Partial sums of the Möbius function

Evaluation of some integrals of sums involving the Möbius function

Integrals of sums involving the Möbius function appear in a variety of problems. In this paper, a divergent integral related to several important properties of the Riemann zeta function is evaluated computationally. The order of magnitude of this integral appears to be compatible with the Riemann hypothesis, and furthermore the value of the multiplicative constant involved seems to be the small...

Partial Sums of the Möbius Function

(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...

Zeros of Partial Sums of the Riemann Zeta-function

We write ρX = βX + iγX for a typical zero of FX(s). The number of these up to height T we denote by NX(T ), and the number of these with βX ≥ σ by NX(σ, T ). We follow the convention that if T is the ordinate of a zero, then NX(T ), say, is defined as limǫ→0+ NX(T + ǫ). There are two natural ways to pose questions about NX(T ), NX(σ, T ), and the distribution of the zeros generally. We can fix ...

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...

Computing the Summation of the Möbius Function