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Let λf(n) be the nth normalized Fourier coefficient of a holomorphic cusp form f for the full modular group. In this paper, we established asymptotic formulae for high power sums of Fourier coefficients of cusp forms and further improved previous results. Moreover, as an application, we studied the signs of the sequences {λf(n)} and {λf(n)λg(n)} in short intervals, and presented some quantitative results for the number of sign changes for n ≤ x. [ABSTRACT FROM AUTHOR]

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change. [ABSTRACT FROM AUTHOR]

Let f be a nonzero cusp form of even integral weight k ⩾ 2 on the Hecke congruence subgroup Γ 0 ( N ) with N squarefree. Suppose that the normalized Fourier coefficients λ f ( n ) of f are real. We prove that the first sign change of λ f ( n ) occurs in the range n ≪ ( k N ) 2 + ε . This improves upon the earlier result of Choie and Kohnen [2] . [ABSTRACT FROM AUTHOR]

Structure theorems for spaces of cusp forms with quadratic characters are presented. It is proved that such spaces of levels N ≠ 3, 17, 19 admit exact cutting if and only if the cutting function is a multiplicative η-product. The cases of the levels N = 3, 17, 19 are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we study the exponential sums involving Fourier coefficients of cusp forms for SL ( 2 , Z ) over primes. We are able to establish some new results. In addition, similar results have been obtained for the coefficients of symmetric-square lifts. [ABSTRACT FROM AUTHOR]

Abstract: Let be the space of all normalized holomorphic Hecke-eigen cusp forms of even integral weight k for the full modular group , and denote by the corresponding symmetric square L-function associated to . In this paper, the lower bound of the higher moment of is established, i.e., for any even number , holds for , where . [Copyright &y& Elsevier]

Abstract: Recently Blomer showed that if denote the normalized Fourier coefficients of any holomorphic cusp form f with integral weight, then holds uniformly in . By an elementary argument we show that independent of q, where could be the normalized Fourier coefficients of any reasonable cusp forms, including Maass cusp forms, holomorphic cusp forms with half-integral or integral weights. [Copyright &y& Elsevier]

Abstract: Lehmer''s conjecture asserts that where τ is the Ramanujan τ-function. This is equivalent to the assertion that for any n. A related problem is to find the distribution of primes p for which . These are open problems. We show that the variant of estimating the number of integers n for which n and do not have a non-trivial common factor is more amenable to study. In particular, we show that the number of such is . We prove a similar result for more general cusp forms. This may be seen as a modular analogue of an old result of Erdős on the Euler ϕ function. [Copyright &y& Elsevier]