Abstract
We investigate the average behavior of the nth normalized Fourier coefficients of the jth (j ≽ 2 be any fixed integer) symmetric power L-function (i.e., L(s,symjf)), attached to a primitive holomorphic cusp form f of weight k for the full modular group \(SL(2,\mathbb{Z})\) over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum
where x is sufficiently large, and
When j = 2, the error term which we obtain improves the earlier known result.
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The authors are also grateful to the anonymous referee for some fruitful comments.
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The first author, Anubhav Sharma, wishes to express his gratitude to the University of Hyderabad for its financial support and IoE’s performance based publication incentive towards his Ph.D. Program.
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Sharma, A., Sankaranarayanan, A. On the average behavior of the Fourier coefficients of jth symmetric power L-function over certain sequences of positive integers. Czech Math J 73, 885–901 (2023). https://doi.org/10.21136/CMJ.2023.0348-22
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DOI: https://doi.org/10.21136/CMJ.2023.0348-22
Keywords
- nonprincipal Dirichlet character
- Holder’s inequality
- jth symmetric power L-function
- holomorphic cusp form