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

By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space…

In this article we obtain, using an expression of the digamma function $\psi(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $\zeta(2p+1)$ for any positive value of $p$. The integrand consists of the…

By using the Wilf-Zeilberger method, we prove a novel finite combinatorial identity related to a bivariate generating function for $\zeta(2+r+2s)$ (an extension of a Bailey-Borwein-Bradley Apery-like formula for even zeta values). Such…

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \frac{\pi}{2} \right]$ may be determined by a finite or infinite sum involving the Riemann…

The alternating zeta function zeta*(s) = 1 - 2^{-s} + 3^{-s} - ... is related to the Riemann zeta function by the identity (1-2^{1-s})zeta(s) = zeta*(s). We deduce the vanishing of zeta*(s) at each nonreal zero of the factor 1-2^{1-s}…

We give a probabilistic interpretation of the Dedekind zeta functions of $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-2})$ using zeta integrals and use this to show that the first two Li coefficients of these zeta functions are positive.…

Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, which states that for any positive integer $K$ and real number…

It is known by a formula of Hasse-Sondow that the Riemann zeta function is given, for any $ s=\sigma+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty} \widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})} \sum_{k=0}^n…

In this paper, we find rational zeta series with $\zeta(2n)$ in terms of $\zeta(2k+1)$ and $\beta(2k)$, the Dirichlet beta function. We then develop a certain family of generalized rational zeta series using the generalized Clausen function…

The Riemann hypothesis, stating that the real part of all non-trivial zero points fo the zeta function must be $\frac{1}{2}$, is one of the most important unproven hypothesises in number theory. In this paper we will proof the Riemann…

It is proved that $$\int_{T}^{2T} \left|\frac{\zeta\left(\frac{1}{2}+{\rm i} t\right)}{\zeta\left(1+2{\rm i} t\right)}\right|^2 {\rm d} t = \frac{1}{\zeta(2)} T \log T + \left( \frac{\log \frac{2}{\pi} + 2\gamma -1 }{\zeta(2)} -4…

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for $\zeta(2m+1)$. The…

It is a well-known fact that Riemann Hypothesis will follows if the function identically equal to -1 can be arbitrarily approximated in the norm $\norma{.}$ of $L^{2}([0,1],dx)$ by functions of the form $f(x)=\sum_{k=1}^{n}a_{k}…

Lindel\"of conjectured that the Riemann zeta function $\zeta(\sigma+it)$ grows more slowly than any fixed positive power of $t$ as $t\rightarrow\infty$ when $\sigma\geq 1/2$. Hardy and Littlewood showed that this is equivalent to the…

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that inserting all cyclic shifts of some fixed blocks of 2's into the multiple zeta value {\zeta}(1,3,...,1,3) gives an explicit rational multiple of a…

In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral…

We compute the Fourier coefficients of the weight one modular form $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$ in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and…

We prove a conjecture of Zagier about the inverse of a $(K-1)\times (K-1)$ matrix $A=A_{K}$ using elementary methods. This formula allows one to express the the product of single zeta values $\zeta(2r)\zeta(2K+1-2r)$, $1\leq r\leq K-1$, in…

In a recent work, Dancs and He found an Euler-type formula for $\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in…

In \cite{TallaWaffo2025arxiv2511.02843} we introduced even polynomials $\Xi_n,\Lambda_n\in\mathbb{Q}[x]$ arising from integral representations of $\beta(2n)/\pi^{2n-1}$ and $\zeta(2n+1)/\pi^{2n}$. In this paper we give explicit closed…