Holder continuity of an alternating Erdos series on prime K-tuples

TL;DR

This paper investigates the convergence and regularity properties of an alternating Erdős series related to prime K-tuples, introducing a new integral representation and analyzing its Holder continuity under the Riemann hypothesis.

Contribution

It presents a novel integral representation of the series, analyzes its convergence using Young's criterion, and establishes Holder continuity assuming the Riemann hypothesis.

Findings

Series converges conditionally under strong Hardy-Littlewood conjecture

Integral representation as Riemann-Stieltjes integral is established

Convergence is proven for λ > 3/2 with sharp error bounds

Abstract

This open problem, first posed by Erd{\omicron}s, was further explored by Terence Tao. Tao work shows that the series can converge conditionally, but only under a sufficiently strong form of the Hardy-Littlewood conjecture for k-primary pairs. Based on this, we offer a new method leading to a representation of the series as a Riemann-Stieltjes integral or a tightly coupled prime counting function. We rigorously analyze this integral by decomposing it into principal and error terms, applying integration by parts in the Stieltjes sense, and defining the error terms. Assuming the Riemann hypothesis, we investigate the H{\omicron}lder continuation of {\psi}(x) in the asymptotic form {\psi}(x) = x+O(x 1/2), and introduce a test function g(x) = e^( i{\pi}x) e^( -{\lambda}x) , which is smooth and Lipschitz. Applying Young's criterion, we show that the integral converges. Moreover , we prove that the integral converges perfectly for {\lambda} > 3 2 , based on sharp bounds on the error terms. Our results are supported by fractional Sobolev integrations and justify the use of Young's inequality under generalized Holder conditions.