Related papers: Resurgent Lambert series with characters

We investigate the analytic properties of a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms for orthogonal groups of signature $(2,n+2)$. Using an orthogonal Eisenstein series of Klingen type, we obtain an…

We introduce an elliptic version of the Grothendieck-Springer sheaf and establish elliptic analogues of the basic results of Springer theory. From a geometric perspective, our constructions specialize geometric Eisenstein series to the…

Various algebraic properties of Heilbronn's exponential sum can be deduced through the use of supercharacter theory, a novel extension of classical character theory due to Diaconis-Isaacs and Andre. This perspective yields a variety of…

We represent the Euler alternating series (sometimes called the "Dirichlet eta function"), and generally $(b^s-b)\zeta(s)/b^s$ for $b>1$ an integer, in the half-plane $\Re s>0$, via series dominated by geometric series, with arbitrarily…

The generating series associated to a certain sequence of limited Taylor expansions coincides with the continuously iterated Invert Transform and displays thus a nice functional equality.

Many observables in quantum field theory can be expressed in terms of trans-series, in which one adds to the perturbative series a typically infinite sum of exponentially small corrections, due to instantons or to renormalons. Even after…

We compute the Hilbert series of the graded algebra of regular functions on a symplectic quotient of a unitary circle representation. Additionally, we elaborate explicit formulas for the lowest coefficients of the Laurent expansion of such…

Some Dirichlet-like functions, attached to a pair (periodic function, polynomial) are introduced and studied. These functions generalize the standard Dirichlet L-functions of Dirichlet characters. They have similar properties, being…

An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…

In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of…

Matrix Dirichlet processes, in reference to their reversible measure, appear in a natural way in many different models in probability. Applying the language of diffusion operators and the method of boundary equations, we describe Dirichlet…

Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let…

In this paper we give a classification of the asymptotic expansion of the $q$-expansion of reciprocals of Eisenstein series $E_k$ of weight $k$ for the modular group $\func{SL}_2(\mathbb{Z})$. For $k \geq 12$ even, this extends results of…

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure…

Let g be a basic classical Lie superalgebra over C. In the case of a typical weight whose every nonnegative integer multiple is also typical, we compute a closed form for the Hilbert series whose coefficients encode the dimensions of…

We review certain classes of iterated integrals that appear in the computation of Feynman integrals that involve elliptic functions. These functions generalise the well-known class of multiple polylogarithms to elliptic curves and are…

Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker--Eisenstein…

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra…

In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and…

We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The…