Values of the Euler phi function not divisible by a prescribed odd prime

Details Values of the Euler phi function not divisible by a prescribed odd prime Values of the Euler phi function not divisible by a prescribed odd prime

2006-11-16

arxiv.org/abs/math/0611509v1

Mathematics

Number Theory

20 pages, 2 tables (each of them page filling)

Scientific paper

Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number E_q(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K. Spearman and K.S. Williams [Ark. Mat. 44 (2006), 166-181]. Furthermore we resolve, under the Generalized Riemann Hypothesis, which of two approximations to E_q(x) is asymptotically superior using recent results of Y. Ihara on the Euler-Kronecker constant of a number field.

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