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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 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]

In this article, we prove an asymptotic formula for the fourth power mean of a general 4-dimensional Kloosterman sum. We use a result of P. Deligne, which counts the number of F p -points on the surface (x - 1) (y - 1) (z - 1) (1 - x y z) - u x y z = 0 , u ≠ 0 , and then take an average of the error terms over u to prove the asymptotic formula. We also find the number of solutions of certain congruence equations mod p which are used to prove our main result. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic methods, the solutions of the congruence equation mod p and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 3-dimensional Kloostermann sums mod p, and give a sharp asymptotic formula for it. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 2-dimensional Kloostermann sums mod p, and give an exact computational formula for it. [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]

A limit theorem involving an increasing modulus of the character is obtained for twists with the Dirichlet character of L-functions of elliptic curves. [ABSTRACT FROM AUTHOR]

Let q be a sufficiently large integer and χ be a Dirichlet character modulo q. In this paper, we extend the product Π L(1, χ) with prime q, arising from the Kummer conjecture, to the products of some general Dirichlet series, and give some meaningful estimates for them. [ABSTRACT FROM AUTHOR]

For integers m, n, q, k, with q, k≧1 and Dirichlet characters $\chi, \chi' \text {\rm \;(mod}\,q)$ we define a generalized Kloosterman sum with a Dirichlet character and a Gauss sum G( a, χ′) as coefficient, where e( z)= e. The aim of this paper is to study the fourth power mean obtaining explicit formulas for M( q). [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]

In the paper, a Voronin-type joint universality theorem for Lerch zeta functions is obtained under weaker assumptions than those in A. Laurinčikas and K. Matsumoto, 'Joint value distribution theorems on Lerch zeta-functions. III,' in Analytic and Probabilistic Methods in Number Theory (TEV, Vilnius, 2007), pp. 87-98. [ABSTRACT FROM AUTHOR]

Enlarged major arcs in the Waring-Goldbach problem are studied by using large sieve estimates for Dirichlet polynomials and estimates for exponential sums over primes. [ABSTRACT FROM AUTHOR]

We obtain lower bounds for the fractional moments of linear combinations of analogs of the Hardy function. In addition, we apply these estimates to the Karatsuba problem of finding a lower bound for the number of zeros of the linear combination of analogs of Hardy functions on the interval (0, T]. [ABSTRACT FROM AUTHOR]

We obtain a limit theorem for L-functions of elliptic curves twisted by Dirichlet characters with increasing modulus. [ABSTRACT FROM AUTHOR]

We prove functional independence and joint functional independence for a set of Hurwitz zeta functions with periodic coefficients and parameters algebraically independent over the field of rational numbers. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean. [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]

Suppose that g(n) is an additive real-valued function, W(N) = 4+min λ(λ² + ∑p

Suppose that $g(n)$ is a real-valued additive function and $\backslash tau(n)$ is the number of divisors of $n$. In this paper, we prove that there exists a constant $C$ such that