Relations among Some Conjectures on the Möbius Function and the Riemann Zeta-function

Abstract. We discuss the multiplicity of the non-trivial zeros of the Riemann zetafunction 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 is that whether all zeros of the Riemann zeta-function are simple or not. The other problem is that whether M(x) ≪ x holds or not. First, we consider the former problem. It is known that the assertion M(x) = o(x log x) is a sufficient condition for the proof of the simplicity of zeros. However, proving this assertion is 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 Mτ (x) = Γ(1 + τ ) −1 ∑ n≤x μ(n)(1 − n x ) . We have that the assertion Mτ (x) = o(x 1/2 log x) for a non-negative fixed τ is a sufficient condition for the simplicity of zeros. Also, we obtain an explicit formula for Mτ (x). By observing the formula, we propose a conjecture, in which τ 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) ≪ x fails, but this is not yet disproved. In this paper we study the mean values ∫ x 1 M(u) u du for any real κ under the weak Mertens Hypothesis ∫ x 1 (M(u)/u)du ≪ log x. We obtain the upper bound of ∫ x 1 M(u) u du under the weak Mertens Hypothesis. We also have Ω-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.