Oscillations in Mertens Theorems and Other Finite Sums and Products

TL;DR

This paper simplifies the proof of prime product oscillations in Martens Theorem, provides a quantitative error term, and extends oscillation results to finite sums and products involving primes.

Contribution

It offers a simplified proof and quantitative analysis of prime product oscillations, extending results to various finite sums and products involving primes.

Findings

Quantitative expression for the error term in prime product oscillation

Extended oscillation results to finite sums of prime reciprocals

Simplified proof of prime product oscillation in Martens Theorem

Abstract

This note simplifies the proof of a recent result on the oscillation of the prime product in Martens Theorem, and provides a quantitative expression for the error term. In addition, the corresponding oscillation results for the finite sums of the reciprocal of the primes up to a fixed number and other finite sums and products are also given.