Related papers: The Kronecker limit formulas via the distribution …

We evaluate the classic sum $\sum_{n\in\mathbb{Z}} e^{-\pi n^2}$. The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuations. Even the $\Gamma$ function, in…

Let $p$ be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form $\frac{E_k^{\ast}}{V(E_k^{\ast})}$ where $E_k^{\ast}$ is a classical,…

In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations…

In this paper, the second Kronecker ``limit" formula for a real quadratic field is established for the first time. More precisely, we obtain the second Kronecker limit formula of Zagier's zeta function. Using the reduction theory of Zagier,…

Linear forms in logarithms have an important role in the theory of Diophantine equations. In this article, we prove explicit $p$-adic lower bounds for linear forms in $p$-adic logarithms of rational numbers using Pad\'e approximations of…

We establish integrality and congruence properties for the Eisenstein-Kronecker cocycle of Bergeron, Charollois and Garc\'ia introduced in [arXiv:2107.01992v2 [math.NT]]. As a consequence, we recover the integrality of the critical values…

$\Theta$ function is defined based upon Kronecher symbol. In light of the principle of inclusion-exclusion, $\Theta$ function of sine function is used to denote the distribution of composites and primes. The structure of Goldbach Conjecture…

We prove an analogue of Kronecker's second limit formula for a continuous family of "indefinite zeta functions". Indefinite zeta functions were introduced in the author's previous paper as Mellin transforms of indefinite theta functions, as…

We give new bounds and asymptotic estimates on the largest Kronecker and induced multiplicities of finite groups. The results apply to large simple groups of Lie type and other groups with few conjugacy classes.

We prove Haynes' version of the Duffin--Schaeffer conjecture for the $p$-adic numbers. In addition, we prove several results about an associated related but false conjecture, related to $p$-adic approximation in the spirit of Jarn\'ik and…

We describe Rudin-Keisler preorders and distribution functions of numbers of limit models for quite o-minimal Ehrenfeucht theories. Decomposition formulas for these distributions are found.

We generalise the proof of the $p$-adic regulator formula for Asai--Flach classes to the finite slope case, without using finite polynomial cohomology. Moreover, we simplify the analogous computation for diagonal classes, relying on a…

A binary linear error correcting codes represented by two code families Kronecker products sum are considered. The dimension and distance of new code is investigated. Upper and lower bounds of distance are obtained. Some examples are given.…

In this paper, we establish Kronecker limit type formulas for the Mordell-Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the second as well as the third arguments. As an application of these formulas, we obtain results of…

Let $p$ be a prime number. In this article, we prove that the $p$-adic Hahn series $\sum_{k=1}^\infty p^{-1/p^k}$, which is the mixed-characteristic analogue of Abhyankar's solution $\sum_{k=1}^\infty t^{-1/p^k}$ to the Artin-Schreier…

In this paper, using $p$-adic analysis and $p$-adic L-functions, we show how to extend classical congruences (due to Wilson, Gauss, Dirichlet, Jacobi, Wolstenholme, Glaisher, Morley, Lemher and other people) to modulo $p^k$ for any $k>0$.

In this paper we investigate the distribution properties of hybrid sequences which are made by combining Halton sequences in the ring of polynomials and digital Kronecker sequences. We give a full criterion for the uniform distribution and…

We prove a version of van der Corput's Lemma for polynomials over the p-adic numbers.

We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We…

The Rotar central limit theorem is a remarkable theorem in the non-classical version since it does not use the condition of asymptotic infinitesimality for the independent individual summands, unlike the theorems named Lindeberg's and…