Distribution function of values of random Euler products at 1
Résumé
In the aim of attacking the first conjecture of Montgomery-Vaughan on the distribution of values of the automorphic symmetric power $L$-functions at 1, we study the distribution function of the truncated random Euler products $$L(1,{\rm sym}^mg^\natural(\omega), y) := \prod_{p\le y} \det\big(I-p^{-1}{\rm sym}^m g_p^\natural(\omega)\big)^{-1},$$ where $g^\natural(\omega):=\{g^\natural_p(\omega)\}_p$ is a sequence of independent random variables, with values in the set of conjugacy classes of $SU(2)$ endowed with the Sato-Tate measure.