Zeta functions over zeros of general zeta and $L$-functions

On zeroes and poles of Helson zeta functions

We show that the analytic continuations of Helson zeta functions $ ζ_χ(s)= \sum_1^{\infty}χ(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole critical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis.

Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling

We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $ζ(s)$ close to the one-line $σ:=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. Theorem 4 provides a zero density estimate, which is a complement to known results for the Selberg class. Note that density results for the Selberg class rely on use of the functional equation of $ζ$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Turán, extending its range…

Zeta functions over zeros of Zeta functions and an exponential-asymptotic view of the Riemann Hypothesis

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the Keiper--Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann Hypothesis being true or not, and the outcome is a sharp exponential-asymptotic …

Zeta functions for Riemann zeros

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structure, plus countably many special values) are explicitly displayed.

Approximating the Riemann Zeta-function by Polynomials with Restricted Zeros

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

亚纯函数零点与极点的 zeta 函数、零点不存在性判据及其在 Cauchy 核之和中的应用

The zeta function of zeros and poles of a meromorphic function, the criterion for the absence of zeros, and its application to sums of Cauchy kernels

该研究通过联系亚纯函数零点与极点的 zeta 函数，提出零点不存在判据，并给出 Cauchy 核之和零点存在的充分条件。