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Harald Cramér and the distribution of prime numbers
 Scandanavian Actuarial J
, 1995
"... “It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the ..."
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Cited by 20 (2 self)
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“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and MittagLeffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,
Eigenvalues, expanders and gaps between primes
, 2005
"... I kept myself positive, by not getting all negative. ..."
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Cited by 8 (4 self)
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I kept myself positive, by not getting all negative.
Riemann and his zeta function
, 2005
"... An exposition is given, partly historical and partly mathematical, of the Riemann zeta function ζ(s) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count nontrivial zeros of ζ(s). Relevance of these investigations to the theory of ..."
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An exposition is given, partly historical and partly mathematical, of the Riemann zeta function ζ(s) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count nontrivial zeros of ζ(s). Relevance of these investigations to the theory of the distribution of prime numbers is discussed.
Expander Graphs and Gaps between Primes
"... The explicit construction of infinite families of dregular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps betwee ..."
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The explicit construction of infinite families of dregular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost ” Ramanujan graphs for almost every value of d. More precisely, for any fixed ǫ> 0 and for almost every value of d (in the sense of natural density), there are infinitely many dregular graphs such that all the nontrivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ǫ) √ d − 1. 1