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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,
About the formalization of some results by Chebyshev in number theory
 Proceedings of TYPES’08, Vol. 5497 of LNCS
, 2009
"... Abstract. We discuss the formalization, in the Matita Interactive Theorem Prover, of a famous result by Chebyshev concerning the distribution of prime numbers, essentially subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime numb ..."
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Cited by 8 (4 self)
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Abstract. We discuss the formalization, in the Matita Interactive Theorem Prover, of a famous result by Chebyshev concerning the distribution of prime numbers, essentially subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development of number theory. Differently from other recent formalizations of other results in number theory, our proof is entirely arithmetical. It makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, factorization, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base. 1
Dedekind Zeta Functions and the Complexity of Hilbert’s Nullstellensatz
, 2008
"... Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the ..."
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Cited by 5 (4 self)
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Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication HN ̸∈P = ⇒ P ̸=NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.
The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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Cited by 4 (4 self)
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.
A proof of Bertrand’s postulate
"... We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery ..."
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We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development of number theory. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base. 1.
THE WIENER–IKEHARA THEOREM BY COMPLEX ANALYSIS JAAP KOREVAAR
"... Abstract. The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman’s contour integration method can be adapted to establish the Wiener–Ikehara theorem. A simple special case suffices for the PNT. But what about the twinprime p ..."
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Abstract. The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman’s contour integration method can be adapted to establish the Wiener–Ikehara theorem. A simple special case suffices for the PNT. But what about the twinprime problem? 1.
An Epic Drama: The Development of the Prime Number Theorem
, 2010
"... The prime number theorem, describing the aymptotic density of the prime numbers, has often been touted as the most surprising result in mathematics. The statement and development of the theorem by Legendre, Gauss and others and its eventual proof by Hadamard and de al ValléePoussin span the whole ..."
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The prime number theorem, describing the aymptotic density of the prime numbers, has often been touted as the most surprising result in mathematics. The statement and development of the theorem by Legendre, Gauss and others and its eventual proof by Hadamard and de al ValléePoussin span the whole nineteenth century and encompass the growth of a brand new field in analytic number theory. As an outgrowth of the techniques of the proof is the Riemann hypothesis which today is perhaps the outstanding open problem in mathematics. These ideas and occurences certainly constitute an epic drama within the history of mathematics and one that is not as well known among the general mathematical community as it should be. In the present paper we trace out the paper, the development of the proof and a raft of other ideas, results and concepts that come from the prime number theorem.