Let f (n) be the n-th normalized Fourier coecient of the Fourier series associated with a holomorphic cusp form f for the full modular group of even weight k and let Af (x) = X n?x f (n). During the ELAZ 2014 conference in Hildesheim, Germany, K.-L. Kong (University of Hong Kong) presented his result, proved in his Master thesis, that Z X 2 2(t)(?t)dt = C(?)X7/4 + O" ? X7/4 , for some explicit > 0, C(?), where ? > 0 is fixed and (x) is the error term in the Dirichlet divisor problem. A problem posed by Professor Ivi′c at this conference was to obtain a formula analogous to the above formula for the sum Af (x) and especially to discuss the sign of C(?) in the new setting. In this paper, we will solve Ivi′c’s problem and prove that for any " > 0, we have Z X 2 A2 f (t)Af (?t)dt = Cf (?)X7/4 + O?," ? X 41 24 +" , for some constant Cf (?) depending on only f, ? and defined by Cf (?) = ?1/4 28?3 X (i0,i1)2{0,1}2 X +1 n,m,l=1 pn+(1)i0pm+(1)i1 p?l=0 f (n)f (m)f (l) (nml)3/4 , where ? > 0 is a fixed constant. Our result is new and throws light on the behavior of the classical function Af (x).
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