Related papers: On a class number formula for real quadratic numbe…

An asymptotic formula for the sum $\sum L(1,\chi)$ is established for a family of hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$ as $g\rightarrow\infty$ making use of the analogue of the approximate functional…

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a…

This is a survey of a connection between the distribution of certain power residues modulo $p$, $p$ a prime, and relative class numbers. The focus lies on quadratic residues and sixth power residues. Dirichlet's class number formula yields…

We give some formulas of poly-Cauchy numbers by the $r$-Stirling transform. In the case of the classical or poly-Bernoulli numbers, the formulas are with Stirling numbers of the first kind. In our case of the classical or poly-Cauchy…

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime and whose…

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for…

Let $p$ be a prime. The $2$-primary part of the class group of the pure quartic field $\mathbb{Q}(\sqrt[4]{p})$ has been determined by Parry and Lemmermeyer when $p \not\equiv \pm 1\bmod 16$. In this paper, we improve the known results in…

We solve the Dirichlet problem in the unit disc and derive the Poisson formula using very elementary methods and explore consequent simplifications in other foundational areas of complex analysis.

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not…

Let $\chi$ be a Dirichlet character modulo $q$, let $L(s, \chi)$ be the attached Dirichlet $L$-function, and let $L^\prime(s, \chi)$ denotes its derivative with respect to the complex variable $s$. The main purpose of this paper is to give…

We deal with negative square moments of Dirichlet $L$-functions. Summing over characters modulo $q$, we obtain an asymptotic formula for the negative second moment of $L(1,\chi)$ involving conductors. As an application, we give the improved…

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.

An explicit formula for the mean value of $\vert L(1,\chi)\vert^2$ is known, where $\chi$ runs over all odd primitive Dirichlet characters of prime conductors $p$. Bounds on the relative class number of the cyclotomic field ${\mathbb…

In this paper, we investigate new class of sequences related to fully degenerate Bernoulli numbers and polynomials. From those sequences, we derive some formulae for the degenerate Bernoulli and Euler polynomials.

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class…

Let $K=\mathbb{Q}(\sqrt[4]{pd^{2}})$ be a real pure quartic number field and $k=\mathbb{Q}(\sqrt{p})$ its real quadratic subfield, where $p\equiv 5\pmod 8$ is a prime integer and $d$ an odd square-free integer coprime to $p$. In this work,…

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic…

Let $\chi$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, \chi)$ denote its associated Dirichlet $L$-function. We show that $|L(1, \chi)| \geq 1/(9.12255 \log(q/\pi))$ for sufficiently large $q$, and…

We focus on the generating series for the rational special values of Pellarin's $L$-series in $1 \leq s \leq 2(q-1)$ indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the…