firstly there is a “effective Laplace method” lemma that gives effective bounds on integrals of the form {\int_I e^{\phi(x)} \psi(x)\ dx} if the real part of {\phi(x)} is either monotone with large derivative, or has a critical point and is decreasing on both sides of that critical point. In principle, all one has to do is manipulate expressions such as {\tilde A - A}, {\tilde B - B}, {\tilde C - C} by change of variables, contour shifting and integration by parts until it is of the form to which the above lemma can be profitably applied. As one may imagine though the computations are messy, particularly for the {\tilde C} term. As a warm up, I have begun by trying to estimate integrals of the form

\displaystyle \int_C \exp( s (1+u-e^u) + \frac{t}{16} (u+b)^2 )\ du

for smallish complex numbers {b}, as these sorts of integrals appear in the form of {\tilde A, \tilde B, \tilde C}. As of this time of writing, there are effective bounds for the {b=0} case, and I am currently working on extending them to the {b \neq 0} case, which should give enough control to approximate {\tilde A - A} and {\tilde B-B}. The most complicated task will be that of upper bounding {\tilde C}, but it also looks eventually doable.

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

where

\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})})

where

\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))

and

\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})})

where

\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

where

\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 {t<0} case. Unfortunately this is not possible in the {t>0} 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)

where

\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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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} )

is

\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.

Apoorva Khare and I have updated our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“, announced at this post from last month. The quantitative results are now sharpened using a new monotonicity property of ratios {s_{\lambda}(u)/s_{\mu}(u)} of Schur polynomials, namely that such ratios are monotone non-decreasing in each coordinate of {u} if {u} is in the positive orthant, and the partition {\lambda} is larger than that of {\mu}. (This monotonicity was also independently observed by Rachid Ait-Haddou, using the theory of blossoms.) In the revised version of the paper we give two proofs of this monotonicity. The first relies on a deep positivity result of Lam, Postnikov, and Pylyavskyy, which uses a representation-theoretic positivity result of Haiman to show that the polynomial combination

\displaystyle s_{(\lambda \wedge \nu) / (\mu \wedge \rho)} s_{(\lambda \vee \nu) / (\mu \vee \rho)} - s_{\lambda/\mu} s_{\nu/\rho} \ \ \ \ \ (1)

of skew-Schur polynomials is Schur-positive for any partitions {\lambda,\mu,\nu,\rho} (using the convention that the skew-Schur polynomial {s_{\lambda/\mu}} vanishes if {\mu} is not contained in {\lambda}, and where {\lambda \wedge \nu} and {\lambda \vee \nu} denotes the pointwise min and max of {\lambda} and {\nu} respectively). It is fairly easy to derive the monotonicity of {s_\lambda(u)/s_\mu(u)} from this, by using the expansion

\displaystyle s_\lambda(u_1,\dots, u_n) = \sum_k u_1^k s_{\lambda/(k)}(u_2,\dots,u_n)

of Schur polynomials into skew-Schur polynomials (as was done in this previous post).

The second proof of monotonicity avoids representation theory by a more elementary argument establishing the weaker claim that the above expression (1) is non-negative on the positive orthant. In fact we prove a more general determinantal log-supermodularity claim which may be of independent interest:

Theorem 1 Let {A} be any {n \times n} totally positive matrix (thus, every minor has a non-negative determinant). Then for any {k}-tuples {I_1,I_2,J_1,J_2} of increasing elements of {\{1,\dots,n\}}, one has

\displaystyle \det( A_{I_1 \wedge I_2, J_1 \wedge J_2} ) \det( A_{I_1 \vee I_2, J_1 \vee J_2} ) - \det(A_{I_1,J_1}) \det(A_{I_2,J_2}) \geq 0

where {A_{I,J}} denotes the {k \times k} minor formed from the rows in {I} and columns in {J}.

For instance, if {A} is the matrix

\displaystyle A = \begin{pmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{pmatrix}

for some real numbers {a,\dots,p}, one has

\displaystyle a h - de\geq 0

(corresponding to the case {k=1}, {I_1 = (1), I_2 = (2), J_1 = (4), J_2 = (1)}), or

\displaystyle \det \begin{pmatrix} a & c \\ i & k \end{pmatrix} \det \begin{pmatrix} f & h \\ n & p \end{pmatrix} - \det \begin{pmatrix} e & h \\ i & l \end{pmatrix} \det \begin{pmatrix} b & c \\ n & o \end{pmatrix} \geq 0

(corresponding to the case {k=2}, {I_1 = (2,3)}, {I_2 = (1,4)}, {J_1 = (1,4)}, {J_2 = (2,3)}). It turns out that this claim can be proven relatively easy by an induction argument, relying on the Dodgson and Karlin identities from this previous post; the difficulties are largely notational in nature. Combining this result with the Jacobi-Trudi identity for skew-Schur polynomials (discussed in this previous post) gives the non-negativity of (1); it can also be used to directly establish the monotonicity of ratios {s_\lambda(u)/s_\mu(u)} by applying the theorem to a generalised Vandermonde matrix.

(Log-supermodularity also arises as the natural hypothesis for the FKG inequality, though I do not know of any interesting application of the FKG inequality in this current setting.)

In one of the earliest posts on this blog, I talked about the ability to “arbitrage” a disparity of symmetry in an inequality, and in particular to “amplify” such an inequality into a stronger one. (The principle can apply to other mathematical statements than inequalities, with the “hypothesis” and “conclusion” of that statement generally playing the role of the “right-hand side” and “left-hand side” of an inequality, but for sake of discussion I will restrict attention here to inequalities.) One can formalise this principle as follows. Many inequalities in analysis can be expressed in the form

\displaystyle A(f) \leq B(f) \ \ \ \ \ (1)

for all {f} in some space {X} (in many cases {X} will be a function space, and {f} a function in that space), where {A(f)} and {B(f)} are some functionals of {f} (that is to say, real-valued functions of {f}). For instance, {B(f)} might be some function space norm of {f} (e.g. an {L^p} norm), and {A(f)} might be some function space norm of some transform of {f}. In addition, we assume we have some group {G} of symmetries {T: X \rightarrow X} acting on the underlying space. For instance, if {X} is a space of functions on some spatial domain, the group might consist of translations (e.g. {Tf(x) = f(x-h)} for some shift {h}), or perhaps dilations with some normalisation (e.g. {Tf(x) = \frac{1}{\lambda^\alpha} f(\frac{x}{\lambda})} for some dilation factor {\lambda > 0} and some normalisation exponent {\alpha \in {\bf R}}, which can be thought of as the dimensionality of length one is assigning to {f}). If we have

\displaystyle A(Tf) = A(f)

for all symmetries {T \in G} and all {f \in X}, we say that {A} is invariant with respect to the symmetries in {G}; otherwise, it is not.

Suppose we know that the inequality (1) holds for all {f \in X}, but that there is an imbalance of symmetry: either {A} is {G}-invariant and {B} is not, or vice versa. Suppose first that {A} is {G}-invariant and {B} is not. Substituting {f} by {Tf} in (1) and taking infima, we can then amplify (1) to the stronger inequality

\displaystyle A(f) \leq \inf_{T \in G} B(Tf).

In particular, it is often the case that there is a way to send {T} off to infinity in such a way that the functional {B(Tf)} has a limit {B_\infty(f)}, in which case we obtain the amplification

\displaystyle A(f) \leq B_\infty(f) \ \ \ \ \ (2)

of (1). Note that these amplified inequalities will now be {G}-invariant on both sides (assuming that the way in which we take limits as {T \rightarrow \infty} is itself {G}-invariant, which it often is in practice). Similarly, if {B} is {G}-invariant but {A} is not, we may instead amplify (1) to

\displaystyle \sup_{T \in G} A(Tf) \leq B(f)

and in particular (if {A(Tf)} has a limit {A_\infty(f)} as {T \rightarrow \infty})

\displaystyle A_\infty(f) \leq B(f). \ \ \ \ \ (3)

If neither {A(f)} nor {B(f)} has a {G}-symmetry, one can still use the {G}-symmetry by replacing {f} by {Tf} and taking a limit to conclude that

\displaystyle A_\infty(f) \leq B_\infty(f),

though now this inequality is not obviously stronger than the original inequality (1) (for instance it could well be trivial). In some cases one can also average over {G} instead of taking a limit as {T \rightarrow \infty}, thus averaging a non-invariant inequality into an invariant one.

As discussed in the previous post, this use of amplification gives rise to a general principle about inequalities: the most efficient inequalities are those in which the left-hand side and right-hand side enjoy the same symmetries. It is certainly possible to have true inequalities that have an imbalance of symmetry, but as shown above, such inequalities can always be amplified to more efficient and more symmetric inequalities. In the case when limits such as {A_\infty} and {B_\infty} exist, the limiting functionals {A_\infty(f)} and {B_\infty(f)} are often simpler in form, or more tractable analytically, than their non-limiting counterparts {A(f)} and {B(f)} (this is one of the main reasons why we take limits at infinity in the first place!), and so in many applications there is really no reason to use the weaker and more complicated inequality (1), when stronger, simpler, and more symmetric inequalities such as (2), (3) are available. Among other things, this explains why many of the most useful and natural inequalities one sees in analysis are dimensionally consistent.

One often tries to prove inequalities (1) by directly chaining together simpler inequalities. For instance, one might attempt to prove (1) by by first bounding {A(f)} by some auxiliary quantity {C(f)}, and then bounding {C(f)} by {B(f)}, thus obtaining (1) by chaining together two inequalities

\displaystyle A(f) \leq C(f) \leq B(f). \ \ \ \ \ (4)

A variant of the above principle then asserts that when proving inequalities by such direct methods, one should, whenever possible, try to maintain the symmetries that are present in both sides of the inequality. Why? Well, suppose that we ignored this principle and tried to prove (1) by establishing (4) for some {C} that is not {G}-invariant. Assuming for sake of argument that (4) were actually true, we could amplify the first half {A(f) \leq C(f)} of this inequality to conclude that

\displaystyle A(f) \leq \inf_{T \in G} C(Tf)

and also amplify the second half {C(f) \leq B(f)} of the inequality to conclude that

\displaystyle \sup_{T \in G} C(Tf) \leq B(f)

and hence (4) amplifies to

\displaystyle A(f) \leq \inf_{T \in G} C(Tf) \leq \sup_{T \in G} C(Tf) \leq B(f). \ \ \ \ \ (5)

Let’s say for sake of argument that all the quantities involved here are positive numbers (which is often the case in analysis). Then we see in particular that

\displaystyle \frac{\sup_{T \in G} C(Tf)}{\inf_{T \in G} C(Tf)} \leq \frac{B(f)}{A(f)}. \ \ \ \ \ (6)

Informally, (6) asserts that in order for the strategy (4) used to prove (1) to work, the extent to which {C} fails to be {G}-invariant cannot exceed the amount of “room” present in (1). In particular, when dealing with those “extremal” {f} for which the left and right-hand sides of (1) are comparable to each other, one can only have a bounded amount of non-{G}-invariance in the functional {C}. If {C} fails so badly to be {G}-invariant that one does not expect the left-hand side of (6) to be at all bounded in such extremal situations, then the strategy of proving (1) using the intermediate quantity {C} is doomed to failure – even if one has already produced some clever proof of one of the two inequalities {A(f) \leq C(f)} or {C(f) \leq B(f)} needed to make this strategy work. And even if it did work, one could amplify (4) to a simpler inequality

\displaystyle A(f) \leq C_\infty(f) \leq B(f) \ \ \ \ \ (7)

(assuming that the appropriate limit {C_\infty(f) = \lim_{T \rightarrow \infty} C(Tf)} existed) which would likely also be easier to prove (one can take whatever proofs one had in mind of the inequalities in (4), conjugate them by {T}, and take a limit as {T \rightarrow \infty} to extract a proof of (7)).

Here are some simple (but somewhat contrived) examples to illustrate these points. Suppose one wishes to prove the inequality

\displaystyle xy \leq x^2 + y^2 \ \ \ \ \ (8)

for all {x,y>0}. Both sides of this inequality are invariant with respect to interchanging {x} with {y}, so the principle suggests that when proving this inequality directly, one should only use sub-inequalities that are also invariant with respect to this interchange. However, in this particular case there is enough “room” in the inequality that it is possible (though somewhat unnatural) to violate this principle. For instance, one could decide (for whatever reason) to start with the inequality

\displaystyle 0 \leq (x - y/2)^2 = x^2 - xy + y^2/4

to conclude that

\displaystyle xy \leq x^2 + y^2/4

and then use the obvious inequality {x^2 + y^2/4 \leq x^2+y^2} to conclude the proof. Here, the intermediate quantity {x^2 + y^2/4} is not invariant with respect to interchange of {x} and {y}, but the failure is fairly mild (changing {x} and {y} only modifies the quantity {x^2 + y^2/4} by a multiplicative factor of {4} at most), and disappears completely in the most extremal case {x=y}, which helps explain why one could get away with using this quantity in the proof here. But it would be significantly harder (though still not impossible) to use non-symmetric intermediaries to prove the sharp version

\displaystyle xy \leq \frac{x^2 + y^2}{2}

of (8) (that is to say, the arithmetic mean-geometric mean inequality). Try it!

Similarly, consider the task of proving the triangle inequality

\displaystyle |z+w| \leq |z| + |w| \ \ \ \ \ (9)

for complex numbers {z, w}. One could try to leverage the triangle inequality {|x+y| \leq |x| + |y|} for real numbers by using the crude estimate

\displaystyle |z+w| \leq |\hbox{Re}(z+w)| + |\hbox{Im}(z+w)|

and then use the real triangle inequality to obtain

\displaystyle |\hbox{Re}(z+w)| \leq |\hbox{Re}(z)| + |\hbox{Re}(w)|

and

\displaystyle |\hbox{Im}(z+w)| \leq |\hbox{Im}(z)| + |\hbox{Im}(w)|

and then finally use the inequalities

\displaystyle |\hbox{Re}(z)|, |\hbox{Im}(z)| \leq |z| \ \ \ \ \ (10)

and

\displaystyle |\hbox{Re}(w)|, |\hbox{Im}(w)| \leq |w| \ \ \ \ \ (11)

but when one puts this all together at the end of the day, one loses a factor of two:

\displaystyle |z+w| \leq 2(|z| + |w|).

One can “blame” this loss on the fact that while the original inequality (9) was invariant with respect to phase rotation {(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}, the intermediate expressions we tried to use when proving it were not, leading to inefficient estimates. One can try to be smarter than this by using Pythagoras’ theorem {|z|^2 = |\hbox{Re}(z)|^2 + |\hbox{Im}(z)|^2}; this reduces the loss from {2} to {\sqrt{2}} but does not eliminate it completely, which is to be expected as one is still using non-invariant estimates in the proof. But one can remove the loss completely by using amplification; see the previous blog post for details (we also give a reformulation of this amplification below).

Here is a slight variant of the above example. Suppose that you had just learned in class to prove the triangle inequality

\displaystyle (\sum_{n=1}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=1}^\infty |a_n|^2)^{1/2} + (\sum_{n=1}^\infty |b_n|^2)^{1/2} \ \ \ \ \ (12)

for (say) real square-summable sequences {(a_n)_{n=1}^\infty}, {(b_n)_{n=1}^\infty}, and was tasked to conclude the corresponding inequality

\displaystyle (\sum_{n \in {\bf Z}} |a_n+b_n|^2)^{1/2} \leq (\sum_{n \in {\bf Z}} |a_n|^2)^{1/2} + (\sum_{n \in {\bf Z}} |b_n|^2)^{1/2} \ \ \ \ \ (13)

for doubly infinite square-summable sequences {(a_n)_{n \in {\bf Z}}, (b_n)_{n \in {\bf Z}}}. The quickest way to do this is of course to exploit a bijection between the natural numbers {1,2,\dots} and the integers, but let us say for sake of argument that one was unaware of such a bijection. One could then proceed instead by splitting the integers into the positive integers and the non-positive integers, and use (12) on each component separately; this is very similar to the strategy of proving (9) by splitting a complex number into real and imaginary parts, and will similarly lose a factor of {2} or {\sqrt{2}}. In this case, one can “blame” this loss on the abandonment of translation invariance: both sides of the inequality (13) are invariant with respect to shifting the sequences {(a_n)_{n \in {\bf Z}}}, {(b_n)_{n \in {\bf Z}}} by some shift {h} to arrive at {(a_{n-h})_{n \in {\bf Z}}, (b_{n-h})_{n \in {\bf Z}}}, but the intermediate quantities caused by splitting the integers into two subsets are not invariant. Another way of thinking about this is that the splitting of the integers gives a privileged role to the origin {n=0}, whereas the inequality (13) treats all values of {n} equally thanks to the translation invariance, and so using such a splitting is unnatural and not likely to lead to optimal estimates. On the other hand, one can deduce (13) from (12) by sending this symmetry to infinity; indeed, after applying a shift to (12) we see that

\displaystyle (\sum_{n=-N}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=-N}^\infty |a_n|^2)^{1/2} + (\sum_{n=-N}^\infty |b_n|^2)^{1/2}

for any {N}, and on sending {N \rightarrow \infty} we obtain (13) (one could invoke the monotone convergence theorem here to justify the limit, though in this case it is simple enough that one can just use first principles).

Note that the principle of preserving symmetry only applies to direct approaches to proving inequalities such as (1). There is a complementary approach, discussed for instance in this previous post, which is to spend the symmetry to place the variable {f} “without loss of generality” in a “normal form”, “convenient coordinate system”, or a “good gauge”. Abstractly: suppose that there is some subset {Y} of {X} with the property that every {f \in X} can be expressed in the form {f = Tg} for some {T \in G} and {g \in Y} (that is to say, {X = GY}). Then, if one wishes to prove an inequality (1) for all {f \in X}, and one knows that both sides {A(f), B(f)} of this inequality are {G}-invariant, then it suffices to check (1) just for those {f} in {Y}, as this together with the {G}-invariance will imply the same inequality (1) for all {f} in {GY=X}. By restricting to those {f} in {Y}, one has given up (or spent) the {G}-invariance, as the set {Y} will in typical not be preserved by the group action {G}. But by the same token, by eliminating the invariance, one also eliminates the prohibition on using non-invariant proof techniques, and one is now free to use a wider range of inequalities in order to try to establish (1). Of course, such inequalities should make crucial use of the restriction {f \in Y}, for if they did not, then the arguments would work in the more general setting {f \in X}, and then the previous principle would again kick in and warn us that the use of non-invariant inequalities would be inefficient. Thus one should “spend” the symmetry wisely to “buy” a restriction {f \in Y} that will be of maximal utility in calculations (for instance by setting as many annoying factors and terms in one’s analysis to be {0} or {1} as possible).

As a simple example of this, let us revisit the complex triangle inequality (9). As already noted, both sides of this inequality are invariant with respect to the phase rotation symmetry {(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}. This seems to limit one to using phase-rotation-invariant techniques to establish the inequality, in particular ruling out the use of real and imaginary parts as discussed previously. However, we can instead spend the phase rotation symmetry to restrict to a special class of {z} and {w}. It turns out that the most efficient way to spend the symmetry is to achieve the normalisation of {z+w} being a nonnegative real; this is of course possible since any complex number {z+w} can be turned into a nonnegative real by multiplying by an appropriate phase {e^{i\theta}}. Once {z+w} is a nonnegative real, the imaginary part disappears and we have

\displaystyle |z+w| = \hbox{Re}(z+w) = \hbox{Re}(z) + \hbox{Re}(w),

and the triangle inequality (9) is now an immediate consequence of (10), (11). (But note that if one had unwisely spent the symmetry to normalise, say, {z} to be a non-negative real, then one is no closer to establishing (9) than before one had spent the symmetry.)

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