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A lower bound for the prime counting function

Published online by Cambridge University Press:  23 January 2015

Daniel Shiu  and
Peter Shiu
Daniel Shiu
Affiliation:
16 St Luke's Place, Cheltenham, GL53 7HP
Peter Shiu
Affiliation:
353 Fulwood Road, Sheffield, S10 3BQ e-mail:, p.shiu@yahoo.co.uk
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Extract

Let π (x) count the primes p ≤ x, where x is a large real number. Euclid proved that there are infinitely many primes, so that π (x) → ∞ as x → ∞; in fact his famous argument ([1: Section 2.2]) can be used to show that

There was no further progress on the problem of the distribution of primes until Euler developed various tools for the purpose; in particular he proved in 1737 [1: Theorem 427] that

Type
Articles
Information
The Mathematical Gazette , Volume 95 , Issue 534 , November 2011 , pp. 433 - 436
Copyright
Copyright © The Mathematical Association 2011

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References

1. Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (4th edn.), Oxford University Press (1960).Google Scholar