Advertisement

Ramanujan’s Theory of Prime Numbers

  • Bruce C. Berndt

Abstract

In his famous letters of 16 January 1913 and 29 February 1913 to G. H. Hardy, Ramanujan [23, pp. xxiii-xxx, 349–353] made several assertions about prime numbers, including formulas for π(x), the number of prime numbers less than or equal to x. Some of those formulas were analyzed by Hardy [3], [5, pp. 234–238] in 1937. A few years later, Hardy [7, Chapter II], in a very penetrating and lucid presentation, thoroughly discussed most of the results on primes found in these letters. In particular, Hardy related Ramanujan’s fascinating, but unsound, argument for deducing the prime number theorem. Generally, Ramanujan thought that his formulas for π(x) gave better approximations than they really did. As Hardy [7, p. 19] (Ramanujan [23, p. xxiv]) pointed out, some of Ramanujan’s faulty thinking arose from his assumption that all of the zeros of the Riemann zeta-function ζ(s) are real.

Keywords

Prime Number Arithmetic Progression Tauberian Theorem Prime Number Theorem Lost Notebook 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Bruce C. Berndt
    • 1
  1. 1.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations