ON THE DISTRIBUTION OF PRIME AND NONPRIME RESIDUES MOD-P

Let P be an odd prime, denote by p(n) (q(n)) the n(th) prime not equal P with (Pn/P) = 1(= -1), d(n) = q(n) = p(n). We discuss the question whether d(n) changes sign infinitely often or not. Without using Turan 's power sum method the following theorem is proved. Suppose that the L-function L(s, chi), defined by the real primitive character mod P, h as no real root sigma with 1/2 < sigma < 1. Then the numbers d(n) chan ge sign infinitely often. The hypothesis is known to be true for all P with 2 < P less than or equal to 227 (J. B. Rosser. J. of Research of the Nat. Bureau of Standards 45, 505-514 (1950)).