Related papers: Revisiting The Riemann Zeta Function at Positive E…

Given a periodic function $f$, we study the convergence almost everywhere and in norm of the series $\sum_{k} c_k f(kx)$. Let $f(x)= \sum_{m=1}^\infty a_m \sin {2\pi m x}$ where $\sum_{m=1}^\infty a_{m }^2d(m) <\infty$ and $d(m)=\sum_{d|m}…

By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, We find that this special integral is two times of another special integral. By using this fact we obtain the…

In this paper, we use the Riemann zeta function $\zeta(x)$ and the Bessel zeta function $\zeta_{\mu}(x)$ to study the log-behavior of combinatorial sequences. We prove that $\zeta(x)$ is log-convex for $x>1$. As a consequence, we deduce…

In this paper, we evaluate some series via the WZ method, and confirm several previous conjectures. For example, we prove the following two identities conjectured by the second author: $$\sum_{k=0}^{\infty} \frac{(28k^2 + 10k + 1)…

On the one hand the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand the Fourier transform representation of the gamma and…

By using the related results in the WZ theory, a new (as far as I know) formula for the values of Dirichlet beta function $\beta (s) = \sum\limits_{n = 1}^{+ \infty} {\frac{(-1)^{n - 1}}{(2n - 1)^s}} $ (where $Re(s) > 0$) at odd positive…

We consider the representation growth of split extensions of $SL_2^m(O)$. We prove that the corresponding representation zeta functions factor as a product of the representation zeta function of $SL_2^m(O)$ and the relative representation…

In this paper we will study the double zeta values $\zeta(k,m)$ using Picard-Fuchs equation. We will give a very efficient method to evaluate $\zeta(k,1)$ (resp. $\zeta(k,2)$) in terms of the products of zeta values…

In this short note, we aim to prove that the Fourier coefficients of the modular function $ \frac{\eta^{24}(\tau)}{\eta^{24}(2\tau)} $ possess a sign-change property.

In this paper we introduce and study double tails of multiple zeta values. We show, in particular, that they satisfy certain recurrence relations and deduce from this a generalization of Euler's classical formula…

We present explicit formulas for the following family of parametric binomial sums involving harmonic numbers for $p=0,1,2$ and $|t|\leq1$. $$ \sum_{k=1}^{\infty}\frac{H_{k-1}t^k}{k^p\binom{n+k}{k}}\quad \mbox{and}\quad…

The evaluation of the Riemann zeta function at negative integers is a classical result typically obtained through analytic continuation or contour integration. In this paper, we present a novel and concise derivation of these special values…

It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=\pi/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=\pi/3$. In this paper we obtain their p-adic analogues such as…

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete…

We obtain several expansions for $\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities…

In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of…

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the…

A general technique for proving the irrationality of the zeta constants $\zeta(s)$ for odd $s = 2n + 1 \geq 3$ from the known irrationality of the beta constants $L(2n+1)$ is developed in this note. The results on the irrationality of the…

A generalization of a well-known relation between the Riemann zeta function $\zeta(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli…

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the…