A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION
نویسندگان
چکیده
منابع مشابه
A Remark on Partial Sums Involving the Mobius Function
Let 〈P〉 ⊂ N be a multiplicative subsemigroup of the natural numbers N= {1, 2, 3, . . .} generated by an arbitrary set P of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈P〉:n≤x (μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈P (1− (1/p)) (the case where P is all the primes is a we...
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(1) M(x) ≪ x 1 2. Conversely, the estimate M(x) ≪ x 12+ǫ implies, by partial summation, the convergence of the series ∑∞ n=1 μ(n)n −s = 1/ζ(s) for any σ > 1/2, and therefore RH. Subsequently, E. Landau [5] showed that, assuming RH, (1) is valid with ǫ ≪ log log log x/ log log x, and E.C. Titchmarsh [13] improved this to ǫ ≪ 1/ log log x. H. Maier and H.L. Montgomery [7] obtained a substantial i...
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 2010
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972709000884