Unexpected Irregularities in the Distribution of Prime Numbers

Andrew Granville
Additional contact information
Andrew Granville: University of Georgia, Department of Mathematics

A chapter in Proceedings of the International Congress of Mathematicians, 1995, pp 388-399 from Springer

Abstract: 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 ) .

Date: 1995
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-9078-6_32

Ordering information: This item can be ordered from
http://www.springer.com/9783034890786

DOI: 10.1007/978-3-0348-9078-6_32

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().