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Recently, the authors gave asymptotic formulas for L(s, χ), which associated with a primitive Dirichlet character χ, in terms of the generalized Bernoulli numbers. In this paper, based on the aforementioned asymptotic formulas we describe a method for calculating Dirichlet L-functions that can be used to validate the generalized Riemann hypothesis up to a height T > 0. Our method is a refinement of the one presented by Davies and Haselgrove. [ABSTRACT FROM AUTHOR]

Let be a Dirichlet character (mod ) with conductor . In a quite recent paper Zhao and Cao deduced the identity , which reduces to Menon's identity if is the principal character (mod ). We generalize the above identity by considering even functions (mod ), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums. [ABSTRACT FROM AUTHOR]

Let be the Euler's totient function and be the number of divisors of . Menon's beautiful identity states that Here we extend this identity to Dirichlet characters mod . [ABSTRACT FROM AUTHOR]

Structural theorems for spaces of modular forms with respect to congruence subgroups are proved. The Dedekind η-function plays an important role in our study. [ABSTRACT FROM AUTHOR]

We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums. [ABSTRACT FROM AUTHOR]

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a 'finite coloring'). Is it true that, in this case, there are x, y ∈ ℤ such that x + y, xy, and x have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group ℤ/ pℤ, where p is a prime, and obtain an even stronger density result. [ABSTRACT FROM AUTHOR]

By introducing the Rademacher-Menchov device, we prove 'maximal' analogs of principal bounds of character sums. This allows us to present the Burgess method so as to separate the main idea of this method from the technical issues. [ABSTRACT FROM AUTHOR]

Abstract: The main purpose of this paper is using the estimate for character sums and the analytic method to study the mean value of the Dirichlet L-functions with the weight of character sums, and give an interesting mean value theorem. [Copyright &y& Elsevier]

Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2,4 and 8. [ABSTRACT FROM AUTHOR]

In this paper we prove that, with at most $$ O{\left( {N^{{\frac{5} {{12}} + \in }} } \right)} $$ exceptions, all positive odd integers n ≤ N with n ≡ 0 or 1(mod 3) can be written as a sum of a prime and two squares of primes. [ABSTRACT FROM AUTHOR]

In the famous work of Dirichlet on class number formula, L(s, χ) at s = 1 has been expressed as a finite sum, where L(s, χ) is the Dirichlet L-series of a real Dirichlet character. We show that this expression with obvious modification is valid for the general primitive Dirichlet character χ. [ABSTRACT FROM AUTHOR]