Relations among Some Conjectures on the M\"obius Function and the Riemann Zeta-Function

We discuss the multiplicity of the non-trivial zeros of the Riemann zeta-function and the summatory function $M(x)$ of the M\"obius function. The purpose of this paper is to consider two open problems under some conjectures. One is that whether all zeros of the Riemann zeta-function are simple or not. The other problem is that whether $M(x) \ll x^{1/2}$ holds or not. First, we consider the former problem. It is known that the assertion $M(x) = o(x^{1/2}\log{x})$ is a sufficient condition for the proof of the simplicity of zeros. However, proving this assertion is presently difficult.%at present. Therefore, we consider another sufficient condition for the simplicity of zeros that is weaker than the above assertion in terms of the Riesz mean $M_{\tau}(x) = {\Gamma(1+\tau)}^{-1}\sum_{n \leq x}\mu(n)(1 - \frac{n}{x})^{\tau}$. We conclude that the assertion $M_{\tau}(x) = o(x^{1/2}\log{x})$ for a non-negative fixed $\tau$ is a sufficient condition for the simplicity of zeros. Also, we obtain an explicit formula for $M_{\tau}(x)$. By observing the formula, we 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 zeros. Next, we consider the latter problem. Many mathematicians believe that the estimate $M(x) \ll x^{1/2}$ fails, but this is not yet disproved. In this paper we study the mean values $\int_{1}^{x}\frac{M(u)}{u^{\kappa}}du$ for any real $\kappa$ under the weak Mertens Hypothesis $\int_{1}^{x}( M(u)/u)^2du \ll \log{x}$. We obtain the upper bound of $\int_{1}^{x}\frac{M(u)}{u^{\kappa}}du$ under the weak Mertens Hypothesis. We also have $\Omega$-result of this integral unconditionally, and so we find that the upper bound which is obtained in this paper of this integral is the best possible estimation.