The mean value of symmetric square L-functions
Copyright policy )The mean value of symmetric square L-functions
This paper studies the first moment of symmetric-square $L$-functions at the critical point in the weight aspect. Asymptotics with the best known error term $O(k^{-1/2})$ were obtained independently by Fomenko in 2005 and by Sun in 2013. We prove that there is an extra main term of size $k^{-1/2}$ in the asymptotic formula and show that the remainder term decays exponentially in $k$. The twisted first moment was evaluated asymptotically by Ng Ming Ho with the error bounded by $lk^{-1/2+ε}$. We improve the error bound to $l^{5/6+ε}k^{-1/2+ε}$ unconditionally and to $l^{1/2+ε}k^{-1/2}$ under the Lindelöf hypothesis for quadratic Dirichlet $L$-functions.
30 pages; Remark on page 4 and new reference added
- University of Turku Finland
- National Research University Higher School of Economics Russian Federation
symmetric square \(L\)-functions, Mathematics - Number Theory, ta111, Liouville-Green method, Liouville–Green method, 11F12, 33C05, 34E20, Classical hypergeometric functions, \({}_2F_1\), Mathematics - Classical Analysis and ODEs, 34E05, Langlands \(L\)-functions; one variable Dirichlet series and functional equations, weight aspect, Gauss hypergeometric function, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Singular perturbations, turning point theory, WKB methods for ordinary differential equations, WKB approximation, Number Theory (math.NT), Asymptotic expansions of solutions to ordinary differential equations, symmetric square $L$-functions
symmetric square \(L\)-functions, Mathematics - Number Theory, ta111, Liouville-Green method, Liouville–Green method, 11F12, 33C05, 34E20, Classical hypergeometric functions, \({}_2F_1\), Mathematics - Classical Analysis and ODEs, 34E05, Langlands \(L\)-functions; one variable Dirichlet series and functional equations, weight aspect, Gauss hypergeometric function, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Singular perturbations, turning point theory, WKB methods for ordinary differential equations, WKB approximation, Number Theory (math.NT), Asymptotic expansions of solutions to ordinary differential equations, symmetric square $L$-functions
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