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In this paper, we present a quantitative result for the number of sign changes for the sequences {a(n j )} n≥1 , j = 2, 3, 4 of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group SL 2 (Z). We also prove a similar kind of quantitative result for the number of sign changes of the q-exponents c(p) (p vary over primes) of certain generalized modular functions for the congruence subgroup Γ 0 (N ), where N is square-free.

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Abstract. We study the signs of the Fourier coefficients of a newform. Let f be a normalized newform of weight k for Γ 0 (N ). Let a f (n) be the nth Fourier coefficient of f . For any fixed positive integer m, we study the distribution of the signs of {a f (p m )} p , where p runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power Lfunctions.

In this paper, we show that any additive complex valued function over non-zero Gaussian integers which vanishes on the shifted Gaussian primes is necessarily identically zero.