[Paper Review] The Riemann Hypothesis

Ilgar Sh. Jabbarov|arXiv (Cornell University)|May 31, 2010

Analytic Number Theory Research10 references3 citations

TL;DR

This paper claims to prove the Riemann Hypothesis by introducing a novel measure on the infinite-dimensional unit cube to approximate the Riemann zeta function using partial Euler products of Dirichlet series. Using Voronin's universality lemma and Rouché's theorem, the author argues that the zeta function has no zeros in the critical strip except on the critical line Res = 1/2, thereby establishing the hypothesis as true.

ABSTRACT

In the paper the well known Riemann Hypothesis is proven. The proof is based on uniform approximation of the zeta function discs of the critical strip placed to the right from the critical line.The basic moment is a use of a new mesure introduced in the infinite dimensional unite cube different from the Haar or product measures

Motivation & Objective

• To establish a rigorous proof of the Riemann Hypothesis, one of the most prominent unsolved problems in mathematics.
• To develop a new measure on the infinite-dimensional unit cube to analyze the distribution of fractional parts of sequences related to zeta function approximation.
• To extend partial Euler product approximations of the zeta function in the right half of the critical strip using probabilistic and ergodic methods.
• To demonstrate that the zeta function has no non-trivial zeros outside the critical line by applying complex analysis tools like Rouché's theorem.
• To unify concepts from value distribution theory, Dirichlet series universality, and analytic number theory to resolve the hypothesis.

Proposed method

• Introduces a new measure on the infinite-dimensional unit cube to study the distribution of fractional parts of sequences (\{t\lambda_n\})_n.
• Uses S.M. Voronin's lemma to approximate the Riemann zeta function in a disc within the right half of the critical strip by partial products of Euler-type Dirichlet series.
• Applies a special structure of divergence sets of certain series to extend the approximation from a disc to the entire right half of the critical strip.
• Employs Rouché's theorem to compare the zeta function with its approximating partial product, showing they have the same number of zeros inside a contour.
• Demonstrates that the approximating function has no zeros in a neighborhood of Re(s) = 3/4, implying the same for the zeta function.
• Uses the fact that the approximating function is a product over primes with complex exponential terms, constructed via Haar or Lebesgue measure on the infinite product space.

Experimental results

Research questions

• RQ1 Can the Riemann zeta function be uniformly approximated in the right half of the critical strip by partial Euler products of Dirichlet series under a new measure on the infinite-dimensional unit cube?
• RQ2 Does the absence of zeros in the approximating function imply the absence of zeros in the zeta function via Rouché's theorem?
• RQ3 Can the distribution of fractional parts of sequences (\{t\lambda_n\})_n in the infinite-dimensional cube be used to control the behavior of the zeta function?
• RQ4 Is the zeta function's zero-free region in the critical strip extendable to the entire right half via ergodic and measure-theoretic methods?
• RQ5 Can the universality properties of zeta and L-functions, as developed by Voronin and Bagchi, be leveraged to prove the Riemann Hypothesis?

Key findings

• The zeta function has no zeros in any disc centered at Re(s) = 3/4 with radius r < 1/4, as shown by Rouché's theorem applied to a convergent approximation.
• The partial product approximation $ F_n(s; \bar{\theta}_n) $, constructed via the infinite product over primes, has no zeros in the right half of the critical strip for almost all $ \theta $, under the new measure.
• The approximation error $ |\zeta(s) - F_n(s; \bar{\theta}_n)| $ is bounded by 0.25 times the minimum of $ |\zeta(s)| $ on a contour, satisfying the condition for Rouché's theorem.
• The method establishes that the zeta function has no zeros in the region $ |\text{Re}(s) - 3/4| < r $ for any $ 0 < r < 1/4 $, implying the absence of zeros in a full neighborhood of the critical line.
• The proof relies on the convergence of the approximation and the topological structure of the parameter space, which allows the extension of local results to the entire right half of the critical strip.
• The author concludes that all non-trivial zeros of the zeta function must lie on the critical line $ \text{Re}(s) = 1/2 $, thus proving the Riemann Hypothesis.