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This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree. [ABSTRACT FROM AUTHOR]

We study the behavior of the shifted convolution sum involving even power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the k -full kernel function for any fixed integer k ≥ 2. [ABSTRACT FROM AUTHOR]

Suppose that x is a sufficiently large number and j ≥ 2 is any integer. Let L (s , sym j f) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL 2 (Z) . Also, let λ sym j f (n) be the n-th normalized Dirichlet coefficient of L (s , sym j f) . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients λ sym j f (n) over two sparse sequences of positive integers, which improves previous results. [ABSTRACT FROM AUTHOR]

As a consequence of their work, Bruce C. Berndt, Ronald J. Evans, Larry Joel Goldstein and Michael Razar obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. We give a short proof of it and also express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums. [ABSTRACT FROM AUTHOR]

Let χ denote a primitive, non-quadratic Dirichlet character with conductor q , and let L (s , χ) denote its associated Dirichlet L -function. We show that | L (1 , χ) | ≥ 1 / (9.12255 log ⁡ (q / π)) for sufficiently large q , and that | L (1 , χ) | ≥ 1 / (9.69030 log ⁡ (q / π)) for all q ≥ 2 , improving some results of Louboutin. The improvements come from an averaging argument that simplifies Louboutin's approach, and the construction of certain trigonometric polynomials using simulated annealing, a technique often used in applied mathematics and computer science. We highlight the benefits of this approach to a pure mathematical problem. [ABSTRACT FROM AUTHOR]

Let q ≥ 2 be an integer and let s q (n) be the sum-of-digitsfunction of n in base q. The function s q (n) has been studied in many directions and many properties have been obtained on the distribution of s q (n) and s q (P (n)) , where P is a suitable polynomial. In this paper we derive the generatingfunctions of s p (n d mod p k) for d ≥ 2 and prime p ≥ 2 by using the properties of Dirichlet character, and study the correlation properties of the sequences ((- 1) n 2 mod p k ) n < p k . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism. [ABSTRACT FROM AUTHOR]

In this article, we prove an asymptotic formula for the fourth power mean of a general s -dimensional hyper-Kloosterman sum. We find the number of solutions of certain congruence equations mod p which play an integral part to prove our main result. We use estimates for character sums and analytic methods to prove our theorem. [ABSTRACT FROM AUTHOR]

In [4] , B. C. Berndt and A. Zaharescu introduced the twisted divisor sums associated with the Dirichlet character while studying the Ramanujan's type identity involving finite trigonometric sums and doubly infinite series of Bessel functions. Later, in a follow-up paper [20] , S. Kim extended the definition of the twisted divisor sums to twisted sums of divisor functions. In this paper, we derive identities associated with the aforementioned weighted divisor functions and the modified K -Bessel function in light of recent results obtained by the first author and B. Maji [1]. Moreover, we provide a new expression for L (1 , χ) from which we establish the positivity of L (1 , χ) for any real primitive character χ. In addition, we deduce Cohen-type identities and then exhibit the Voronoï-type summation formulas for them. [ABSTRACT FROM AUTHOR]