Mean values of \(L\)-functions and symmetry.

*(English)*Zbl 1035.11038N. M. Katz and P. Sarnak [Random matrices, Frobenius eigenvalues, and monodromy. AMS Colloquium Publications, 45, AMS, Providence, RI (1999; Zbl 0958.11004), Bull. Am. Math. Soc., New Ser. 36, 1–26 (1999; Zbl 0921.11047)] have introduced the idea of studying zero distributions within families of \(L\)-functions and have conjectured that these coincide with the eigenvalue distributions of the classical compact group. The symmetry of a family of \(L\)-functions determines the classical group (unitary, symplectic or orthogonal). In this paper the authors give evidence that the symmetry type of a family of \(L\)-functions governs the behavior of mean values of \(L\)-functions. In particular, they conjecture that in general, as \(\mathcal Q\to \infty\),
\[
\frac1{\mathcal Q^*} \sum _{\substack{ f\in\mathcal F\\ c(f)\leq\mathcal Q}} \biggl(L_f \Bigl(\frac12\Bigr)\biggr)^k \sim g_k\frac{a_k}{\Gamma(1+B(k))}(\log \mathcal Q^A)^{B(k)},
\]
where \(L\)-functions are normalized to have a functional equation \(s\leftrightarrow1-s\); \(V(z)=| z| ^2\) for unitary symmetry and \(V(z)=z\) for the orthogonal or symplectic case; the family \(\mathcal F\) is considered to be partially ordered by the conductor, \(c(f)\), of each \(L\)-function; the sum is over the \(\mathcal Q^*\) elements with \(c(f)\leq\mathcal Q\); \(A\) and \(a_k\) depend on the family; and the parameters \(g_k\) and \(B(k)\) depend only on the symmetry type. J. Keating and N. Snaight have conjectured formulas for \(g_k\). For unitary symmetry type the conjecture is that
\[
g_{k,U}=\Gamma(1+B_U(k))\lim_{N\to\infty} N^{-B_U(k)}\prod_{j=1}^N\frac{\Gamma(j)\Gamma(j+2k)}{\Gamma(j+k)^2},
\]
where \(B_U(k)=k^2\). An example of a family \(\mathcal{F}\) with this symmetry type are Dirichlet \(L\)-functions \(L(s, \chi)\) for \(\chi\) a Dirichlet character \(\operatorname{mod} q\). The conjecture in the orthogonal case is
\[
g_{k,O}=\Gamma(1+B_O(k)) \lim_{N\to\infty} N^{-B_O(k)} 2^{2Nk-1} \prod_{j=1}^N\frac{\Gamma(N+j-1) \Gamma(j-1/2+k)} {\Gamma(N+j-1+k) \Gamma(j-1/2)},
\]
where \(B_O(k)=(1/2)k(k+1)\). An example of \(\mathcal F\) for this case is the \(L\)-function \(L_f(s)\) associated with cusp forms \(f\in S_m(\Gamma_0(1))\) of weight \(m\) for the full modular group. And the conjecture in the symplectic case is
\[
g_{k,Sp}= \Gamma(1+B_{Sp}(k)) \lim_{N\to\infty} N^{-B_{Sp}(k)}2^{2Nk} \prod_{j=1}^N \frac{\Gamma(N+j+1) \Gamma(j+1/2+k)} {\Gamma(N+j+1+k)\Gamma(j+1/2)},
\]
with \(B_{Sp}(k)=(1/2)k(k+1)\). An example of \(\mathcal F\) for the symplectic case are Dirichlet \(L\)-functions \(L(s, \chi_d)\), where \(\chi_d\) is a quadratic Dirichlet character \(\operatorname {mod} | d| \).

Altogether the paper presents nine examples of families of \(L\)-functions. The authors also shortly discuss the mollified mean squares for \(L\)-functions in different symmetry type families.

The remaining part of the work is devoted to develop the properties of the constants \(g_k\) (\(k\in\mathbb C\)). The authors prove that \(g_k/\Gamma(1+B(k))\) is a nonvanishing meromorphic function of order 2, give its pole locations, and express it in terms of the Barnes-Vignéras double \(\Gamma\)-function. They show that \(g_k\) is an integer for positive integer \(k\), establish asymptotic formulas for \(g_k\) as \(k\to\infty\), and consider its prime factorization.

Altogether the paper presents nine examples of families of \(L\)-functions. The authors also shortly discuss the mollified mean squares for \(L\)-functions in different symmetry type families.

The remaining part of the work is devoted to develop the properties of the constants \(g_k\) (\(k\in\mathbb C\)). The authors prove that \(g_k/\Gamma(1+B(k))\) is a nonvanishing meromorphic function of order 2, give its pole locations, and express it in terms of the Barnes-Vignéras double \(\Gamma\)-function. They show that \(g_k\) is an integer for positive integer \(k\), establish asymptotic formulas for \(g_k\) as \(k\to\infty\), and consider its prime factorization.

Reviewer: Ramūnas Garunkštis (Vilnius)