1 6 N ov 2 00 6 Values of the Euler phi function not divisible by a prescribed odd prime Pieter Moree
نویسنده
چکیده
Let φ denote Euler’s phi function. For a fixed odd prime q we give an asymptotic series expansion in the sense of Poincaré for the number Eq(x) of n ≤ x such that q ∤ φ(n). Thereby we improve on a recent theorem by 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 Eq(x) is asymptotically superior using recent results of Y. Ihara on the Euler-Kronecker constant of a number field.
منابع مشابه
Values of the Euler Φ-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
Let φ denote Euler’s phi function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n 6 x such that q ∤ φ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime k-tuples and a conjecture o...
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