On algebraic differential equations concerning the Riemann-zeta function and the Euler-gamma function

Difference independence of the Riemann zeta function

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 $r\to \infty$

Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analog…

On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

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 general statement of the functional equation for the Riemann zeta-function

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.

Relations among the Riemann zeta and Hurwitz zeta functions as well as their products

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.

X-Ray of Riemann zeta-function

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).

黎曼ζ函数 $ζ(s)$ 的一种新表示

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。