Related papers: Dirichlet series associated to quartic fields with…

We develop a new theory of $L$-series based on replacing Dirichlet characters mod $N$ by symmetric functions mod $N$ in order to calculate explicitly the sums of infinite series more closely related to $\zeta(2n+1)$, specifically…

We study the $4$-rank of the ideal class group of $K_n := \mathbb{Q}(\sqrt{-n}, \sqrt{n})$. Our main result is that for a positive proportion of the squarefree integers $n$ we have that the $4$-rank of $\text{Cl}(K_n)$ equals $\omega_3(n) -…

This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…

For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and…

We derive a general formula for the product of two Dirichlet series that satisfy Hecke's functional equation. Several examples are provided to demonstrate the applicability of the formula. In addition, we discuss prior work on similar…

Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of…

In this paper, we consider the Dirichlet problem for a new class of augmented Hessian equations. Under sharp assumptions that the matrix function in the augmented Hessian is regular and there exists a smooth subsolution, we establish global…

This paper investigates lower bounds on the number of zeros and poles of a general Dirichlet series in a disk of radius $r$ and gives, as a consequence, an affirmative answer to an open problem of Bombieri and Perelli on the bound.…

Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square…

We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.

We give an axiomatic characterization of multiple Dirichlet series over the function field $\mathbb F_q(T)$, generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the…

We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding…

In this paper, we prove a large sieve inequality for quartic Dirichlet characters. The result is analogous to large sieve inequalities for the quadratic and cubic Dirichlet characters.

A desmic quartic surface is a birational model of the Kummer surface of the self-product of an elliptic curve. We recall the classical geometry of these surfaces and study their analogs in arbitrary characteristic. Moreover, we discuss the…

We study superpositions and direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces,…

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,\alpha)$-summable to the function in the norm of the space, provided that $\alpha>1/2$. We…

Asymptotic expansions as well as necessary and sufficient conditions are provided for the pointwise convergence of the spherical partial integrals of the associated Fourier transforms on the real hyperbolic space. The proposed method…

We establish an adelic version of Dirichlet's approximation theorem on spheres. Let $K$ be a number field, $E$ be a rigid adelic space over $K$ and $q\colon E\to K$ be a quadratic form. Let $v$ be a place of $K$ and $\alpha\in…

We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $\mathbb{C}^2$ and use Tauberian methods to…

We present other proofs, generalizations and analogues of the identities concerning multiple Dirichlet series by Tahmi and Derbal (2022). As applications, we obtain asymptotic formulas with remainder terms for certain related sums.