The Lindelöf Hypothesis for Zeta Zero Ordinates

TL;DR

The paper studies conditional and unconditional asymptotics for the exponential sums $\sum_{0<\gamma<x} \gamma^{-i\tau}$ taken over the ordinates of nontrivial zeta zeros, linking these sums to a Lindelöf-type hypothesis for the zeros, LH$(\gamma)$. It develops two analytic approaches: Stieltjes integration, using the Riemann–von Mangoldt formula to extract a main term $\frac{x^{1-i\tau}}{2\pi(1-i\tau)}\left(\log\frac{x}{2\pi}-\frac{1}{1-i\tau}\right)$ with error $\mathcal{E}$ bounded by $O(|\tau|\, (\log x)^2)$, and contour integration, which yields asymptotics in different $\tau$-regimes and connects to Dirichlet sums over primes via $\Lambda(n)$. The work then analyzes the secondary zeta-function $G(s)=\sum_{\gamma>0} \gamma^{-s}$, showing that LH-type bounds for the zero-ordinate sums are equivalent to growth bounds for $G(1/2+it)$, and establishing the meromorphic structure of $G(s)$ with a double pole at $s=1$ and simple poles at $s=1-2n$. Overall, the results illuminate how LH-type behavior for zeta zeros governs the size of zero-ordinate sums and how this translates into properties of secondary Dirichlet series.

Abstract

We provide conditional and unconditional asymptotic formulae for the exponential sums $\sum_γ\,γ^{-iτ}$, where the summation is over the ordinates of the nontrivial zeros $ρ=β+iγ$ of the Riemann zeta-function. In particular, the obtained results are related to the Lindelöf Hypothesis for these ordinates (in the sense of Gonek et al. [10]).

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Lemma 1
• proof : Proof of Theorem \ref{['unconditional1']}