Fekete polynomials, quadratic residues, and arithmetic

arXiv (Cornell University), 2022

For each prime number p one can associate a Fekete polynomial with coefficients −1 or 1 except the constant term, which is 0. These are classical polynomials that have been studied extensively in the framework of analytic number theory. In a recent paper, we showed that these polynomials also encode interesting arithmetic information. In this paper, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be composite numbers. We then investigate the appearance of cyclotomic factors in these generalized Fekete polynomials. Based on this investigation, we introduce a compact version of Fekete polynomials as well as their trace polynomials. We then study the Galois groups of these Fekete polynomials using modular techniques. In particular, we discover some surprising extra symmetries which imply some restrictions on the corresponding Galois groups. Finally, based on both theoretical and numerical data, we propose a precise conjecture on the structure of these Galois groups.

TURKISH JOURNAL OF MATHEMATICS

In this paper, by applying the p-adic q-integrals to a family of continuous differentiable functions on the ring of p-adic integers, we construct new generating functions for generalized Apostol-type numbers and polynomials attached to the Dirichlet character of a finite abelian group. By using these generating functions with their functional equations, we derive various new identities and relations for these numbers and polynomials. These results are generalizations of known identities and relations including some well-known families of special numbers and polynomials such as the generalized Apostol-type Bernoulli, the Apostol-type Euler, the Frobenius-Euler numbers and polynomials, the Stirling numbers, and other families of numbers and polynomials. Moreover, by the help of these generating functions, we also construct other new families of numbers and polynomials with their generating functions. By using these functions, we investigate some fundamental properties of these numbers and polynomials. Finally, we also give explicit formulas for computing the Apostol-Bernoulli and Apostol-Euler numbers.

Transactions of the American Mathematical Society, 1985

We study the zeros (mod p) of the polynomial ßp(X) = ¿Zk(Bk/k)(Xp~l~k -1) for p an odd prime, where Bk denotes the k\\\ Bernoulli number and the summation extends over 1 < k < p -2. We establish a reciprocity law which relates the congruence ßp(r) = 0 (mod p) to a congruence f (n) = 0 (mod r) for r a prime less than/7 and n e Z. The polynomial/ (x) is the irreducible polynomial over Q of the number Tr^(il f, where f is a primitive p2 th root of unity and L c Q(f ) is the extension of degree p over Q. These congruences are closely related to the prime divisors of the indices 1(a) = (0 : T\a]), where 0 is the integral closure in L and a e 0 is of degree p over Q. We establish congruences (mod p ) involving the numbers /(a) and show that their prime divisors r =£ p are closely related to the congruence rr~l ■ 1 (mod p2 ).

Math Sci (2014) 8:131

In the present paper, we introduce the Eulerian polynomials attached to χ by using p-adic q-integral on ℤ_{p}. Also, we give some new interesting identities via the generating functions of Dirichlet's type of Eulerian polynomials. In addition, by applying Mellin transformation to the generating function of Dirichlet's type of Eulerian polynomials, we define Eulerian-L function which interpolates of Dirichlet's type of Eulerian polynomials at negative integers.

Class Groups of Number Fields and Related Topics, 2020

Let p be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo p which are multiples of 2 or 3 or 4 and lying in the interval [1, p − 1], by applying the Dirichlet's class number formula for the imaginary quadratic field Q(√ −p).

Bulletin of the Australian Mathematical Society, 2002

For an even Dirichlet character ψ, we obtain a formula for L(1, ψ) in terms of a sum of Dirichlet L-series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coefficients. We then derive a class number formula for real quadratic number fields by taking L(s, ψ) to be the quadratic L-series associated with these fields. ∞ k=1 2k k −1 k −n and related sums, J. Number Theory, 20 (1985), 92-102.

Algebra & Number Theory, 2011

HAL (Le Centre pour la Communication Scientifique Directe), 2021

Arxiv preprint math/0502019, 2005

Abstract: By using $ q $-Volkenborn integration and uniform differentiable on $\ mathbb {Z}% _ {p} $, we construct $ p $-adic $ q $-zeta functions. These functions interpolate the $ q $-Bernoulli numbers and polynomials. The value of $ p $-adic $ q $-zeta functions at ...

Journal of Number Theory, 1976

An elementary proof is given for the existence of the Kubota-Leopoldt padic L-functions. Also, an explicit formula is obtained for these functions, and a relationship between the values of the padic and classical L-functions at positive integers is discussed.