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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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Cited by 28 (6 self)
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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.
Zero Spacing Distributions for Differenced LFunctions
, 2006
"... The paper studies the local zero spacings of deformations of the Riemann ξfunction under certain averaging and differencing operations. For real h we consider the entire functions Ah(s): = 1 2 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1 ..."
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Cited by 5 (2 self)
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The paper studies the local zero spacings of deformations of the Riemann ξfunction under certain averaging and differencing operations. For real h we consider the entire functions Ah(s): = 1 2 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1
Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution. See http://arXiv.org/abs/0805.4682 (posted May 30
, 2008
"... Abstract. We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the ktuple conjecture and more general problems of polynomial representation of primes. We show that the “singular series ” for the ..."
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Cited by 2 (1 self)
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Abstract. We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the ktuple conjecture and more general problems of polynomial representation of primes. We show that the “singular series ” for the ktuple conjecture have a limiting distribution when taken over ktuples with (distinct) entries of growing size. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in suitable short intervals are asymptotically Poisson distributed. 1.
Sieving and the ErdősKac Theorem
, 2006
"... We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. ..."
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Cited by 2 (0 self)
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We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.
THE DISTRIBUTION OF PRIME NUMBERS
, 2006
"... What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consec ..."
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What follows is an expanded version of my lectures at the NATO School on Equidistribution. I have tried to keep the informal style of the lectures. In particular, I have sometimes oversimplified matters in order to convey the spirit of an argument. Lecture 1: The Cramér model and gaps between consecutive primes The prime number theorem tells us that π(x), the number of primes below x, is ∼ x / logx. Equivalently, if pn denotes the nth smallest prime number then pn ∼ n log n. What is the distribution of the gaps between consecutive primes, pn+1 − pn? We have just seen that pn+1 − pn is approximately log n “on average”. How often do we get a gap of size 2 logn, say; or of size 1 log n? One way to make this question precise 2 is to fix an interval [α, β] (with 0 ≤ α < β) and ask for
Mathematisches Forschungsinstitut Oberwolfach Report No. 46/2004 Theory of the Riemann Zeta and Allied Functions
, 2004
"... Introduction by the Organisers This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a h ..."
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Introduction by the Organisers This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a hike in the mountains and piano recitals by Peter
Riffle shuffles with biased cuts
"... Abstract. The wellknown GilbertShannonReeds model for riffle shuffles assumes that the cards are initially cut ‘about in half ’ and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and Li ..."
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Abstract. The wellknown GilbertShannonReeds model for riffle shuffles assumes that the cards are initially cut ‘about in half ’ and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and Linfinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function. Résumé. Le modèle de GilbertShannonReeds pour melange de cartes suppose que les cartes sont d’abord coupés énviron de moitié’, puis intescaler ensemble. Nous analysons une variante naturelle, où la coupe initiale est biaisé. Nous propons une extension des résultats de Fulman (1998), nous montrent une forte coupure dans les distances de séparation et Linfinity. Cette analyse est possible grâce à l’étroite relation entre brassage et fonctions quasisymmetric.