Functiones et Approximatio Commentarii Mathematici

On exponential sums with Hecke series at central points

Aleksandar Ivić

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Abstract

Upper bound estimates for the exponential sum $$\sum_{K<\kappa_j\le K^\prime<2K} \alpha_j H_j^3(\tfrac{1}{2}) \cos(\kappa_j\log(\frac{4eT}{\kappa_j}))\qquad(T^\varepsilon \le K \le T^{1/2-\varepsilon})$$ are considered, where $\alpha_j = |\rho_j(1)|^2(\cosh\pi\kappa_j)^{-1}$, and $\rho_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue $\lambda_j = \kappa_j^2 + \tfrac{1}{4}$ to which the Hecke series $H_j(s)$ is attached. The problem is transformed to the estimation of a classical exponential sum involving the binary additive divisor problem. The analogous exponential sums with $H_j(\tfrac{1}{2})$ or $H_j^2(\tfrac{1}{2})$ replacing ${H_j^3(\tfrac{1}{2})}$ are also considered. The above sum is conjectured to be $\ll_\varepsilon K^{3/2+\varepsilon}$, which is proved to be true in the mean square sense.

Article information

Source
Funct. Approx. Comment. Math., Volume 37, Number 2 (2007), 233-261.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229619651

Digital Object Identifier
doi:10.7169/facm/1229619651

Mathematical Reviews number (MathSciNet)
MR2363824

Zentralblatt MATH identifier
1223.11067

Subjects
Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M06: $\zeta (s)$ and $L(s, \chi)$

Keywords
Hecke series Riemann zeta-function hypergeometric function exponential sums

Citation

Ivić, Aleksandar. On exponential sums with Hecke series at central points. Funct. Approx. Comment. Math. 37 (2007), no. 2, 233--261. doi:10.7169/facm/1229619651. https://projecteuclid.org/euclid.facm/1229619651


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