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Ramūnas Garunkštis

Publications and source records attributed to Ramūnas Garunkštis.

11 recordsLinked to original sources
arXiv2026

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

Let $t_k$ be the Gram points, and let $d\geq1$ be a fixed integer. Put $η_d=1/(2^{d+1}-2)$, $σ_d=1-η_d/d$, and $σ_d^*=7/8$ for $d=2$ and $σ_d^*=σ_d$ otherwise. We prove that, uniformly on compact subsets of $σ_d^*<\Re s<1$, $$ \frac1N\sum_{k=2}^{N}|ζ(s+it_k^d)|^2 \longrightarrow ζ(2\Re s). $$ 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 $d\geq1$. As an application, we obtain a discrete universality theorem along the same Gram-power shifts, in the same unconditional and conditional regions.

math.NT↗
arXiv2026

Second moments of general Dirichlet series and Selberg zeta functions at $σ= 1$

In this paper, we prove the existence of the second moment at $σ=1$ for a general Dirichlet series with positive coefficients whose partial sum of the coefficients up to $x$ satisfies $\varrho x+O(x/(\log x)^{1+\varepsilon})$ for some $\varepsilon>0$. The result extends a theorem of Broucke and Hilberdink. As applications, we establish the existence of the second moment for Beurling zeta functions satisfying a prime number theorem with a power-saving error term and for the Selberg zeta function for every cofinite Fuchsian group.

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arXiv2026

Higher derivatives of the Gram function

We study the nonlinear phenomenon of the asymptotics of higher-order derivatives of the Gram function $t_u$. We also prove the uniform distribution and continuous uniform distribution modulo one of $t^n_u$ for $n = 1, 2, \dots$. We use AI to formalize our proofs in Lean.

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arXiv2025

Mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis

Assuming the Riemann hypothesis, we obtain asymptotic formulas for $\sum_{0<γ<T}ζ(ρ+δ)ζ(1-ρ+\overlineδ)$ in the region $-\frac{a}{\log T} \leq \Re δ\leq \frac{1}{2}+\frac{a}{\log T}$, $|\Im δ|\ll 1$. Unconditionally, this asymptotic formula was recently obtained by Garunkštis and Novikas in essentially the same region, with a slight incompleteness. Assuming RH, we obtain a sharper error term, and we also correct an inaccuracy in the unconditional error term there.

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arXiv2025

The Lindelöf Hypothesis for Zeta Zero Ordinates

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]).

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arXiv2023

Notes on Universality in Short Intervals and Exponential Shifts

We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurin\v cikas in a problem session of a recent workshop on universality.

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arXiv2019

Zeros of the extended Selberg class zeta-functions and of their derivatives

Levinson and Montgomery proved that the Riemann zeta-function $ζ(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $ζ'(1/2+it)=0$ implies $ζ(1/2+it)=0$. Here we obtain that in small areas located to the left of the critical line and near it the functions $ζ(s)$ and $ζ'(s)$ have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class $S$. We also consider zero trajectories of a certain family of zeta-functions from $S$.

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arXiv2019

Zeros of the Lerch zeta-function and of its derivative for equal parameters

A. Speiser proved that the Riemann hypothesis is equivalent to the absence of non-real zeros of the derivative of the Riemann zeta-function left of the critical line. His result has been extended by N. Levinson and H.L. Montgomery to the statement that the Riemann zeta-function and its derivative have approximately the same number of non-real zeros left of the critical line. We obtain the Levinson-Montgomery type result for the Lerch zeta-function with equal parameters. For the Lerch zeta-function, the analogue of the Riemann hypothesis is usually not true and its zeros usually are distributed asymmetrically with respect to the critical line. However, for equal parameters, the symmetry of the zeros is almost restored.

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arXiv2019

Symmetry of zeros of Lerch zeta-function for equal parameters

For most values of parameters $λ$ and $α$, the zeros of the Lerch zeta-function $L(λ, α, s)$ are distributed very chaotically. In this paper we consider the special case of equal parameters $L(λ, λ, s)$ and show by calculations that the nontrivial zeros either lie extremely close to the critical line $σ= 1/2$ or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically.

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