Related papers: A fast algorithm to compute the Ramanujan-Deninger…

We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler-Kronecker constants $\mathfrak{G}_q$ for the prime cyclotomic fields $\mathbb{Q}(\zeta_q)$, where $q$…

We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of…

We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus…

We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…

The aim of the paper is to relate computational and arithmetic questions about Euler's constant $\gamma$ with properties of the values of the $q$-logarithm function, with natural choice of $q$. By these means, we generalize a classical…

Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non trivial primitive Dirichlet character $\chi$ defined $\bmod\ q$, where $q$ is an odd prime. In this paper we introduce a fast method to compute $\vert L(1,\chi) \vert$ using…

For $\chi_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,\chi_k)$, expressed as linear combinations of Hurwitz $\zeta$ functions, are found for $s=2,3$ and some…

We provide a new algorithm for evaluating the gamma function at any (rational) point and a new infinite product representation free from the presence of Euler and Mascheroni constant.Formulae and inequalities seemingly new are obtained as…

An algorithm to compute Dirichlet $L$-functions for many quadratic characters is derived. The algorithm is optimal (up to logarithmic factors) provided that the conductors of the characters under consideration span a dyadic window.

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function:…

For a primitive Dirichlet character $\chi$ modulo $q$, we define $M(\chi)=\max_{t } |\sum_{n \leq t} \chi(n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\geq 3$. Our main result provides a further…

A finite, connected, $(d+1)$-regular graph $G$ is called Ramanujan if every its eigenvalue $\lambda$ satisfies either $\lambda=\pm (d+1)$ or $|\lambda|\leq 2\sqrt{d}$. The Ramanujan condition corresponds to the optimal rate of decay of…

Suppose $q$ is a fixed odd prime power, $F(\vec{x})$ is a non-degenerate quadratic form over $\mathbb{F}_q[t]$ of discriminant $\Delta$ in $d\geq 5$ variables $\vec{x}$, and $f,g\in\mathbb{F}_q[t]$,…

We discuss the best methods available for computing the gamma function $\Gamma(z)$ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small…

Let $\epsilon \in (0, 1)$ and $n, \Delta \in \mathbb N$ be such that $\Delta = \Omega\left(\max\left\{\frac{\log n}{\epsilon},\, \left(\frac{1}{\epsilon}\log \frac{1}{\epsilon}\right)^2\right\}\right)$. Given an $n$-vertex $m$-edge simple…

We prove a number of results regarding odd values of the Ramanujan $\tau$-function. For example, we prove the existence of an effectively computable positive constant $\kappa$ such that if $\tau(n)$ is odd and $n \ge 25$ then either \[…

In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of…

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the $q$-aspect for the logarithmic derivative $\left(L'/L\right)\left(\sigma,\chi\right)$ of Dirichlet $L$-functions, where $\chi$ is a primitive character modulo…

We prove that $q+1$-regular Morgenstern Ramanujan graphs $X^{q,g}$ (depending on $g\in\mathbb{F}_q[t]$) have diameter at most $\left(\frac{4}{3}+\varepsilon\right)\log_{q}|X^{q,g}|+O_{\varepsilon}(1)$ (at least for odd $q$ and irreducible…

Recently, the degenerate gamma functions are introduced as a degenerate version of the usual gamma function by Kim-Kim. In this paper, we investigate several properties of them. Namely, we obtain an analytic continuation as a meromorphic…