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Estimates for Sums of Coefficients of Dirichlet Series with Functional Equation

Authors:	Murty V Kumar

Institution:	(1) Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, ON, M5S 3G3, Canada

Abstract:	Suppose we have a Dirichlet series L(s) = _{n = 1} a _n n ^–s such that it, and its twists by Dirichlet characters have analytic continuation and a functional equation of a specific kind. Suppose also that the root numbers of the twists are equidistributed on the unit circle. The purpose of this note is to get an estimate for the quantity $\sum\limits_{\mathop {n \leqslant x}\limits_{n \equiv \ell (\bmod p)} } {a_n - \frac{1}{{\phi (p)}}} \sum\limits_{\mathop {n \leqslant x}\limits_{(n,p){\text{ = }}1} } {a_n }$ for a prime modulus p.We use a modification of the method of Chandrasekharan and Narasimhan and we use in an essential way a Rankin-Selberg type estimate for the average of \|a _n\|².

Keywords:	Dirichlet series functional equation Fourier coefficients of modular forms
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