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On Riesz means of the coefficients of the Rankin-Selberg series. (English) Zbl 0958.11065

Let \(a(n)\) denote the Fourier coefficients of a holomorphic cusp form of weight \(\kappa \) for the full modular group. Then \(Z(s)=\zeta (2s) \sum_{n=1}^{\infty }|a(n)|^2n^{1-\kappa -s}=\sum_{n=1}^{\infty} c_nn^{-s}\) is the corresponding Rankin-Selberg series. More than sixty years ago, Rankin proved an asymptotic formula with a linear main term and an error term \(O(x^{3/5})\) for the sum function of \(c_n\), and this has resisted all attempts of improvement. The authors attach the weight \((x-n)^{\rho }\) for some fixed \(\rho \geq 0\) to the terms of the sum function to make it smoother and more amenable to an analytic treatment. Then, for \(\rho > 1/2\), such a Riesz mean can be expressed by a convergent series of the Voronoi type. A truncated version of this formula is worked out, in particular for \(\rho =1\). The authors show that if the error term for the Riesz mean with \(\rho =1\) is \(\ll x^{\alpha }\), then the error term in the original sum (without a weight) is \(\ll x^{\alpha /2}\). The authors show further that \(\alpha =6/5\) is admissible, which implies Rankin’s classical result again.
Reviewer: M.Jutila (Turku)

MSC:

11N37 Asymptotic results on arithmetic functions
11F30 Fourier coefficients of automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F60 Hecke-Petersson operators, differential operators (several variables)
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