Limiting distributions of the classical error terms of prime number theory

Abstract

In this article we prove a general theorem which establishes the existence of limiting distributions for a wide class of error terms from prime number theory. As a corollary to our main theorem, we deduce previous results of Wintner (1935), Rubinstein and Sarnak (1994), and of Ng (2004). In addition, we establish limiting distribution results for the error term in the prime number theorem for an automorphic $L$-function, weighted sums of the M\"{o}bius function, weighted sums of the Liouville function, the sum of the M\"{o}bius function in an arithmetic progression, and the error term in Chebotarev's density theorem.