Short-interval averages of sums of Fourier coefficients of cusp forms

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• 作者：Thomas A. Hulse^a ; ¹ ; ^{tahulse@colby.edu} ; Chan Ieong Kuan^b ; ^{chan.i.kuan@maine.edu} ; David Lowry-Duda^c ; ² ; ^{djlowry@math.brown.edu} ; Alexander Walker^c ; ^{alexander_walker@brown.edu}
• 关键词：11F30 ; 11F03
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：173
• 期：Complete
• 页码：394-415
• 全文大小：445 K
• 卷排序：173

文摘

Let f be a weight k   holomorphic cusp form of level one, and let Sf(n)Sf(n) denote the sum of the first n Fourier coefficients of f  . In analogy with Dirichlet's divisor problem, it is conjectured that Sf(X)≪Xk−12+14+ϵ. Understanding and bounding Sf(X)Sf(X) has been a very active area of research. The current best bound for individual Sf(X)Sf(X) is Sf(X)≪Xk−12+13(log⁡X)−0.1185 from Wu [17].Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for Sf(X)Sf(X) holds on average over intervals of length X. Jutila [10] improved this result to show that the Classical Conjecture for Sf(X)Sf(X) holds on average over short intervals of length X34+ϵ. Building on the results and analytic information about ∑|Sf(n)|2n−(s+k−1)∑|Sf(n)|2n−(s+k−1) from our recent work [9], we further improve these results to show that the Classical Conjecture for Sf(X)Sf(X) holds on average over short intervals of length X23(log⁡X)16.