Moments of restricted divisor functions

Hyperelliptic sigma functions and the Kadomtsev–Petviashvili equation

Abstract We study Severi curves parametrizing rational bisections of elliptic fibrations associated to general pencils of plane cubics. Our main results show that these Severi curves are connected and reduced, and we give an upper bound on their geometric genus using quasi-modular forms. We conjecture that these Severi curves are eventually reducible, and we formulate a precise conjecture for their degrees in $\mathbb{P}^{2}$, featuring a divisor sum formula for collision multiplicities of branch points.

Abstract Let $f $ be a normalized Hecke eigenform of even weight $k \geq 2$ for $SL_2(\mathbb {Z})$ . In this article, we establish an asymptotic formula for the shifted convolution sum of a general divisor function, where the sum involves the Fourier coefficients of a multi-folded L-function weighted with a kernel function.

Sums of the higher divisor function of diagonal homogeneous forms at prime arguments

Abstract Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function over number fields, which were inspired by analogous results over function fields proven by the authors. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.

Correction: Arithmetic properties of MacMahon-type sums of divisors

In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference α, an integer with the Voronoĭ function weight Vk. In the case of V1(x)=e−x, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The α=0 case is the divisor function, while the α=1 case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan.

An asymptotic formula involving the triple divisor function

Let [Formula: see text] denote the coefficients of an automorphic [Formula: see text]-function on [Formula: see text]. Let [Formula: see text] be a multiplicative function which satisfies the following conditions: For every prime number [Formula: see text] outside of an exceptional set [Formula: see text], [Formula: see text], where [Formula: see text] is a non-zero integer. Also for every positive integer [Formula: see text], [Formula: see text], where [Formula: see text] is the [Formula: see text]-dimensional divisor function. Moreover, the exceptional set [Formula: see text] satisfies [Formula: see text] for any [Formula: see text]. Then under GRH and GRC, we get a squareroot cancelation for [Formula: see text] in almost all short intervals, provided [Formula: see text] as [Formula: see text], [Formula: see text] and [Formula: see text] for any fixed [Formula: see text]. Let [Formula: see text] be an even weight [Formula: see text], level [Formula: see text] primitive, Hecke-normalized holomorphic cusp form without complex multiplication. For the symmetric power [Formula: see text]-functions [Formula: see text] we only need to assume GRH. And we find that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are within the scope of our results, which generalize the work of Chinis. In the course of the proof we unconditionally obtain an upper bound on the summation of [Formula: see text] over smooth numbers.

Note on a sum involving the divisor function

Hurwitz integrality of the power series expansion of the sigma function for a telescopic curve

Let [Formula: see text] denote the kth divisor function and [Formula: see text], [Formula: see text] be a general positive definite integral quadratic form. In this paper, we study the asymptotic formula of the sum [Formula: see text] with the box [Formula: see text] and [Formula: see text] is an integer. Previously only the diagonal case is studied.

A unified approach to Rohrlich-type divisor sums

Arithmetic properties of MacMahon-type sums of divisors

In this paper, we introduce alternative representations for the generating function of the number of cubic partitions of an integer. These new representations lead to novel formulas and provide a fresh combinatorial interpretation of cubic partitions as color partitions into distinct parts. We also obtain analogous results regarding the number of parts of size d colored identically in the cubic partitions of n and the number of cubic partitions of n that exclude parts of size d colored identically. Additionally, a new connection between an alternative sum of divisors and the 2-adic valuation is established. Furthermore, we present two open problems related to the positivity of truncated theta series within this framework.

We offer an arithmetic proof of a result from the recent paper [1]. A more general result is provided, too.

In Kuperberg and Lalín [Forum Math. 34 (2022), pp. 711–747], the authors studied the mean-square of certain sums of the divisor function d k ( f ) d_k(f) over the function field F q [ T ] \mathbb {F}_q[T] in the limit as q → ∞ q \to \infty and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [Math. Z. 288 (2018), pp. 167–198] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields.

Abstract Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$ . Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$ . That family is called the $(e,q)$ -curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$ . Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$ , and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

In this paper, we investigate the average behavior of a hybrid arithmetic function, i.e. [Formula: see text] over a certain sequence of positive integers, where [Formula: see text] is the [Formula: see text] normalized Fourier coefficients of the [Formula: see text] ([Formula: see text] is any fixed integer) symmetric power [Formula: see text]-function (i.e. [Formula: see text]), [Formula: see text] and [Formula: see text] are the sum of divisors function and the Euler totient function, respectively. Precisely, we prove an asymptotic formula with an error term for the sum [Formula: see text] where [Formula: see text] is sufficiently large, and [Formula: see text]