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Prime numbers and Lfunctions
"... Abstract. The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this t ..."
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Abstract. The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk I will show a panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory.
Finding and Counting MSTD sets
"... Abstract We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends ..."
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Abstract We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends to zero, and ‘explicit ’ constructions of large families of MSTD sets. We conclude with some new constructions and results of generalized MSTD sets, including among other items results on a positive percentage of sets having a given linear combination greater than another linear combination, and a proof that a positive percentage of sets are kgenerational sumdominant (meaning A, A+A,..., kA = A+ · · ·+A are each sumdominant).
ADDITIVE PROBLEMS WITH PRIME VARIABLES THE CIRCLE METHOD OF HARDY, RAMANUJAN AND LITTLEWOOD
, 2012
"... In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sket ..."
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In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sketch of the proof of Vinogradov’s threeprime Theorem.
Lebanese University
, 2012
"... The Fundamental Theorem of Arithmetic shows the importance of prime numbers. A wellknown result is that the set of prime numbers is infinite (the subset of even prime numbers is obviously finite while that of odd prime numbers is therefore infinite). The subset of Ramanujan primes is infinite. The ..."
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The Fundamental Theorem of Arithmetic shows the importance of prime numbers. A wellknown result is that the set of prime numbers is infinite (the subset of even prime numbers is obviously finite while that of odd prime numbers is therefore infinite). The subset of Ramanujan primes is infinite. The set of triplet prime numbers is finite while it is not known whether or not the subset of twin prime numbers is infinite even though it is so conjectured. We give many results involving the different types of prime numbers. 1
Computational Sieving Applied to some Classical NumberTheoretic
, 1998
"... Computational sieving applied to some classical numbertheoretic problems H.J.J. te Riele Modelling, Analysis and Simulation (MAS) ..."
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Computational sieving applied to some classical numbertheoretic problems H.J.J. te Riele Modelling, Analysis and Simulation (MAS)