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The Prime Number Theorem

In: Numbers from all Angles

Author

Listed:
  • J. J. P. Veerman

    (Portland State University)

Abstract

In 1850, Chebyshev seemed awfully close to proving the prime number theorem (Theorem 2.19 ). But to bridge that last brook, a whole new approach to the problem was needed. That approach was the connection with analytic functions in the complex domain pioneered by Riemann in 1859 [101]. A very weak version of the Riemann hypothesis (Conjecture 2.21 ), namely the absence of zeroes of $$\zeta (z)$$ζ(z) in $$\mathrm{{Re\,}}z\ge 1$$Rez≥1 turns out to be an essential step. We look at this in Sect. 13.3 and in particular Lemma 13.13. It would take another 37 years after Riemann’s monumental contribution before that step was finally taken by de la Vallée Poussin [39, 40] and Hadamard [60] in 1896. The version we prove is a highly streamlined derivative of that proof, the last stage of which was achieved by Newman in 1982 [91].

Suggested Citation

  • J. J. P. Veerman, 2026. "The Prime Number Theorem," Springer Books, in: Numbers from all Angles, chapter 0, pages 269-292, Springer.
    • Handle: RePEc:spr:sprchp:978-3-032-10000-9_13
    DOI: 10.1007/978-3-032-10000-9_13
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