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Unexpected Irregularities in the Distribution of Prime Numbers

In: Proceedings of the International Congress of Mathematicians

Author

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  • Andrew Granville

    (University of Georgia, Department of Mathematics)

Abstract

In 1849 the Swiss mathematican ENCKE wrote to GAUSS, asking whether he had ever considered trying to estimate Π(x), the number of primes up to x, by some sort of “smooth” function. On Christmas Eve 1849, GAUSS replied that “he had pondered this problem as a boy” and had come to the conclusion that “at around x, the primes occur with density 1/log x.” Thus, he concluded, π(ϰ) could be approximated by $$Li(x): = \int_2^x {\frac{{dt}}{{\log t}} = \frac{x}{{\log x}} + \frac{x}{{{{\log }^2}x}} + O(\frac{x}{{{{\log }^3}x}})}$$ L i ( x ) : = ∫ 2 x d t log t = x log x + x log 2 x + O ( x log 3 x ) .

Suggested Citation

  • Andrew Granville, 1995. "Unexpected Irregularities in the Distribution of Prime Numbers," Springer Books, in: S. D. Chatterji (ed.), Proceedings of the International Congress of Mathematicians, pages 388-399, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-9078-6_32
    DOI: 10.1007/978-3-0348-9078-6_32
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