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We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty [24] to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an L -function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL ( 2 ) holomorphic cusp forms, GL ( 2 ) Maass forms, and GL ( 3 ) Maass forms. Building on [13,14] , we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL ( 2 ) holomorphic cusp forms and Maass forms. [ABSTRACT FROM AUTHOR]

Let f and g be weight k holomorphic cusp forms and let S f ( n ) and S g ( n ) denote the sums of their first n Fourier coefficients. Hafner and Ivić [9] proved asymptotics for ∑ n ≤ X | S f ( n ) | 2 and proved that the Classical Conjecture, that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ , holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series D ( s , S f × S g ) = ∑ S f ( n ) S g ( n ) ‾ × n − ( s + k − 1 ) and D ( s , S f × S g ‾ ) = ∑ n S f ( n ) S g ( n ) n − ( s + k − 1 ) . We then prove asymptotics for the smoothed second moment sums ∑ S f ( n ) S g ( n ) ‾ e − n / X , giving a smoothed generalization of [9] . We also attain asymptotics for analogous sums of normalized Fourier coefficients. Our methodology extends to a wide variety of weights and levels, and comparison with [4] indicates very general cancellation between the Rankin–Selberg L -function L ( s , f × g ) and convolution sums of the coefficients of f and g . In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on | S f ( n ) | 2 is true on short intervals, and to prove sign change results on { S f ( n ) } n ∈ N . [ABSTRACT FROM AUTHOR]

Let f be a weight k holomorphic cusp form of level one, and let S f ( n ) denote the sum of the first n Fourier coefficients of f . In analogy with Dirichlet's divisor problem, it is conjectured that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ . Understanding and bounding S f ( X ) has been a very active area of research. The current best bound for individual S f ( X ) is S f ( X ) ≪ X k − 1 2 + 1 3 ( log ⁡ X ) − 0.1185 from Wu [17] . Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for S f ( X ) holds on average over intervals of length X . Jutila [10] improved this result to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 3 4 + ϵ . Building on the results and analytic information about ∑ | S f ( n ) | 2 n − ( s + k − 1 ) from our recent work [9] , we further improve these results to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 2 3 ( log ⁡ X ) 1 6 . [ABSTRACT FROM AUTHOR]