[Paper Review] Estimates of Some Functions Over Primes without R.H.

Pierre Dusart|arXiv (Cornell University)|Feb 2, 2010

Analytic Number Theory Research1 references90 citations

TL;DR

This paper provides explicit, effective bounds for Chebyshev's functions ϑ(x) and ψ(x) without assuming the Riemann Hypothesis, leveraging verified zero-free regions and numerical computations of zeta zeros. It establishes tighter error estimates for π(x), the k-th prime pₖ, and prime gaps, improving on prior results using extensive numerical verification of zeta zeros up to 10¹³.

ABSTRACT

Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory classical functions which are closely linked to zeta zeroes like psi(x), theta(x), pi(x) or the k-th prime number.

Motivation & Objective

• To derive effective, explicit upper and lower bounds for ϑ(x) and ψ(x) without assuming the Riemann Hypothesis.
• To improve estimates for the k-th prime number pₖ using bounds on ϑ(x) and ψ(x).
• To establish explicit intervals containing at least one prime, improving on known prime gap estimates.
• To provide tighter, numerically verified bounds for π(x) using explicit constants.

Proposed method

• Utilizes the identity ψ(x) = ∑ₖ₌₁^∞ ϑ(x¹ᐟᵏ) to relate ψ(x) and ϑ(x), enabling recursive estimation.
• Employs known zero-free regions of the Riemann zeta function and numerical verification of nontrivial zeros up to 10¹³ to refine error terms.
• Applies results from Rosser, Schoenfeld, and others on ψ(x) and ϑ(x) with explicit error bounds, particularly for x ≥ 3,594,641.
• Uses the relation π(x) ≈ x / ln x and derives tighter upper and lower bounds with explicit constants in the error terms.
• Employs asymptotic expansions of pₖ in terms of ln k and ln₂ k, refining coefficients using bounds on ϑ(x).
• Validates results numerically using extensive tables of ϑ(x), π(x), and pₖ up to 8×10¹¹, with constants determined per interval.

Experimental results

Research questions

• RQ1 What are the sharpest explicit bounds for ϑ(x) − x that hold without assuming the Riemann Hypothesis?
• RQ2 How can improved bounds on ϑ(x) and ψ(x) be used to derive tighter estimates for the k-th prime pₖ?
• RQ3 What is the minimal length of an interval [x, x + x/(25 ln²x)] that guarantees at least one prime for x ≥ 396,738?
• RQ4 How can explicit, numerically verified bounds for π(x) be constructed with improved constants in the error terms?
• RQ5 What are the optimal constants a₅, b₅, a₆, b₆, a₇, b₇ such that π(x) lies within the specified bounds for x up to 8×10¹¹?

Key findings

• For all x > 0, |ϑ(x) − x| < x / 36,260, providing a universal, absolute error bound.
• For x ≥ 3,594,641, |ϑ(x) − x| ≤ 0.2x / ln²x, significantly improving on prior explicit bounds.
• For k ≥ 688,383, pₖ ≤ k(ln k + ln₂k − 1 + (ln₂k − 2)/ln k), refining the asymptotic expansion of pₖ.
• For k ≥ 3, pₖ ≥ k(ln k + ln₂k − 1 + (ln₂k − 2.1)/ln k), establishing a tighter lower bound.
• For x ≥ 396,738, the interval [x, x + x/(25 ln²x)] contains at least one prime, improving prime gap estimates.
• For x ≥ 2,953,652,287, π(x) ≤ x / ln x × (1 + 1/ln x + 2.334 / ln²x), providing a sharp upper bound with explicit constant.