[Paper Review] On the first sign change of $ heta(x) - x$

David J. Platt, Tim Trudgian|arXiv (Cornell University)|Jul 8, 2014

Analytic Number Theory Research14 references3 citations

TL;DR

This paper establishes that θ(x) < x for all x < 1.39 × 10^17, improving prior bounds, and proves the existence of an x < exp(727.951332668) where θ(x) > x, using explicit formulas, Riemann zeta zeros, and numerical integration with Gaussian kernels. The result confirms the first sign change of θ(x) − x occurs below exp(727.951332668), with over 10^152 consecutive integers satisfying θ(x) > x.

ABSTRACT

Let $\ heta(x) = \\sum_{p\\leq x} \\log p$. We show that $\ heta(x)&lt;x$ for $2&lt;x&lt; 1.39\\cdot 10^{17}$. We also show that there is an $x&lt;\\exp(727.951332668)$ for which $\ heta(x) &gt;x.$

Motivation & Objective

• To determine the first sign change of θ(x) − x, where θ(x) = ∑_{p≤x} log p.
• To improve the upper bound for the first x where θ(x) > x, beyond previous estimates.
• To extend the range where θ(x) < x, building on Schoenfeld and Dusart’s work.
• To provide an effective, computationally verified bound for θ(x) < x up to 1.39 × 10^17.

Proposed method

• Uses the explicit formula for ψ(x) and relates it to θ(x) via ψ(x) = θ(x) + θ(x^{1/2}) + θ(x^{1/3}) + ⋯.
• Applies a Gaussian kernel K(u−ω) to smooth the integral of θ(e^u)−e^u, enabling estimation of sign changes.
• Employs numerical integration and summation over non-trivial zeta zeros ρ = 1/2 + iγ up to height T ≈ 5000, with error bounds.
• Uses double-precision interval arithmetic and parallelized prime sieving to compute θ(x) efficiently across 10,390 segments up to 1.39×10^17.
• Applies tail bounds from known ψ(x)−x estimates and θ(x)−x error terms to refine the interval where θ(x) > x.
• Validated results using independent checks from Oliveira e Silva’s π(x) tables.

Experimental results

Research questions

• RQ1 What is the smallest x for which θ(x) > x, given that θ(x) < x holds up to 1.39×10^17?
• RQ2 Can the first sign change of θ(x)−x be effectively bounded below exp(727.951332668)?
• RQ3 How many consecutive integers satisfy θ(x) > x in the interval where the first sign change occurs?
• RQ4 To what extent can the error terms in θ(x)−x be controlled using known bounds on ψ(x)−x and ζ(s) zeros?
• RQ5 Can a parallelized, numerically stable algorithm compute θ(x) accurately over very large ranges?

Key findings

• θ(x) < x for all x < 1.39 × 10^17, extending prior bounds from 8×10^11.
• There exists an x < exp(727.951332668) such that θ(x) > x, establishing the first sign change.
• The interval [exp(727.951332642), exp(727.951332668)] contains at least 10^152 consecutive integers where θ(x) > x.
• The sum ∑_{|γ|≤T} e^{iωγ}/ρ exp(−γ²/(2α)) at ω = 727.951332655 lies in [−1.0013360278, −1.0013360277], confirming a sign change.
• The error terms R₁ + R₂ + R₃ + R₄ < 1.7×10⁻⁹, ensuring the positivity of the integral and thus the existence of θ(x) > x.
• The computation used 78,000 node-hours on a 16-core cluster, with 10,390 segments and interval arithmetic to ensure numerical reliability.