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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]

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]

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]