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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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“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,
The distribution of totients
, 1998
"... This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s φfunction. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of φ(x) =m(the “multiplicity ” of m), and Vk(x), the numb ..."
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This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s φfunction. The main functions studied are V (x), the number of totients not exceeding x, A(m), the number of solutions of φ(x) =m(the “multiplicity ” of m), and Vk(x), the number of m ≤ x with A(m) =k. The first of the main results of the paper is a determination of the true order of V (x). It is also shown that for each k ≥ 1, if there is a totient with multiplicity k, thenVk(x)≫V(x). We further show that every multiplicity k ≥ 2 is possible, settling an old conjecture of Sierpiński. An older conjecture of Carmichael states that no totient has multiplicity 1. This remains an open problem, but some progress can be reported. In particular, the results stated above imply that if there is one counterexample, then a positive proportion of all totients are counterexamples. Determining the order of V (x) andVk(x) also provides a description of the “normal ” multiplicative structure of totients. This takes the form of bounds on the sizes of the prime factors of a preimage of a typical totient. One corollary is that the normal number of prime factors of a totient ≤ x is c log log x, wherec≈2.186. Lastly, similar results are proved for the set of values taken by a general multiplicative arithmetic function, such as the sum of divisors function, whose behavior is similar to that of Euler’s function.
Two contradictory conjectures concerning Carmichael numbers
"... Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to ..."
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Erdös [8] conjectured that there are x 1;o(1) Carmichael numbers up to x, whereas Shanks [24] was skeptical as to whether one might even nd an x up to which there are more than p x Carmichael numbers. Alford, Granville and Pomerance [2] showed that there are more than x 2=7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in persontoperson discussions) that Erdös must be correct. Nonetheless, Shanks's skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch's extended new data [14,15]), and so we herein derive conjectures that are consistent with Shanks's observations, while tting in with the viewpoint of Erdös [8] and the results of [2,3].
An asymptotic formula for the number of smooth values of a polynomial
 J. Number Theory
, 1999
"... Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from mult ..."
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Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity
Prime Number Races
 Amer. Math. Monthly
"... 1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that ..."
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1. INTRODUCTION. There’s nothing quite like a day at the races....The quickening of the pulse as the starter’s pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking
Yildirim, Small gaps between products of two primes
 20世纪最伟大数学家埃尔德什开始关注这个问题, 但 也没得出有用结果。最近国际顶尖数学 家 Goldston 等正在研究这个问题。得到国际数学界广泛的支持和关注，但文章都发表在著 名杂志上，但没有得出任何实质性进展，蒋春暄在 2002年[1]就彻底证明了它，但国内外数 学家都读了它，都不说话，看到文献[4]后，我们决定写本文，如不用 Jiang函数，再过两百 年也不一定能证明它，国内更无人研究它，这才是研究方向！2009年 1月 10日蒋春暄为休 息去参加宋正海讲座在公共汽车上发现公式(22),1 月 1
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Affine linear sieve, expanders, and sumproduct
"... This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and ..."
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This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and
Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
Small gaps between primes
"... ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing betwe ..."
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ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.