An exact formula for a Lambert series associated to a cusp form and the Möbius function

In 1981, Zagier conjectured that the constant term of the automorphic form y 12 | Δ ( z ) | 2 , that is, a 0 ( y ) : = y 12 ∑ n = 1 ∞ τ 2 ( n ) exp ( - 4 π n y ) , where τ ( n ) is the n th Fourier coefficient of the Ramanujan cusp form Δ ( z ) , has an asymptotic expansion when y → 0 + and it can b...

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Veröffentlicht in:The Ramanujan journal 2022-02, Vol.57 (2), p.769-784
Hauptverfasser: Juyal, Abhishek, Maji, Bibekananda, Sathyanarayana, Sumukha
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Sathyanarayana, Sumukha
description In 1981, Zagier conjectured that the constant term of the automorphic form y 12 | Δ ( z ) | 2 , that is, a 0 ( y ) : = y 12 ∑ n = 1 ∞ τ 2 ( n ) exp ( - 4 π n y ) , where τ ( n ) is the n th Fourier coefficient of the Ramanujan cusp form Δ ( z ) , has an asymptotic expansion when y → 0 + and it can be expressed in terms of the non-trivial zeros of the Riemann zeta function ζ ( s ) . This conjecture was settled by Hafner and Stopple, and later Chakraborty, Kanemitsu, and the second author have extended this result for any Hecke eigenform over the full modular group. In this paper, we investigate a Lambert series associated to the Fourier coefficients of a cusp form and the Möbius function μ ( n ) . We present an exact formula for the Lambert series and interestingly the main term is in terms of the non-trivial zeros of ζ ( s ) , and the error term is expressed as an infinite series involving the generalized hypergeometric series 2 F 1 ( a , b ; c ; z ) . As an application of this exact form, we also establish an asymptotic expansion of the Lambert series.
doi_str_mv 10.1007/s11139-020-00375-7
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subjects Asymptotic series
Combinatorics
Cusps
Field Theory and Polynomials
Fourier Analysis
Fourier series
Functions of a Complex Variable
Infinite series
Mathematics
Mathematics and Statistics
Number Theory
title An exact formula for a Lambert series associated to a cusp form and the Möbius function
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