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

Nikos Mantzakouras

TL;DR

This work studies the convergence of Erdős’s alternating series by recasting it as a Riemann–Stieltjes integral I = \int_1^\infty g(x) d\psi(x) with g(x) = e^{i\pi x} e^{−\lambda x} and a Chebyshev/prime-counting function \psi. Under the Riemann Hypothesis, the authors prove convergence via Hölder continuity and Young’s inequality, decomposing I into a main term and a controlled error term, and they derive explicit asymptotics for the dominant contribution: I = e^{(iπ − λ)}/(λ − iπ)^2 + O(1/λ^{3/2}). The approach combines Abel summation, fractional Sobolev embeddings, and integration by parts in the Stieltjes sense, offering a new analytic framework for oscillatory prime-series and potential insights into prime-gap behavior and Hardy–Littlewood-type conjectures. If RH holds, the analysis yields rigorous control over the integral, revealing a pole-like dominance as λ → 0^+ and illustrating a novel method to study Erdős-type alternating sums via Hölder-regularity and Stieltjes-interval techniques with potential broader applicability to related series.

Abstract

This open problem, first posed by Erdοs, 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 Hοlder continuation of ψ(x) in the asymptotic form ψ(x) = x+O(x 1/2), and introduce a test function g(x) = e^( iπx) e^( -λx) , which is smooth and Lipschitz. Applying Young's criterion, we show that the integral converges. Moreover , we prove that the integral converges perfectly for λ > 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.

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

TL;DR

This work studies the convergence of Erdős’s alternating series by recasting it as a Riemann–Stieltjes integral I = \int_1^\infty g(x) d\psi(x) with g(x) = e^{i\pi x} e^{−\lambda x} and a Chebyshev/prime-counting function \psi. Under the Riemann Hypothesis, the authors prove convergence via Hölder continuity and Young’s inequality, decomposing I into a main term and a controlled error term, and they derive explicit asymptotics for the dominant contribution: I = e^{(iπ − λ)}/(λ − iπ)^2 + O(1/λ^{3/2}). The approach combines Abel summation, fractional Sobolev embeddings, and integration by parts in the Stieltjes sense, offering a new analytic framework for oscillatory prime-series and potential insights into prime-gap behavior and Hardy–Littlewood-type conjectures. If RH holds, the analysis yields rigorous control over the integral, revealing a pole-like dominance as λ → 0^+ and illustrating a novel method to study Erdős-type alternating sums via Hölder-regularity and Stieltjes-interval techniques with potential broader applicability to related series.

Abstract

This open problem, first posed by Erdοs, 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 Hοlder continuation of ψ(x) in the asymptotic form ψ(x) = x+O(x 1/2), and introduce a test function g(x) = e^( iπx) e^( -λx) , which is smooth and Lipschitz. Applying Young's criterion, we show that the integral converges. Moreover , we prove that the integral converges perfectly for λ > 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.
Paper Structure (14 sections, 1 theorem, 52 equations)

This paper contains 14 sections, 1 theorem, 52 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Main results
  3. Hölder Continuity of g(x) and psi(x)
  4. Regularity of the Chebyshev function
  5. Regularity of g(x)
  6. Young's Criterion and Convergence
  7. Asymptotic Evaluation
  8. Main Term Evaluation of Integral
  9. Error Term I2
  10. Final Asymptotics
  11. Functional Regularity and Embedding
  12. Integration by Parts in the Riemann-Stieltjes Sense
  13. Evaluation of the Main Term I1
  14. Bounding the Error Term I2

Key Result

Theorem 2.1

Let $f \in W_p$ and $g \in W_q$, where $p, q > 0$ and $\frac{1}{p} + \frac{1}{q} > 1$. If $f$ and $g$ have no common discontinuities, then the Riemann-Stieltjes integral exists in the Riemann sense.

Theorems & Definitions (2)

  • Theorem 2.1: Existence of the Riemann-Stieltjes Integral
  • proof