Identities for the shifted harmonic numbers and binomial coeffecients

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6071-6086
Author(s):  
Ce Xu
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Harmonic Numbers ◽  
Riemann Zeta

We develop new closed form representations of sums of (n+?)th shifted harmonic numbers and reciprocal binomial coeffecients through ?th shifted harmonic numbers and Riemann zeta function with positive integer arguments. Some interesting new consequences and illustrative examples are considered.

scholarly journals Explicit evaluation of quadratic Euler sums

2017 ◽  
Vol 13 (03) ◽  
pp. 655-672 ◽  
Author(s):  
Ce Xu ◽  
Yingyue Yang ◽  
Jianwen Zhang
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Harmonic Numbers ◽  
Euler Sums ◽  
Explicit Evaluation ◽  
Explicit Formulae ◽  
Riemann Zeta ◽  
The Given ◽  
Double Nonlinear

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.

Fourier transforms related to ζ(s)

2021 ◽  
pp. 1-17
Author(s):  
Pablo A. Panzone
Keyword(s):  
Fourier Transform ◽  
Zeta Function ◽  
Closed Form ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Fourier Transforms ◽  
Riemann Zeta ◽  
The Fourier Transform

Abstract Using some formulas of S. Ramanujan, we compute in closed form the Fourier transform of functions related to Riemann zeta function $\zeta (s)=\sum \nolimits _{n=1}^{\infty } {1}/{n^{s}}$ and other Dirichlet series.

scholarly journals An integral involving the generalized zeta function

1990 ◽  
Vol 13 (3) ◽  
pp. 453-460 ◽  
Author(s):  
E. Elizalde ◽  
A. Romeo
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Surfaces ◽  
The Riemann Zeta Function ◽  
Special Integral ◽  
Riemann Zeta ◽  
Compact Riemann Surfaces ◽  
Determinants Of Laplacians ◽  
Generalized Zeta Function

A general value for∫abdtlogΓ(t), fora,bpositive reals, is derived in terms of the Hurwitzζfunction. That expression is checked for a previously known special integral, and the case whereais a positive integer andbis half an odd integer is considered. The result finds application in calculating the numerical value of the derivative of the Riemann zeta function at the point−1, a quantity that arises in the evaluation of determinants of Laplacians on compact Riemann surfaces.

On the distribution of zeros of derivatives of the Riemann ξ-function

2020 ◽  
Vol 32 (1) ◽  
pp. 1-22
Author(s):  
Amita Malik ◽  
Arindam Roy
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Real Number ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Density Estimate ◽  
Distribution Of Zeros ◽  
Riemann Zeta ◽  
Derivatives Of

AbstractFor the completed Riemann zeta function {\xi(s)}, it is known that the Riemann hypothesis for {\xi(s)} implies the Riemann hypothesis for {\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence {(\alpha\gamma_{m})}, where α is any fixed non-zero real number and {\gamma_{m}} runs over the imaginary parts of the zeros of {\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of {\xi^{(m)}(s)}. In particular, all our results hold uniformly for {0\leq m\leq g(T)}, where the function {g(T)} tends to infinity with T and {g(T)=o(\log\log T)}.

scholarly journals On a question of Ramachandra

1982 ◽  
Vol Volume 5 ◽  
Author(s):  
hugh L Montgomery
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Sigma 1 ◽  
International Audience ◽  
Riemann Zeta

International audience For each positive integer $k$, let $$a_k(n)=(\sum_p p^{-s})^k=\sum_{n=1}^{\infty} a_k(n)n^{-s},$$ where $\sigma={\rm Re}(s)>1$, and the sum on the left runs over all primes $p$. This paper is devoted to proving the following theorem: If $1/2<\sigma<1$, then $$\max_k(\sum_{n\leq N} a_k(n)^2n^{-2\sigma})^{1/2k}\approx (\log N)^{1-\sigma}/\log\log N$$ and $$(\sum_{n=1}^{\infty} a_k(n)^2n^{-2\sigma})^{1/2k} \approx k^{1-\sigma}/(\log k)^{\sigma}.$$ The constants implied by the $\approx$ sign may depend upon $\sigma$. This theorem has applications to the Riemann zeta function.

scholarly journals On the values of the Riemann zeta-function at rational arguments

2001 ◽  
Vol Volume 24 ◽  
Author(s):  
S Kanemitsu ◽  
Y Tanigawa ◽  
M Yoshimoto
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Riemann Zeta Function ◽  
Companion Paper ◽  
Hurwitz Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Ramanujan’S Formula ◽  
Form Evaluation ◽  
Rational Arguments

International audience In a companion paper, ``On multi Hurwitz-zeta function values at rational arguments, Acta Arith. {\bf 107} (2003), 45-67'', we obtained a closed form evaluation of Ramanujan's type of the values of the (multiple) Hurwitz zeta-function at rational arguments (with denominator even and numerator odd), which was in turn a vast generalization of D. Klusch's and M. Katsurada's generalization of Ramanujan's formula. In this paper we shall continue our pursuit, specializing to the Riemann zeta-function, and obtain a closed form evaluation thereof at all rational arguments, with no restriction to the form of the rationals, in the critical strip. This is a complete generalization of the results of the aforementioned two authors. We shall obtain as a byproduct some curious identities among the Riemann zeta-values.

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