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Mean values of $$L$$-functions and symmetry. (English) Zbl 1035.11038
N. 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.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11Y35 Analytic computations
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