On estimates of the Mertens function

Abstract

Assuming the simplicity of the zeros of the Riemann zeta function ∂(s), Gonek and Hejhal studied the sum J-k(T) := Σ0 < γ≤T |∂'(ρ)|-2k for real number k and conjectured that J-k(T) Lt; T(log T)(k-1)2 for any real k. Assuming Riemann hypothesis and J-1(T) Lt; T, Ng [11] proved that the Mertens function M(x) Lt; √ x(log x)3/2. He also pointed out that with the additional hypothesis of J-12 (T) Lt; T(log T)1/4 one gets M(x) Lt; √ x(log x)5/4. Here we show that M(x) Lt; √ x(log x)a for any real number a ϵ [5/4, 3/2], under similar hypotheses.