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Localized factorizations of integers
 Proc. London Math. Soc. (2010); doi: 10.1112/plms/pdp056
"... Abstract. We determine the order of magnitude of H (k+1) (x, y, 2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1,..., log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a ..."
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Cited by 4 (2 self)
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Abstract. We determine the order of magnitude of H (k+1) (x, y, 2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1,..., log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear in a (k + 1)dimensional multiplication table as well as how many distinct sums of k + 1 Farey fractions there are modulo 1. 1.
The distribution of integers with at least two divisors in a short interval, Quart
 J. Math. Oxford
"... Abstract. We estimate the density of integers which have more than one divisor in an interval (y, z] with z ≈ y + y/(log y) log 4−1. As a consequence, we determine the precise range of z such that most integers which have at least one divisor in (y, z] have exactly one such divisor. 1. ..."
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Cited by 3 (1 self)
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Abstract. We estimate the density of integers which have more than one divisor in an interval (y, z] with z ≈ y + y/(log y) log 4−1. As a consequence, we determine the precise range of z such that most integers which have at least one divisor in (y, z] have exactly one such divisor. 1.
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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Cited by 1 (1 self)
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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.
On Artin's Conjecture for Primitive Roots
, 1993
"... Various generalizations of the Artin's Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field ) valid for the range l < log x is proven. ..."
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Various generalizations of the Artin's Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field ) valid for the range l < log x is proven. A uniform asymptotic formula for the number of primes up to x for which there exists a primitive root less than s is established. Lower bounds for the exponent of the class group of imaginary quadratic fields valid for density one sets of discriminants are determined. RESUM E Nous considerons di#erentes generalisations de la conjecture d'Artin pour les racines primitives. Nous demontrons que pour au moins la moitie des nombres premiers p, les premiers log p nombres premiers engendrent une racine primitive. Nous demontrons une version uniforme du Theoreme de Densite de Chebotarev pour le corps Q(# l , 2 ) pour l'intervalle l < log x. On etablit une formule asymptotique uniforme pour les nombres de premiers plus petits que x tels qu' il existe une racine primitive plus petite que s. Nous determinons des minorants pour l'exposant du groupe de classe des corps quadratiques imaginaires valides pour ensembles de discriminants de densite 1. Contents
On the Order of Finitely Generated Subgroups of Q*(mod ρ) and Divisors of ρ1
, 1996
"... Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote t ..."
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Introduction Let r be a positive integer. We say that r nonzero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote the subgroup of Q* generated by a 1 , ..., a r and let 1 p  denote the order of such a group 1 (mod p). In the case r=1, 1=(a), let ord p (a) denote the order of a (mod p). The famous Artin Conjecture for primitive roots (see [1]) states that ord p (a)=p&1 for infinitely many primes p. Artin's Conjecture has been proved under the assumption of the Generalized Riemann Hypothesis by C. Hooley (See [13]). In his paper it is implicitly shown (unconditionally) that ord p (a)>p#log p (1.1) for all but O(x#log x) primes p#x. article no. 0044 207 0022314X#96 #18.00 Copyright # 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * Supported in part by C.I.C.M.A.
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.
ON THE NUMBER OF INTEGERS IN A GENERALIZED MULTIPLICATION TABLE
"... Abstract. Motivated by the Erdős multiplication table problem we study the following question: Given numbers N1,..., Nk+1, how many distinct products of the form n1 · · · nk+1 with 1 ≤ ni ≤ Ni for i ∈ {1,..., k + 1} are there? Call Ak+1(N1,..., Nk+1) the quantity in question. Ford established the ..."
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Abstract. Motivated by the Erdős multiplication table problem we study the following question: Given numbers N1,..., Nk+1, how many distinct products of the form n1 · · · nk+1 with 1 ≤ ni ≤ Ni for i ∈ {1,..., k + 1} are there? Call Ak+1(N1,..., Nk+1) the quantity in question. Ford established the order of magnitude of A2(N1, N2) and the author of Ak+1(N,..., N) for all k ≥ 2. In the present paper we generalize these results by establishing the order of magnitude of Ak+1(N1,..., Nk+1) for arbitrary choices of N1,..., Nk+1 when 2 ≤ k ≤ 5. Moreover, we obtain a partial answer to our question when k ≥ 6. Lastly, we develop a heuristic argument which explains why the limitation of our method is k = 5 in general and we suggest ways of improving the results of this paper.
LOCALIZED FACTORIZATIONS OF INTEGERS
, 809
"... Abstract. We determine the order of magnitude of H (k+1) (x, y,2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1,...,log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a c ..."
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Abstract. We determine the order of magnitude of H (k+1) (x, y,2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1,...,log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear in a (k + 1)dimensional multiplication table as well as how many distinct sums of k + 1 Farey fractions there are modulo 1. 1.
Article 13.6.7 A Generalization of the GcdSum Function
"... In this paper we consider the generalization Gd(n) of the Broughan gcdsum function, i.e., the sum of such gcd’s that are divisors of the positive integer d. Examples of Dirichlet series and asymptotic relations for Gd and related functions are given. 1 ..."
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In this paper we consider the generalization Gd(n) of the Broughan gcdsum function, i.e., the sum of such gcd’s that are divisors of the positive integer d. Examples of Dirichlet series and asymptotic relations for Gd and related functions are given. 1