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Higher correlations of divisor sums related to primes, II: Variations of . . .
, 2007
"... We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the ..."
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We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet Lfunctions, we obtain an Ω±result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ωresults for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes.
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.
Dense Admissible Sets
"... Call a set of integers fb1 ; b2 ; : : : ; bk g admissible if for any prime p, at least one congruence class modulo p does not contain any of the b i . Let ae (x) be the size of the largest admissible set in [1; x]. The Prime ktuples Conjecture states that any for any admissible set, there are infin ..."
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Call a set of integers fb1 ; b2 ; : : : ; bk g admissible if for any prime p, at least one congruence class modulo p does not contain any of the b i . Let ae (x) be the size of the largest admissible set in [1; x]. The Prime ktuples Conjecture states that any for any admissible set, there are infinitely many n such that n+b1 ; n+b2 ; : : : n+bk are simultaneously prime. In 1974, Hensley and Richards [3] showed that ae (x) ? ß(x) for x sufficiently large, which shows that the Prime ktuples Conjecture is inconsistent with a conjecture of Hardy and Littlewood that for all integers x; y 2, ß(x + y) ß(x) + ß(y): In this paper we examine the behavior of ae (x), in particular, the point at which ae (x) first exceeds ß(x), and its asymptotic growth.
The Life and Work of R. A. Rankin (19152001)
"... decades, one of the world’s foremost experts in modular forms, died on January 27, 2001 in Glasgow at the age of 85. He was one of the founding editors of The Ramanujan Journal. For this and the next issue of the The Ramanujan Journal, many wellknown mathematicians have prepared articles in Rankin’ ..."
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decades, one of the world’s foremost experts in modular forms, died on January 27, 2001 in Glasgow at the age of 85. He was one of the founding editors of The Ramanujan Journal. For this and the next issue of the The Ramanujan Journal, many wellknown mathematicians have prepared articles in Rankin’s memory. In this opening paper, we provide a short biography of Rankin and discuss some of his major contributions to mathematics. At the conclusion of this article, we provide a complete list of Rankin’s doctoral students and a complete bibliography of all of Rankin’s writings divided into
PRIMES IN TUPLES II
, 2007
"... We prove that liminf n→∞ pn+1 − pn √ log pn(log log pn) 2 where pn denotes the n th prime. Since on average pn+1 −pn is asymptotically log pn, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of valu ..."
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We prove that liminf n→∞ pn+1 − pn √ log pn(log log pn) 2 where pn denotes the n th prime. Since on average pn+1 −pn is asymptotically log pn, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p − p′ between primes which includes the small gap result above.