Result on the Mobius Function over Shifted Primes

2022/06/26 by Carella, N. A.
#FOS: Mathematics #General Mathematics (math.GM) #Primary 11N37 #Secondary 11L20

paper · doi:10.48550/arxiv.2206.12956

Abstract

This article provides new asymptotic results for the summatory Mobius function ∑_{p ≤ x} μ(p+a) =O (x(log x)^-c ) and the summatory Liouville function ∑_{p ≤ x} λ(p+a) =O (x(log x)^-c ) over the shifted primes, where a≠0 is a fixed parameter, and c>1 is an arbitrary constant. These results improve the current estimates ∑_{p ≤ x} μ(p+a)=(1-δ)π(x), and ∑_{p ≤ x} λ(p+a)=(1-δ)π(x) for δ>0, respectively. Furthermore, a conditional proof for the autocorrelation function ∑_{p ≤ x} μ(p+a)μ(p+b) =O (x(log x)^-c ), and an unconditional proof for the autocorrelation function ∑_{p ≤ x} λ(p+a)λ(p+b) =O (x(log x)^-c ) over the shifted primes, where a≠ b, are also included.

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