[Paper Review] A computational history of prime numbers and Riemann zeros

Pieter Moree, Izabela Petrykiewicz+1 more|arXiv (Cornell University)|Oct 11, 2018

Analytic Number Theory Research44 references11 citations

TL;DR

This paper provides a computational and historical survey of prime number distribution and the non-trivial zeros of the Riemann zeta function, tracing developments from Gauss’s early estimates to modern random matrix theory connections. It highlights how the Riemann Hypothesis, if true, would yield sharp bounds on the prime counting function π(x), with key advances driven by numerical computations and theoretical insights from analytic number theory and random matrix theory.

ABSTRACT

We give an informal survey of the historical development of computations related to prime number distribution and zeros of the Riemann zeta function.

Motivation & Objective

• To trace the historical evolution of computational methods in prime number theory, from manual prime tables to modern numerical verification.
• To explain the significance of the Riemann Hypothesis in enabling sharp estimates for π(x), the prime counting function.
• To explore the deep connection between the distribution of non-trivial zeta zeros and random matrix theory (RMT), particularly in predicting moments of the zeta function.
• To emphasize the role of extensive numerical computation—especially Odlyzko’s work on zeros near 10^22—in validating theoretical conjectures.
• To caution researchers against overrelying on numerical data due to the subtle growth of iterated logarithmic functions, which can mislead without theoretical grounding.

Proposed method

• Tracing the historical development of prime counting via manual and algorithmic computation, from Gauss’s chiliade method to Kulik’s 4,212-page factor table.
• Using the logarithmic integral Li(x) as a central approximation for π(x), with asymptotic equivalence π(x) ~ x/log x proven via the Prime Number Theorem.
• Applying the Riemann Hypothesis to derive a sharp bound: |π(x) − Li(x)| < (1/(8π))√x log x for x > 2,657.
• Employing random matrix theory (RMT) to model the statistical distribution of non-trivial zeta zeros, leading to conjectures on moments I_k(T) of the zeta function.
• Deriving conjectural integer values for f(k) in the asymptotic formula I_k(T) ~ (f(k)a(k)/k²!) T (log T)^{k²}, with f(3)=42 and f(4)=24024 confirmed via RMT.
• Validating these RMT-based conjectures through extensive numerical computation, particularly using Odlyzko’s high-precision zeros near T = 10^22.

Experimental results

Research questions

• RQ1 How did historical computations of prime numbers evolve from manual tables to modern numerical methods?
• RQ2 What is the significance of the Riemann Hypothesis in enabling precise approximation of π(x) by Li(x), and what bounds does it imply?
• RQ3 How do random matrix theory predictions for the moments of the Riemann zeta function compare with analytic number theory results?
• RQ4 What role do high-precision numerical computations of zeta zeros—especially near T = 10^22—play in validating theoretical conjectures?
• RQ5 Why is caution required when drawing conjectures from numerical data, especially regarding slowly growing functions like log log log x?

Key findings

• Gauss’s early conjecture that π(x) ≈ Li(x) led to the Prime Number Theorem, later proven independently by Hadamard and de la Vallée-Poussin.
• The Riemann Hypothesis implies the sharp bound |π(x) − Li(x)| < (1/(8π))√x log x for all x > 2,657, significantly improving error estimates.
• Random matrix theory successfully predicted the integer values f(3) = 42 and f(4) = 24024 for the asymptotic moments of the zeta function, consistent with earlier analytic results.
• Odlyzko’s numerical computation of zeta zeros near T = 10^22 revealed that the zeta function only exhibits its true statistical behavior at such large heights.
• The connection between L-functions and random matrix theory is robust in function fields, where the Riemann Hypothesis is proven, and has led to theorems in number theory via the RMT dictionary.
• Despite strong numerical and heuristic support, no proof exists that the RMT dictionary always holds, underscoring the continued importance of computational number theory.