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On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters.
, 1993
"... : We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exist ..."
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: We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that `(x; q; a) ¸ 2x=OE(q) for each a 62 H and `(x; q; a) = o(x=OE(q)), otherwise. From here, we deduce that if the second case holds at all, then it holds only for the multiples of some fixed integer q 0 ? 1. Actually, even if the Prime Number Theorem for arithmetic progressions, modulo q, does hold, these methods allow us to deduce the behaviour of a possible `Siegel zero' from Selberg's formula. We also propose a new method for determining explicit upper and lower bounds on `(x; q; a), which uses only elementary number theoretic computations. 1. Introduction. Define `(x) = P px log p, where p only denot...
Bull. London Math. Soc. 34 (2002) 385–402 C ❢ 2002 London Mathematical Society DOI: 10.1112/S002460930200111X G. H. HARDY
"... Despite a true antipathy to the subject, Hardy contributed deeply to modern probability. His work with Ramanujan begat probabilistic number theory. His work on Tauberian theorems and divergent series has probabilistic proofs and interpretations. Finally, Hardy spaces are a central ingredient in stoc ..."
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Despite a true antipathy to the subject, Hardy contributed deeply to modern probability. His work with Ramanujan begat probabilistic number theory. His work on Tauberian theorems and divergent series has probabilistic proofs and interpretations. Finally, Hardy spaces are a central ingredient in stochastic calculus. This paper reviews his prejudices and accomplishments, through these examples. 1. Hardy G. H. Hardy was a great hard analyst, who worked in the first half of the twentieth century. He was born on 7th February, 1877, in Surrey, and died on 1st December, 1947, at Cambridge. C. P. Snow’s biographical essay on Hardy, which opens modern editions of Hardy’s A mathematician’s apology [51], gives a good nonmathematical picture of Hardy. Titchmarsh’s obituary, which gives more of a mathematical overview, is reprinted in his collected works [52], which comprise seven thick volumes, containing Hardy’s 350 papers. Hardy had a close connection to the London Mathematical Society, serving as its President in 1926–1928, and almost continuously on its council. He left his estate to