Perron's formula and resulting explicit bounds on sums

Abstract

By working with Perron’s formula we prove an explicit bound on ∑n≤x an/ns, where an,s ∈ C. We then prove a second explicit bound on this sum for the special case where s = 0: These bounds apply to specific sums that are involved in the Prime Number Theorem. Moreover, they are particularly useful in cases where a variant of the Riemann von-Mangoldt explicit formula is not unconditionally available. We choose to implement our bounds on M(x) =∑n≤x μ(n) and m(x) =∑n≤x μ(n/)n (with μ(n) denoting the Möbius function). This gives constants C > 0; c > 0 and x0 > 0 for which |M(x)|≤Cxexp(−c√logx) if x > x0 and a similar kind of bound for m(x): We believe that explicit bounds for M(x) and m(x) like these have never before been published.