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I’ve just uploaded to the arXiv my paper “The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjecture“. As the title suggests, this paper links together the Elliott-Halberstam conjecture from sieve theory with the conjecture of Vinogradov concerning the least quadratic nonresidue {n(p)} of a prime {p}. Vinogradov established the bound

\displaystyle  n(p) \ll p^{\frac{1}{2\sqrt{e}}} \log^2 p \ \ \ \ \ (1)

and conjectured that

\displaystyle  n(p) \ll p^\varepsilon \ \ \ \ \ (2)

for any fixed {\varepsilon>0}. Unconditionally, the best result so far (up to logarithmic factors) that holds for all primes {p} is by Burgess, who showed that

\displaystyle  n(p) \ll p^{\frac{1}{4\sqrt{e}}+\varepsilon} \ \ \ \ \ (3)

for any fixed {\varepsilon>0}. See this previous post for a proof of these bounds.

In this paper, we show that the Vinogradov conjecture is a consequence of the Elliott-Halberstam conjecture. Using a variant of the argument, we also show that the “Type II” estimates established by Zhang and numerically improved by the Polymath8a project can be used to improve a little on the Vinogradov bound (1), to

\displaystyle  n(p) \ll p^{(\frac{1}{2}-\frac{1}{34})\frac{1}{\sqrt{e}} + \varepsilon},

although this falls well short of the Burgess bound. However, the method is somewhat different (although in both cases it is the Weil exponential sum bounds that are the source of the gain over (1)) and it is conceivable that a numerically stronger version of the Type II estimates could obtain results that are more competitive with the Burgess bound. At any rate, this demonstrates that the equidistribution estimates introduced by Zhang may have further applications beyond the family of results related to bounded gaps between primes.

The connection between the least quadratic nonresidue problem and Elliott-Halberstam is follows. Suppose for contradiction we can find a prime {q} with {n(q)} unusually large. Letting {\chi} be the quadratic character modulo {q}, this implies that the sums {\sum_{n \leq x} \chi(n)} are also unusually large for a significant range of {x} (e.g. {x < n(q)}), although the sum is also quite small for large {x} (e.g. {x > q}), due to the cancellation present in {\chi}. It turns out (by a sort of “uncertainty principle” for multiplicative functions, as per this paper of Granville and Soundararajan) that these facts force {\sum_{n\leq x} \chi(n) \Lambda(n)} to be unusually large in magnitude for some large {x} (with {q^C \leq x \leq q^{C'}} for two large absolute constants {C,C'}). By the periodicity of {\chi}, this means that

\displaystyle  \sum_{n\leq x} \chi(n) \Lambda(n+q)

must be unusually large also. However, because {n(q)} is large, one can factorise {\chi} as {f * 1} for a fairly sparsely supported function {f = \chi * \mu}. The Elliott-Halberstam conjecture, which controls the distribution of {\Lambda} in arithmetic progressions on the average can then be used to show that {\sum_{n \leq x} (f*1)(n) \Lambda(n+q)} is small, giving the required contradiction.

The implication involving Type II estimates is proven by a variant of the argument. If {n(q)} is large, then a character sum {\sum_{N\leq n \leq 2N} \chi(n)} is unusually large for a certain {N}. By multiplicativity of {\chi}, this shows that {\chi} correlates with {\chi * 1_{[N,2N]}}, and then by periodicity of {\chi}, this shows that {\chi(n)} correlates with {\chi * 1_{[N,2N]}(n+jq)} for various small {q}. By the Cauchy-Schwarz inequality (cf. this previous blog post), this implies that {\chi * 1_{[N,2N]}(n+jq)} correlates with {\chi * 1_{[N,2N]}(n+j'q)} for some distinct {j,j'}. But this can be ruled out by using Type II estimates.

I’ll record here a well-known observation concerning potential counterexamples to any improvement to the Burgess bound, that is to say an infinite sequence of primes {p} with {n(p) = p^{\frac{1}{4\sqrt{e}} + o(1)}}. Suppose we let {a(t)} be the asymptotic mean value of the quadratic character {\chi} at {p^t} and {b(t)} the mean value of {\chi \Lambda}; these quantities are defined precisely in my paper, but roughly speaking one can think of

\displaystyle  a(t) = \lim_{p \rightarrow \infty} \frac{1}{p^t} \sum_{n \leq p^t} \chi(n)

and

\displaystyle  b(t) = \lim_{p \rightarrow \infty} \frac{1}{p^t} \sum_{n \leq p^t} \chi(n) \Lambda(n).

Thanks to the basic Dirichlet convolution identity {\chi(n) \log(n) = \chi * \chi\Lambda(n)}, one can establish the Wirsing integral equation

\displaystyle  t a(t) = \int_0^t a(u) b(t-u)\ du

for all {t \geq 0}; see my paper for details (actually far sharper claims than this appear in previous work of Wirsing and Granville-Soundararajan). If we have an infinite sequence of counterexamples to any improvement to the Burgess bound, then we have

\displaystyle  a(t)=b(t) = 1 \hbox{ for } t < \frac{1}{4\sqrt{e}}

while from the Burgess exponential sum estimates we have

\displaystyle  a(t) = 0 \hbox{ for } t > \frac{1}{4}.

These two constraints, together with the Wirsing integral equation, end up determining {a} and {b} completely. It turns out that we must have

\displaystyle  b(t) = -1 \hbox{ for } \frac{1}{4\sqrt{e}} \leq t \leq \frac{1}{4}

and

\displaystyle  a(t) = 1 - 2 \log(4 \sqrt{e} t) \hbox{ for } \frac{1}{4\sqrt{e}} \leq t \leq \frac{1}{4}

and then for {t > \frac{1}{4}}, {b} evolves by the integral equation

\displaystyle  b(t) = \int_{1/4\sqrt{e}}^{1/4} b(t-u) \frac{2du}{u}.

For instance

\displaystyle  b(t) = 1 \hbox{ for } \frac{1}{4} < t \leq \frac{1}{2\sqrt{e}}

and then {b} oscillates in a somewhat strange fashion towards zero as {t \rightarrow \infty}. This very odd behaviour of {\sum_n \chi(n) \Lambda(n)} is surely impossible, but it seems remarkably hard to exclude it without invoking a strong hypothesis, such as GRH or the Elliott-Halberstam conjecture (or weaker versions thereof).

Suppose one is given a {k_0}-tuple {{\mathcal H} = (h_1,\ldots,h_{k_0})} of {k_0} distinct integers for some {k_0 \geq 1}, arranged in increasing order. When is it possible to find infinitely many translates {n + {\mathcal H} =(n+h_1,\ldots,n+h_{k_0})} of {{\mathcal H}} which consists entirely of primes? The case {k_0=1} is just Euclid’s theorem on the infinitude of primes, but the case {k_0=2} is already open in general, with the {{\mathcal H} = (0,2)} case being the notorious twin prime conjecture.

On the other hand, there are some tuples {{\mathcal H}} for which one can easily answer the above question in the negative. For instance, the only translate of {(0,1)} that consists entirely of primes is {(2,3)}, basically because each translate of {(0,1)} must contain an even number, and the only even prime is {2}. More generally, if there is a prime {p} such that {{\mathcal H}} meets each of the {p} residue classes {0 \hbox{ mod } p, 1 \hbox{ mod } p, \ldots, p-1 \hbox{ mod } p}, then every translate of {{\mathcal H}} contains at least one multiple of {p}; since {p} is the only multiple of {p} that is prime, this shows that there are only finitely many translates of {{\mathcal H}} that consist entirely of primes.

To avoid this obstruction, let us call a {k_0}-tuple {{\mathcal H}} admissible if it avoids at least one residue class {\hbox{ mod } p} for each prime {p}. It is easy to check for admissibility in practice, since a {k_0}-tuple is automatically admissible in every prime {p} larger than {k_0}, so one only needs to check a finite number of primes in order to decide on the admissibility of a given tuple. For instance, {(0,2)} or {(0,2,6)} are admissible, but {(0,2,4)} is not (because it covers all the residue classes modulo {3}). We then have the famous Hardy-Littlewood prime tuples conjecture:

Conjecture 1 (Prime tuples conjecture, qualitative form) If {{\mathcal H}} is an admissible {k_0}-tuple, then there exists infinitely many translates of {{\mathcal H}} that consist entirely of primes.

This conjecture is extremely difficult (containing the twin prime conjecture, for instance, as a special case), and in fact there is no explicitly known example of an admissible {k_0}-tuple with {k_0 \geq 2} for which we can verify this conjecture (although, thanks to the recent work of Zhang, we know that {(0,d)} satisfies the conclusion of the prime tuples conjecture for some {0 < d < 70,000,000}, even if we can’t yet say what the precise value of {d} is).

Actually, Hardy and Littlewood conjectured a more precise version of Conjecture 1. Given an admissible {k_0}-tuple {{\mathcal H} = (h_1,\ldots,h_{k_0})}, and for each prime {p}, let {\nu_p = \nu_p({\mathcal H}) := |{\mathcal H} \hbox{ mod } p|} denote the number of residue classes modulo {p} that {{\mathcal H}} meets; thus we have {1 \leq \nu_p \leq p-1} for all {p} by admissibility, and also {\nu_p = k_0} for all {p>h_{k_0}-h_1}. We then define the singular series {{\mathfrak G} = {\mathfrak G}({\mathcal H})} associated to {{\mathcal H}} by the formula

\displaystyle {\mathfrak G} := \prod_{p \in {\mathcal P}} \frac{1-\frac{\nu_p}{p}}{(1-\frac{1}{p})^{k_0}}

where {{\mathcal P} = \{2,3,5,\ldots\}} is the set of primes; by the previous discussion we see that the infinite product in {{\mathfrak G}} converges to a finite non-zero number.

We will also need some asymptotic notation (in the spirit of “cheap nonstandard analysis“). We will need a parameter {x} that one should think of going to infinity. Some mathematical objects (such as {{\mathcal H}} and {k_0}) will be independent of {x} and referred to as fixed; but unless otherwise specified we allow all mathematical objects under consideration to depend on {x}. If {X} and {Y} are two such quantities, we say that {X = O(Y)} if one has {|X| \leq CY} for some fixed {C}, and {X = o(Y)} if one has {|X| \leq c(x) Y} for some function {c(x)} of {x} (and of any fixed parameters present) that goes to zero as {x \rightarrow \infty} (for each choice of fixed parameters).

Conjecture 2 (Prime tuples conjecture, quantitative form) Let {k_0 \geq 1} be a fixed natural number, and let {{\mathcal H}} be a fixed admissible {k_0}-tuple. Then the number of natural numbers {n < x} such that {n+{\mathcal H}} consists entirely of primes is {({\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}}.

Thus, for instance, if Conjecture 2 holds, then the number of twin primes less than {x} should equal {(2 \Pi_2 + o(1)) \frac{x}{\log^2 x}}, where {\Pi_2} is the twin prime constant

\displaystyle \Pi_2 := \prod_{p \in {\mathcal P}: p>2} (1 - \frac{1}{(p-1)^2}) = 0.6601618\ldots.

As this conjecture is stronger than Conjecture 1, it is of course open. However there are a number of partial results on this conjecture. For instance, this conjecture is known to be true if one introduces some additional averaging in {{\mathcal H}}; see for instance this previous post. From the methods of sieve theory, one can obtain an upper bound of {(C_{k_0} {\mathfrak G} + o(1)) \frac{x}{\log^{k_0} x}} for the number of {n < x} with {n + {\mathcal H}} all prime, where {C_{k_0}} depends only on {k_0}. Sieve theory can also give analogues of Conjecture 2 if the primes are replaced by a suitable notion of almost prime (or more precisely, by a weight function concentrated on almost primes).

Another type of partial result towards Conjectures 1, 2 come from the results of Goldston-Pintz-Yildirim, Motohashi-Pintz, and of Zhang. Following the notation of this recent paper of Pintz, for each {k_0>2}, let {DHL[k_0,2]} denote the following assertion (DHL stands for “Dickson-Hardy-Littlewood”):

Conjecture 3 ({DHL[k_0,2]}) Let {{\mathcal H}} be a fixed admissible {k_0}-tuple. Then there are infinitely many translates {n+{\mathcal H}} of {{\mathcal H}} which contain at least two primes.

This conjecture gets harder as {k_0} gets smaller. Note for instance that {DHL[2,2]} would imply all the {k_0=2} cases of Conjecture 1, including the twin prime conjecture. More generally, if one knew {DHL[k_0,2]} for some {k_0}, then one would immediately conclude that there are an infinite number of pairs of consecutive primes of separation at most {H(k_0)}, where {H(k_0)} is the minimal diameter {h_{k_0}-h_1} amongst all admissible {k_0}-tuples {{\mathcal H}}. Values of {H(k_0)} for small {k_0} can be found at this link (with {H(k_0)} denoted {w} in that page). For large {k_0}, the best upper bounds on {H(k_0)} have been found by using admissible {k_0}-tuples {{\mathcal H}} of the form

\displaystyle {\mathcal H} = ( - p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, - p_{m+1}, -1, +1, p_{m+1}, \ldots, p_{m+\lfloor (k_0+1)/2\rfloor - 1} )

where {p_n} denotes the {n^{th}} prime and {m} is a parameter to be optimised over (in practice it is an order of magnitude or two smaller than {k_0}); see this blog post for details. The upshot is that one can bound {H(k_0)} for large {k_0} by a quantity slightly smaller than {k_0 \log k_0} (and the large sieve inequality shows that this is sharp up to a factor of two, see e.g. this previous post for more discussion).

In a key breakthrough, Goldston, Pintz, and Yildirim were able to establish the following conditional result a few years ago:

Theorem 4 (Goldston-Pintz-Yildirim) Suppose that the Elliott-Halberstam conjecture {EH[\theta]} is true for some {1/2 < \theta < 1}. Then {DHL[k_0,2]} is true for some finite {k_0}. In particular, this establishes an infinite number of pairs of consecutive primes of separation {O(1)}.

The dependence of constants between {k_0} and {\theta} given by the Goldston-Pintz-Yildirim argument is basically of the form {k_0 \sim (\theta-1/2)^{-2}}. (UPDATE: as recently observed by Farkas, Pintz, and Revesz, this relationship can be improved to {k_0 \sim (\theta-1/2)^{-3/2}}.)

Unfortunately, the Elliott-Halberstam conjecture (which we will state properly below) is only known for {\theta<1/2}, an important result known as the Bombieri-Vinogradov theorem. If one uses the Bombieri-Vinogradov theorem instead of the Elliott-Halberstam conjecture, Goldston, Pintz, and Yildirim were still able to show the highly non-trivial result that there were infinitely many pairs {p_{n+1},p_n} of consecutive primes with {(p_{n+1}-p_n) / \log p_n \rightarrow 0} (actually they showed more than this; see e.g. this survey of Soundararajan for details).

Actually, the full strength of the Elliott-Halberstam conjecture is not needed for these results. There is a technical specialisation of the Elliott-Halberstam conjecture which does not presently have a commonly accepted name; I will call it the Motohashi-Pintz-Zhang conjecture {MPZ[\varpi]} in this post, where {0 < \varpi < 1/4} is a parameter. We will define this conjecture more precisely later, but let us remark for now that {MPZ[\varpi]} is a consequence of {EH[\frac{1}{2}+2\varpi]}.

We then have the following two theorems. Firstly, we have the following strengthening of Theorem 4:

Theorem 5 (Motohashi-Pintz-Zhang) Suppose that {MPZ[\varpi]} is true for some {0 < \varpi < 1/4}. Then {DHL[k_0,2]} is true for some {k_0}.

A version of this result (with a slightly different formulation of {MPZ[\varpi]}) appears in this paper of Motohashi and Pintz, and in the paper of Zhang, Theorem 5 is proven for the concrete values {\varpi = 1/1168} and {k_0 = 3,500,000}. We will supply a self-contained proof of Theorem 5 below the fold, the constants upon those in Zhang’s paper (in particular, for {\varpi = 1/1168}, we can take {k_0} as low as {341,640}, with further improvements on the way). As with Theorem 4, we have an inverse quadratic relationship {k_0 \sim \varpi^{-2}}.

In his paper, Zhang obtained for the first time an unconditional advance on {MPZ[\varpi]}:

Theorem 6 (Zhang) {MPZ[\varpi]} is true for all {0 < \varpi \leq 1/1168}.

This is a deep result, building upon the work of Fouvry-Iwaniec, Friedlander-Iwaniec and BombieriFriedlanderIwaniec which established results of a similar nature to {MPZ[\varpi]} but simpler in some key respects. We will not discuss this result further here, except to say that they rely on the (higher-dimensional case of the) Weil conjectures, which were famously proven by Deligne using methods from l-adic cohomology. Also, it was believed among at least some experts that the methods of Bombieri, Fouvry, Friedlander, and Iwaniec were not quite strong enough to obtain results of the form {MPZ[\varpi]}, making Theorem 6 a particularly impressive achievement.

Combining Theorem 6 with Theorem 5 we obtain {DHL[k_0,2]} for some finite {k_0}; Zhang obtains this for {k_0 = 3,500,000} but as detailed below, this can be lowered to {k_0 = 341,640}. This in turn gives infinitely many pairs of consecutive primes of separation at most {H(k_0)}. Zhang gives a simple argument that bounds {H(3,500,000)} by {70,000,000}, giving his famous result that there are infinitely many pairs of primes of separation at most {70,000,000}; by being a bit more careful (as discussed in this post) one can lower the upper bound on {H(3,500,000)} to {57,554,086}, and if one instead uses the newer value {k_0 = 341,640} for {k_0} one can instead use the bound {H(341,640) \leq 4,982,086}. (Many thanks to Scott Morrison for these numerics.) UPDATE: These values are now obsolete; see this web page for the latest bounds.

In this post we would like to give a self-contained proof of both Theorem 4 and Theorem 5, which are both sieve-theoretic results that are mainly elementary in nature. (But, as stated earlier, we will not discuss the deepest new result in Zhang’s paper, namely Theorem 6.) Our presentation will deviate a little bit from the traditional sieve-theoretic approach in a few places. Firstly, there is a portion of the argument that is traditionally handled using contour integration and properties of the Riemann zeta function; we will present a “cheaper” approach (which Ben Green and I used in our papers, e.g. in this one) using Fourier analysis, with the only property used about the zeta function {\zeta(s)} being the elementary fact that blows up like {\frac{1}{s-1}} as one approaches {1} from the right. To deal with the contribution of small primes (which is the source of the singular series {{\mathfrak G}}), it will be convenient to use the “{W}-trick” (introduced in this paper of mine with Ben), passing to a single residue class mod {W} (where {W} is the product of all the small primes) to end up in a situation in which all small primes have been “turned off” which leads to better pseudorandomness properties (for instance, once one eliminates all multiples of small primes, almost all pairs of remaining numbers will be coprime).

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