Related papers: Arithmetic of linear forms involving odd zeta valu…

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of…

Make an exponential transformation in the integral formulation of Riemann's zeta-function zeta(s) for Re(s) > 0. Separately, in addition make the substitution s -> 1 - s and then transform back to s again using the functional equation.…

We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and…

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…

We have used the first 2600 nontrivial zeros gamma_l of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these…

A generalization of the Apery-like numbers, which is used to describe the special values $\zeta_Q(2)$ and $\zeta_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a…

We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and…

Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the…

We revisit Beukers' modular-form proof of the irrationality of $\zeta(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $\Gamma_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to…

In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the…

We put forward a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is a generalization of that of A. Knopfmacher and J. Knopfmacher. Our result implies that there…

We give an explicit representation for the sums of multiple zeta-star values of fixed weight and height in terms of Riemann zeta values.

We prove an evaluation for the stuffle-regularised multiple $t$ value $ t^{\ast,V}(\{\overline{1}\}^a, 1, \{\overline{1}\}^b) $ in terms of $ \log(2) $, $ \zeta(k) $ and $ \beta(k) $. This arises by evaluating the corresponding generating…

One of the many remarkable properties of the Ap\'ery numbers $A (n)$, introduced in Ap\'ery's proof of the irrationality of $\zeta (3)$, is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m)…

The main goal of this article is to present an elementary proof of Ramanujan's identity for odd zeta values. Our proof solely relies on a Mittag-Leffler type expansion for hyperbolic cotangent function and Euler's identity for even zeta…

The renormalization of MZV was until now carried out by algebraic means. We show that renormalization in general, of the multiple zeta functions in particular, is more than mere convention. We show that simple calculus methods allow us to…

Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two…

Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those…

We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results…

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan's classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd…