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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 [On the asymptotic distribution of the remainder term of the prime number theorem,  Amer. J. Math.  57  (1935), 534–538], Rubinstein and Sarnak [Chebyshev's bias,  Experiment. Math.  3  (1994), 173–197] and of Ng [The summatory function of the Möbius function,  Proc. London Math. Soc.  (3)  89  (2004), 361–389]. 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 Möbius function, weighted sums of the Liouville function, the sum of the Möbius function in an arithmetic progression and the error term in Chebotarev's density theorem.

LIMITING DISTRIBUTIONS OF THE CLASSICAL ERROR TERMS OF PRIME NUMBER THEORY