On Euler products and multi-variate Gaussians.
Let \(F(s)=\sum\limits_n a_n n^{-s}\) be a Dirichlet series which converges absolutely for \(\Re s > 1\) and satisfies a certain functional equation. Under assumptions similar to those satisfied by the classical \(L\)-functions, \textit{A. Selberg} [Collected papers, Vol. 2 \S 44 (Berlin: Springer-Verlag) (1991; Zbl 0729.11001)] proved that when \(\dfrac 12 \leq \sigma \leq \dfrac 12 + (\log T)^{-\delta}\) for some \(\delta > 0\), both the real and imaginary parts of \(\kappa (\sigma , t)= (\pi \sum\limits_{p< t} | a_p| ^2 p^{-2\sigma})^{-1/2} \log F(\sigma + it)\) have a normal Gaussian distribution. In the present paper, the author considers the joint distribution in higher dimension of the logarithms of \(J\) such functions \(F_1, F_2, \cdots , F_J\), where each \(F_j\) is, in addition, an Euler product. It is also assumed that each pair of \(\log F_j\) and \(\log F_k\) is independent, that is, \[ \sum\limits_{p\leq x} p^{-1} c_j (p) \overline c_k (p) = \delta_{jk} A_j \log\log X + c_{ij} + O ((\log X)^{-v}) \] for certain positive constants \(A_j\) and \(c_{jk} \in \mathbb C\). The author states in detail the main results and gives an outline of its proof, which goes along the same line as Selberg's original argument.
scientific article; zbMATH DE number 1986152
scientific article