Related papers: Two dimensional value-distribution of cubic Hecke …

Beginning from the resolution of the Dirichlet L function, using the inner product formula between two infinite-dimensional vectors in the complex space, the author proved the baffling problem--Hecke conjecture.

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the…

We construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as…

We evaluate the first three moments of central values of a family of qudratic Hecke $L$-functions in the Gaussian field with power saving error terms. In particular, we obtain asymptotic formulas for the first two moments with error terms…

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for…

In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double…

Let $f$ be a holomorphic cusp form of weight $k$ with respect to $SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper, we shall give the relation of the number of zeros…

We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series,…

Let $k = \mathbb{F}_{q}(T)$ be the rational function field over a finite field $\mathbb{F}_{q}$, where $q$ is a power of $2$. In this paper we solve the problem of averaging the quadratic $L$-functions $L(s, \chi_{u})$ over fundamental…

In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials, which are associated with the space of cusp…

We prove several results on the distribution of values of $L$-functions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power…

For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular…

We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the…

We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family,…

We prove an asymptotic formula for the second moment of $L$-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight.

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in…

This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…

Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According…

We explain how to compute idempotents that correspond to the indecomposable objects in the Hecke category. Closed formulas are provided for some common coefficients that appear in these idempotents. We also explain how to compute…

We evaluate the smoothed first moment of central values of a family of qudratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size…