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Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
DIVISORS OF SHIFTED PRIMES
"... Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the correspo ..."
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Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants. 1.
GENERALIZED AND RESTRICTED MULTIPLICATION TABLES OF INTEGERS BY
"... In 1955 Erdős posed the multiplication table problem: Given a large integer N, how many distinct products of the form ab with a ≤ N and b ≤ N are there? The order of magnitude of the above quantity was determined by Ford. The purpose of this thesis is to study generalizations of Erdős’s question in ..."
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In 1955 Erdős posed the multiplication table problem: Given a large integer N, how many distinct products of the form ab with a ≤ N and b ≤ N are there? The order of magnitude of the above quantity was determined by Ford. The purpose of this thesis is to study generalizations of Erdős’s question in two different directions. The first one concerns the kdimensional version of the multiplication table problem: for a fixed integer k ≥ 3 and a large parameter N, we establish the order of magnitude of the number of distinct products n1 · · · nk with ni ≤ N for all i ∈ {1,..., k}. The second question we shall discuss is the restricted multiplication table problem. More precisely, for B ⊂ N we seek estimates on the number of distinct products ab ∈ B with a ≤ N and b ≤ N. This problem is intimately connected with the available information on the distribution of B in arithmetic progressions. We focus on the special and important case when B = Ps = {p + s: p prime} for some fixed s ∈ Z \ {0}. Ford established upper bounds of the expected order of magnitude for {ab ∈ Ps: a ≤ N, b ≤ N}. We prove the corresponding lower bounds thus determining the size of the quantity in question up to multiplicative constants. ii To my parents, Dimitra and Paris iii Acknowledgements I would like to express my warmest thanks to my advisor Professor Kevin Ford for his help and encouragement over the years.
Different Approaches to the Distribution of Primes
 MILAN JOURNAL OF MATHEMATICS
, 2009
"... In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction. ..."
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In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction.
9 SpringerVerlag 1984 A remark on Artin's conjecture
"... A famous conjecture of E. Artin [t] states that for any integer a 4 = + _ I or a perfect square, there are infinitely many primes p for which a is a primitive root (modp). This conjecture was shown to be true if one assumes the generalized Riemann hypothesis by Hooley [5]. The purpose of this note i ..."
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A famous conjecture of E. Artin [t] states that for any integer a 4 = + _ I or a perfect square, there are infinitely many primes p for which a is a primitive root (modp). This conjecture was shown to be true if one assumes the generalized Riemann hypothesis by Hooley [5]. The purpose of this note is to exhibit a finite set S such that for some a eS, a is a primitive root (modp) for an infinity ofprimesp. To this end, let q, r and s denote three distinct primes. Define the following set:
Asymptotic sieve for primes
, 1998
"... For a long time it was believed that sieve methods might be incapable of achieving the goal for which they had been created, the detection of prime numbers. Indeed, it was shown [S], [B] to be inevitable that, in the sieve’s original framework, no such result was possible although one could come tan ..."
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For a long time it was believed that sieve methods might be incapable of achieving the goal for which they had been created, the detection of prime numbers. Indeed, it was shown [S], [B] to be inevitable that, in the sieve’s original framework, no such result was possible although one could come tantalizingly close. This general limitation is recognized as the “parity problem” of sieve theory. More recently, beginning with the work [IJ], this goal has in certain cases become possible by adapting the sieve machinery to enable it to take advantage of the input of additional analytic data. There have been a number of recent developments in this regard, for example [H], [DFI], [FI], and several recent works of R.C. Baker and G. Harman. The story however is far from finished. In this paper we consider a sequence of real nonnegative numbers (1.1) A = (an) with the purpose of showing an asymptotic formula for (1.2) S(x) = ∑ anΛ(n) n�x where Λ(n) denotes the von Mangoldt function. The sequence A can be quite thin although not arbitrarily so. Setting (1.3) A(x) = ∑ n�x an we require that A(x) be slightly larger than A ( √ x); precisely (1.4) A(x) ≫ A ( √ x)(log x) 2.
An Overview of Sieve Method and its History
, 2005
"... Trying to decompose an integer into a product of integers, we feel irritation. There should dwell the reason why any prime appears like a real gem that one can touch and hold. We thus muse ever and again how and when ancient people discovered the way of sifting out primes and began appreciating them ..."
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Trying to decompose an integer into a product of integers, we feel irritation. There should dwell the reason why any prime appears like a real gem that one can touch and hold. We thus muse ever and again how and when ancient people discovered the way of sifting out primes and began appreciating them. Perhaps those who conceived the divisibility had already some sieves in their minds. Indeed, a wealth of evidences have been excavated supporting our view. The story to be told below must have originated more than five millennia ago 1) , while the primordial intellectual irritation has remained fresh and fundamental till today. The history of Sieve Method is rich and fascinating; we would need a volume to exhaust the story. In the present article we shall instead concentrate onto several pivotal ideas that made the progress possible; so the scope is inevitably limited. Nevertheless, you will encounter instances of precious mathematical achievements that people in the future will certainly continue to relate. Notes are to be read as essential parts, although they are in the style of personal memoranda. Mathematical symbols and definitions are introduced where they are needed for the first time, and will continue to be effective until otherwise stated. Theorems are given somewhat