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On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms

Luca, Florian and Radziwill, Maksym and Shparlinski, Igor E. (2013) On the Typical Size and Cancelations Among the Coefficients of Some Modular Forms. . (Unpublished)

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We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato-Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ≤ n^(11/2) (log n)^((−1/2) + o(1)) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n. In addition we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato--Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally assuming the automorphy of all symmetric powers of the form and seems to be within reach unconditionally using the currently established potential automorphy.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:This paper started during a visit of F. L. to the Department of Computing of Macquarie University in March 2013. F. L. thanks this Institution for the hospitality. During the preparation of this paper, F. L. was partially supported in part by a Marcos Moshinsky fellowship; M. R. was partially supported by NSF Grant DMS-1128155, I. S. was partially supported by ARC Grant DP130100237.
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Marcos Moshinsky fellowshipUNSPECIFIED
Australian Research CouncilDP130100237
Record Number:CaltechAUTHORS:20180614-155813169
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87138
Deposited By: George Porter
Deposited On:14 Jun 2018 23:20
Last Modified:03 Oct 2019 19:52

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