Values of the Euler phi function not divisible by a given odd prime
نویسندگان
چکیده
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...
متن کامل1 6 N ov 2 00 6 Values of the Euler phi function not divisible by a prescribed odd prime Pieter Moree
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 asymptotical...
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• An arithmetic function takes positive integers as inputs and produces real or complex numbers as outputs. • If f is an arithmetic function, the divisor sum Df(n) is the sum of the values of f at the positive divisors of n. • τ (n) is the number of positive divisors of n; σ(n) is the sum of the positive divisors of n. • The Möbius function μ(n) is 1 if n = 1 and 0 if n has a repeated prime fac...
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ure of the nonmonotonicity of f. In particular, F is identically zero if and only if f is strictly increasing . Here we shall take f to be (p, Euler's function, and study the associated function F 4„ which we henceforth call F. We shall show that F(n)/n is asymptotically representable as a function of T(n)/n . Then we shall prove that F(n)/n has a distribution function. We shall study max,,, F(...
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ژورنال
عنوان ژورنال: Arkiv för Matematik
سال: 2006
ISSN: 0004-2080
DOI: 10.1007/s11512-005-0001-6