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16
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.
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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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
The variance of the number of prime polynomials in short intervals and in residue classes
 Int. Math. Res. Not. IMRN, page
, 2012
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On the distribution of imaginary parts of zeros of the Riemann zeta function, II
, 2009
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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
Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution.
, 2008
"... 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 c ..."
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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.
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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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.
ON THE VARIANCE OF SUMS OF ARITHMETIC FUNCTIONS OVER PRIMES IN SHORT INTERVALS AND PAIR CORRELATION FOR LFUNCTIONS IN THE SELBERG CLASS
"... Abstract. We establish the equivalence of conjectures concerning the pair correlation of zeros of Lfunctions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgo ..."
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Abstract. We establish the equivalence of conjectures concerning the pair correlation of zeros of Lfunctions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan [11] for the Riemann zetafunction to other Lfunctions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the HardyLittlewood conjecture for the autocorrelation of the arithmetic functions we consider is not available in general. One of our main findings is that the variances of sums of these arithmetic functions over primes in short intervals have a different form when the degree of the associated Lfunctions is 2 or higher to that which holds when the degree is 1 (e.g. the Riemann zetafunction). Specifically, when the degree is 2 or higher there are two regimes in which the variances take qualitatively different forms, whilst in the degree1 case there is a single regime. 1.