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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]

Menon's identity is a classical identity involving gcd sums and the Euler totient function ϕ . In a recent paper, Zhao and Cao (Int. J. Number Theory13(9) (2017) 2373–2379) derived the Menon-type identity ∑ k = 1 n (k - 1 , n) χ (k) = ϕ (n) τ (n d) , where χ is a Dirichlet character mod n with conductor d. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao–Cao identity can be improved if one uses the method we employ here. [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]