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A Comprehensive Course in Number Theory by Alan Baker

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15

On the distribution of the primes

15.1 The prime-number theorem

In this chapter we shall give an account of classical results on the distribution of the primes both in the sequence of the ordinary integers and, more generally, in arithmetical progressions. To begin with we establish the celebrated prime-number theorem originally conjectured by Legendre and first proved by Hadamard and de la Vallée Poussin independently in 1896. The proofs were based on the theory of functions of a complex variable and could not therefore be considered as elementary. An ‘elementary’ proof was given by Selberg and Erdimages in 1948; we shall mention some details relating ...

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