Related papers: A Lindemann-Weierstrass theorem for $E$-functions

We prove explicit formulae for $\alpha$-points of $L$-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function $\zeta(s)$ to the case of $\alpha$-points of…

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…

We derive new integral representations for objects arising in the classical theory of elliptic functions: the Eisenstein series $E_s$, and Weierstrass' $\wp$ and $\zeta$ functions. The derivations proceed from the Laplace-Mellin…

Notable results on the special values of $L$-functions of Siegel modular forms were obtained by J. Sturm in the case when the degree $n$ is even and the weight $k$ is an integer. In this paper we extend this method to half-integer weights…

Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value $s =…

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…

We prove joint universality theorems on the half plane of absolute convergence for general classes of Dirichlet series with an Euler-product, where in addition to vertical shifts we also allow scaling. This generalizes our recent joint…

Let $(E)$ be a homogeneous linear differential equation Fuchsian of order $n$ over $\mathbb{P}^{1}(\mathbb{C}) $. The idea of Riemann (1857) was to obtain the properties of solutions of ($E$) by studying the local system. Thus, he obtained…

In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic…

Let $\pi$ and $\pi'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function…

In this article, we show a new general linear independence criterion related to values of $G$-functions, including the linear independence of values at algebraic points of contiguous hypergeometric functions, which is not known before. Let…

In this paper we use the Vandermonde matrices and their properties to give a new proof of the classical result of Karl Weierstrass about the approximation of continuous functions $f$ on closed intervals, using a sequence of polynomials. The…

In [S\'eries Gevrey de type arithm\'etique I Th\'eor\'emes de puret\'e et de dualit\'e, Annals of Math. 151 (2000), 705--740], Andr\'e has introduced E-operators, a class of differential operators intimately related to E-functions, and…

We provide the increasing eigenfunctions associated to spectrally negative self-similar Feller semigroups, which have been introduced by Lamperti. These eigenfunctions are expressed in terms of a new family of power series which includes,…

We prove the universality theorem for the iterated integrals of logarithms of $L$-functions in the Selberg class on some line parallel to the real axis.

We continue our investigation of $E$-operators, in particular their connection with $G$-operators; these differential operators are fundamental in understanding the diophantine properties of Siegel's $E$ and $G$-functions. We study in…

Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars $\mathbb{R}$ or $\mathbb{C}$ by a real or complex Dedekind complete unital…

We establish endoscopic and stable trace formulas whose discrete spectral terms are weighted by automorphic $L$-functions, by the use of basic functions that are incorporated into the global spectral and geometric coefficients. This is a…

For a lattice \Lambda in the complex plane, let K_{\Lambda} be the field of \Lambda-elliptic functions. For two relatively prime integers p (respectively q) greater than 1, consider the endomorphisms \psi (resp. \phi) of K_{\Lambda} given…

In this paper, we present a general method for obtaining addition theorems of the Weierstrass elliptic function $\wp(z)$ in terms of given parameters. We obtain the classical addition theorem for the Weierstrass elliptic function as a…