Relations among some conjectures on the Möbius function and the Riemann zeta-function
Volume 191 / 2019
Abstract
We discuss the multiplicity of the non-trivial zeros of the Riemann zeta-function and the summatory function $M(x)$ of the Möbius function. The purpose of this paper is to consider two open problems under some conjectures. One problem is whether all zeros of the Riemann zeta-function are simple or not. The other is whether or not $M(x) \ll x^{1/2}$. Concerning the former problem, it is known that the condition $M(x) = o(x^{1/2}\log{x})$ is a sufficient condition for the simplicity of the zeros. However, to prove this condition is at present difficult. Therefore, we consider another, weaker sufficient condition for the simplicity of the zeros in terms of the Riesz mean $M_{\tau}(x) = {\varGamma(1+\tau)}^{-1}\sum_{n \leq x}\mu(n)(1 - {n}/{x})^{\tau}$. We conclude that $M_{\tau}(x) = o(x^{1/2}\log{x})$ for a fixed non-negative $\tau$ is a sufficient condition for the simplicity of the zeros. Also, we obtain an explicit formula for $M_{\tau}(x)$, which leads us to propose a conjecture, in which $\tau$ is not fixed, but depends on $x$. This conjecture also gives a sufficient condition, which seems easier to approach, for the simplicity of the zeros. Next, we consider the latter problem. Many mathematicians believe that the estimate $ M(x) \ll x^{1/2}$ fails, but this is difficult and not yet disproved. We study the mean values $\int_{1}^{x}({M(u)}/{u^{\kappa}})\,du$ for any real $\kappa$ under the weak Mertens Hypothesis $\int_{1}^{x}( M(u)/u)^2\,du \ll \log{x}$. We obtain an upper bound of $\int_{1}^{x}({M(u)}/{u^{\kappa}})\,du$ under that hypothesis. We also have an $\varOmega$-result for this integral unconditionally, and so we find that the upper bound of this integral which is obtained in this paper is best possible.
