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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.
A DISCRETE MEAN VALUE OF THE DERIVATIVE OF THE RIEMANN ZETA FUNCTION
, 706
"... Abstract. In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of ζ ′ (ρ) where ζ(s) is the Riemann zeta function and ρ is a nontrivial zero of the Riemann zeta function. 1. ..."
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Abstract. In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of ζ ′ (ρ) where ζ(s) is the Riemann zeta function and ρ is a nontrivial zero of the Riemann zeta function. 1.
THE DISTRIBUTION OF PRIMES: CONJECTURES vs. HITHERTO PROVABLES
"... We present the main results, conjectures and ideas concerning the distribution of primes. We recount only the most important milestones and we do not go into technical details. Instead, suggestions for further reading are provided. ..."
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We present the main results, conjectures and ideas concerning the distribution of primes. We recount only the most important milestones and we do not go into technical details. Instead, suggestions for further reading are provided.
THE PARITY PROBLEM FOR REDUCIBLE CUBIC FORMS
, 2005
"... Let f ∈ Z[x, y] be a reducible homogeneous polynomial of degree 3. We show that f(x, y) has an even number of prime factors as often as an odd number of prime factors. ..."
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Let f ∈ Z[x, y] be a reducible homogeneous polynomial of degree 3. We show that f(x, y) has an even number of prime factors as often as an odd number of prime factors.