Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2505.06242

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

Nikos Mantzakouras
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nikos Mantzakouras. 2025-04-27. Holder continuity of an alternating Erdos series on prime K-tuples. https://doi.org/10.13140/rg.2.2.15200.60165

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

math.GM↗
arXiv2026

A quadratic critical-value conjecture for the fifth Bessel moment

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2π^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^σ,2) = κL(f,2)$ with explicit $κ\in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(χ_{-4}\mathrm{Sym}^2 f,2) = \sqrt{15}\,π^2 \langle f,f\rangle_{60}$, together with $L(χ_{-4}\mathrm{Sym}^2 f,3) = π^4\langle f,f\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}} = \frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}} = π^2 L(χ_{-4}\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}} = π^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.

math.GM↗