A. Voros
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.
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Assuming the Riemann hypothesis (RH) and using Littlewood's conditional estimates for the Riemann zeta-function, we provide an estimate related to an approach of Y. Motohashi to the zero-free region.
It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions $ϕ$ with Nevanlinna characteristic satisfying $T(r, ϕ)=o(r)$ as $r\to \infty$
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $ζ(s)$, $s=σ+i t$, $0\leq σ\leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $\sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $Φ(u,v,β)$, $…
A formal description of a functional analysis approach to the Riemann zeta-functional equation that provides in principle an infinity of different proofs based on work by the author on the existence of dilation-invariant unitary operators in L2 of the positive real line.
Several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation if the important results of [11]. Also, a relation derived here provides the starting point of a novel approach which in a series of companion papers yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large $α$ behaviour of the modified Hurwitz zeta function $ζ_1(s,α)$, $s\in \mathbf{C}$, $α\in \mathbf{R}$, which is also presented here.
We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about the zeta function. But it gives a new perspective to many known results. Also it may prove useful to show to students of Complex Variables or Analytic Number Theory. (This was my initial motivation to make the drawings).
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 …
In the paper it is discussed the relations of the Riemann $ζ-$function to classes of generating functions of multiply positive sequences according to Schoenberg (also called Pólya frequency sequences). Key words: multiply positive sequences, totally positive sequences, Pólya frequency sequences, Laguerre-Pólya class, Riemann $ζ-$function.
We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\frac12$ with $\bar{f(\bar{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann's classical explicit formula, these representations use integrals along the critical line ${Re}(s)=\frac12$ and Blaschke zeta zeroes.
Series expansions for the Riemann zeta function
该研究证明了含临界线的复垂直带形区域内黎曼ζ函数的收敛无穷级数表示结论，据此重推已知展开式并发现了基于不完全伽马函数等的新级数表示。
Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.
We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.
We prove some uniqueness results for the Riemann zeta-function and the Euler gamma-function by virtue of shared values using the value distribution theory.
In this paper we treat the classical Riemann zeta function as a function of three variables: one is the usual complex $\adyn$-dimensional, customly denoted as $s$, another two are complex infinite dimensional, we denote it as $\b = \{b_n\}_{n=1}^{\infty}$ and $\z =\{z_n\}_{n=1}^{\infty}$. When $\b = \{1\}_{n=1}^{\infty}$ and $\z = \{\frac{1}{n}\}_{n=1}^{\infty}$ one gets the usual Riemann zeta function. Our goal in this paper is to study the meromorphic continuation of $ζ(\b , \z ,s)$ as a function of the triple $(\a , \z , s)$. Minor corrections, to appear in the Journal of Mathematical Analy…
It is shown that a new series representation of Riemann s Zeta function obtained by Tyagi and Holm leads to an interesting new recursion for Bernoulli numbers of even index as well as new representations of, and infinite series involving, Zeta functions of special (integer) argument.
A New Representation of the Riemann Zeta Function $ζ(s)$
本文推广了黎曼ζ函数与伯努利数的已知关系，提出了一种以伯努利数嵌套级数表示ζ函数的新公式。
The quadratic Mandelbrot set has been referred to as the most complex and beautiful object in mathematics and the Riemann Zeta function takes the prize for the most complicated and enigmatic function. Here we elucidate the spectrum of Mandelbrot and Julia sets of Zeta, to unearth the geography of its chaotic and fractal diversities, combining these two extremes into one intrepid journey into the deepest abyss of complex function space.
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