The discrete second moment of the Riemann zeta-function along powers of Gram points

Abstract

Let $t_k$ be the Gram points, and let $d\ge 1$ be a fixed integer. Put ${\eta }_d=1/(2^{d+1}-2)$, ${\sigma }_d=1-{\eta }_d/d$, and ${\sigma }_d^{\ast }=7/8$ for $d=2$ and ${\sigma }_d^{\ast }={\sigma }_d$ otherwise. We prove that, uniformly on compact subsets of ${\sigma }_d^{\ast }<\Re s<1$, $$\frac{1}{N}\sum _{k=2}^N|\zeta (s+i{t}_k^d){|}^2⟶\zeta (2\Re s).$$ The underlying result is a mean-square approximation by finite Euler products. For the squares of Gram points, grouping equal frequency ratios and applying the double large sieve improves the general derivative-test boundary $11/12$ to $7/8$. Assuming the Lindelöf hypothesis, both the Euler-product approximation and the second-moment formula hold on every compact subset of $1/2<\Re s<1$, for every fixed $d\ge 1$. As an application, we obtain a discrete universality theorem along the same Gram-power shifts, in the same unconditional and conditional regions.

arXiv commentIn Remark 1.1, we note an interesting coincidence related to the constants 7/8 and 11/12