Unconditional discrete second moments of the Riemann zeta-function along polynomial shifts: closing the abscissa gap to the line $\sigma=1/2$

For real polynomials $P_a(n)=a_1n+\cdots+a_dn^d$ of degree $d\ge 2$ with positive coefficients, Sourmelidis (2020) proved that the discrete second moments $\frac{1}{N}\sum_{n\le N}|\zeta(\sigma+iP_a(n))|^2$ and their monomial analogues converge to $\zeta(2\sigma)$ for Lebesgue-almost-every coefficient vector $a$, but only in a strip $\mathbf{S}(d)<\sigma\le 1$ where the abscissas $\mathbf{S}(d)$, $\mathbf{S}_{\mathbf{mo}}(d)$ lie in $(1/2,1)$ and tend to $1$ as $d\to\infty$; the value $1/2$ is reached under the Lindelöf hypothesis. The problem is to prove the same asymptotics unconditionally for every $\sigma>1/2$ (i.e. $\mathbf{S}(d)=\mathbf{S}_{\mathbf{mo}}(d)=1/2$), the obstruction being the known order-of-growth bounds for $\zeta(s)$ inside the critical strip; the linked sub-goal from the same source is the existence and value of the limit for the exceptional resonant coefficient vectors that the source leaves untreated (already for $\sigma>1$).

Background

The Riemann zeta-function $\zeta(s)$, $s=\sigma+it$, continues the Dirichlet series $\sum_{n\ge1}n^{-s}$ meromorphically to the plane. A classical theorem on the continuous mean square says that $\frac{1}{T}\int_0^T|\zeta(\sigma+it)|^2\,dt\to\zeta(2\sigma)$ for every $\sigma>1/2$, unconditionally. This record is about the discrete analogue in which $\zeta$ is sampled along a polynomial sequence of heights: for an integer $d\ge 2$ and positive real coefficients, set $P_a(x)=a_1x+\cdots+a_dx^d$, and ask for $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}|\zeta(\sigma+iP_a(n))|^2$ and its monomial counterpart with $P_a(x)=ax^d$.

For linear polynomials ($d=1$) the question is classical: Reich evaluated the limit, which almost always equals $\zeta(2\sigma)$, and Good obtained asymptotic formulas with error terms; the linear case is also tied to the zero-distribution of $\zeta$ on vertical arithmetic progressions studied by Li and Radziwiłł. For $d\ge 2$ the essential difficulty is the estimation of Weyl sums — finite exponential sums $\sum_{n\le N}e(P(n))$ with $e(x)=\exp(2\pi i x)$ — whose quality depends on the Diophantine properties of the coefficients. Sourmelidis (2020) overcame this for Lebesgue-almost-every coefficient vector by metric (i.e. valid for Lebesgue-almost-all coefficient vectors) mean-value estimates for Weyl sums that follow from the main conjecture in Vinogradov's Mean Value Theorem (the number $J_{h,d}(N)$ of solutions of the diagonal system $\sum_{j\le h}X_j^k=\sum_{j>h}X_j^k$, $1\le k\le d$, proved to satisfy $J_{h,d}(N)\ll_{h,d,\epsilon}N^{h+\epsilon}+N^{2h-d(d+1)/2+\epsilon}$ by Bourgain, Demeter and Guth), together with an approximate functional equation for $\zeta(s)$ — a representation of $\zeta(s)$ as an explicit finite Dirichlet polynomial (a truncated partial sum of $\sum_{n\ge1}n^{-s}$) plus a controlled error — truncated at length $t^{\mu}$ that is valid only for $\sigma$ exceeding a threshold $\mathbf{A}(\mu)$. That threshold is dictated by the best known order-of-growth bound for $\zeta(s)$ in the strip $1/2\le\sigma\le 1$ (the Vinogradov–Korobov-type bound of Ford, $|\zeta(s)|\ll|t|^{\eta(1-\sigma)^{3/2}}(\log|t|)^{2/3}$ with an absolute constant $\eta$), which in turn rests on zero-free regions — explicit subregions of the critical strip $0<\sigma<1$ in which $\zeta(s)$ is known to have no zeros.

The main theorems of the source produce effectively computable abscissas $\mathbf{S}(d)$ and $\mathbf{S}_{\mathbf{mo}}(d)$ in $(1/2,1)$, tending to $1$ from the left as $d\to\infty$, such that both discrete second-moment limits exist and equal $\zeta(2\sigma)$ for all $\mathbf{S}(d)<\sigma\le 1$ and almost every coefficient vector. The Lindelöf hypothesis — the conjecture that $\zeta(1/2+it)\ll_{\epsilon}|t|^{\epsilon}$, equivalent to $\mathbf{A}(\mu)=1/2$ for all $0<\mu<1$ — makes the asymptotics hold for every $\sigma>1/2$ (i.e. the abscissas drop to $1/2$). In the half-plane of absolute convergence $\sigma>1$ the limit equals $\zeta(2\sigma)$ for all coefficient vectors outside a countable family of resonant ones — vectors parallel to $\bigl(2\pi m_1/\log(k_0/\ell_0),\ldots,2\pi m_d/\log(k_0/\ell_0)\bigr)$, where $m\in\mathbb{Z}^d\setminus\{0\}$ and $k_0\neq\ell_0$ are coprime positive integers such that $(k_0/\ell_0)^{1/q}$ is irrational for every integer $q\ge 2$ — for which the limit exists but acquires explicit correction terms; the remaining countably many resonant cases, in which the coefficients of the induced Weyl-sum polynomial $P_a(x)\log(k/\ell)/(2\pi)$ are all rational with at least one non-integer, are left untreated in the source.

Problem Statement

Prove, without unproved hypotheses, that for every integer $d\ge 2$ (the first open case is $d=2$) the discrete second-moment limits

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\Bigl|\zeta\bigl(\sigma+i(a_1n+a_2n^2+\cdots+a_dn^d)\bigr)\Bigr|^2=\zeta(2\sigma)$

and, for Lebesgue-almost-every $a>0$, the monomial analogue

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\bigl|\zeta(\sigma+ian^d)\bigr|^2=\zeta(2\sigma)$

hold for every $\sigma>1/2$ (the value $\sigma=1/2$ itself is excluded since $\zeta(2\sigma)$ has a pole there) and for Lebesgue-almost-every coefficient vector $(a_1,\ldots,a_d)\in(0,\infty)^d$ (respectively, for Lebesgue-almost-every $a>0$ in the monomial case) — that is, that the abscissas of the source satisfy $\mathbf{S}(d)=\mathbf{S}_{\mathbf{mo}}(d)=1/2$ unconditionally, not merely under the Lindelöf hypothesis; that hypothesis is a sufficient condition under which the source obtains the statement and is not the target itself. The common objective — the discrete second-moment asymptotics of $\zeta$ along super-linear polynomial sampling throughout the critical strip — is accompanied by a second, independent question, also left untreated in the same source: for $\sigma>1$, does the limit exist — and what is its value — for each exceptional resonant coefficient vector, that is, for each coefficient vector whose induced Weyl-sum polynomial $P_a(x)\log(k/\ell)/(2\pi)$ has all coefficients rational with at least one non-integer (the countably many resonant cases recorded in the Background), in both the polynomial case (vectors $(a_1,\ldots,a_d)$, $d\ge2$) and the monomial case (coefficients $a>0$).

References

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