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The Riemann-Siegel integral formula for Dirichlet series associated to cusp forms. (English) Zbl 0881.11047

Nowak, W. G. (ed.) et al., Proceedings of the conference on analytic and elementary number theory: a satellite conference of the European Congress on Mathematics ’96, Vienna, July 18–20, 1996. Dedicated to the honour of the 80th birthday of E. Hlawka. Wien: Universität Wien, Institut für Mathematik, Institut für Mathematik und Statistik, 53-69 (1996).
The Riemann-Siegel integral formula is an integral representation for Riemann’s zeta function implying its functional equation and the Riemann-Siegel formula, a precise form of the approximate functional equation. The author proves an analogue of the Riemann-Siegel integral formula for a cusp form \(L\)-function \(\varphi (s)=\sum_{n=1}^{\infty }a(n)n^{-s}\), where the \(a(n)\) are the Fourier coefficients of a cusp form \(f(z)\) for the full modular group; let us assume for simplicity that these coefficients are real. Let \(\psi (x)=f(ix)\), \(X(s)=(2\pi )^{2s-k}\Gamma (k-s)/\Gamma (s)\), and \(T(s)=(2\pi )^s\Gamma (s)^{-1}\int_0^{\infty }\psi (x)(i+x)^{s-1} dx\). Then \[ \varphi (s)=T(s)+(-1)^{k/2}X(s)\overline{T(k-\overline s)}; \] this is a preliminary version of the Riemann-Siegel integral formula for \(\varphi (s)\).
The main result follows if \(T(s)\) here is written as a loop integral involving the Laplace transform \(F(z)=\int_0^{\infty}e^{2\pi xz}\psi (x) dx\). This expression is \(T(s)=(2\pi i)^{-1}\int_{\Lambda }e^{2\pi iz}F(z)z^{-s} dz\), where \(\Lambda \) is a loop around the positive imaginary axis starting from \(e^{-3\pi i/2}\infty \), running to \(e^{-3\pi i/2}\varepsilon \) (with \(0<\varepsilon <1\)), encircling the origin in the positive direction to \(e^{\pi i/2}\varepsilon\), and finally returning to \(e^{\pi i/2}\infty \). In addition, these results are generalized to the case of congruence subgroups of the modular group.
For the entire collection see [Zbl 0868.00042].
Reviewer: M.Jutila (Turku)

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11M41 Other Dirichlet series and zeta functions
11F11 Holomorphic modular forms of integral weight
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