Let the summatory function of the Möbius function be denoted M (x). We deduce in this article conditional results concerning M (x) assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. The main results shown are that the weak Mertens conjecture and the existence of a limiting distribution of e −y/2 M (e y ) are consequences of the aforementioned conjectures. By probabilistic techniques, we present an argument that suggests M (x) grows as large positive and large negative as a constant times ± √ x(log log log x) 5 4 infinitely often, thus providing evidence for an unpublished conjecture of Gonek's.

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ABSTRACT. Let φ : [0, ∞) → R and let y 0 be a non-negative constant. Let (λ n ) n∈N be a nondecreasing sequence of positive numbers which tends to infinity, let (r n ) n∈N be a complex sequence, and c a real number. Assume that φ is square-integrable on [0, y 0 ] and for y ≥ y 0 , φ can be expressed as φ(y) = c + ℜ λn≤X r n e iλny + E(y, X),We prove that, under certain assumptions on the exponents λ n and the coefficients r n , φ(y) is a B 2 -almost periodic function and thus possesses a limiting distribution. Also if {λ n } n∈N is linearly independent over Q, we explicitly calculate the Fourier transform of the limiting distribution measure. Moreover, we prove general versions of the above results for vector-valued functions. Finally, we illustrate some applications of our general theorems by applying them to several classical error terms which occur in prime number theory. Examples include the error term in the prime number theorem for an automorphic L-function, weighted sums of the Möbius function, weighted sums of the Liouville function, the sum of the Möbius function in an arithmetic progression, and the error term in Chebotarev's density theorem.

In a previous paper, the second author proved that the equation [Formula: see text] had no integral solutions for prime p > 211 and (A,B,C) = 1. In the present paper, we explain how to extend this result to smaller exponents, and to the related equation [Formula: see text]

We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the zeta function whose spacing is three times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.