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 L-functions, 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 Ω-results 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. 1.

In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of the Riemann zeta-function. One of his main accomplishments was to determine partially the pair correlation of zeros, and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros. Perhaps more

THE prime number theorem for arithmetic progressions tells us that, for integers a, q s = 1 with (a, q) = 1 we have

In this paper, we study a more general pair correlation function, Fh(x, T), of the zeros of the Riemann zeta function. It provides information on the distribution of larger differences between the zeros. 1

Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) =π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x)−li(x). However, there is a heuristic approximate formula for averages of the remainders ω2r(x) which is supported by numerical results. 1.

Abstract. Taking r>0, let π2r(x) denote the number of prime pairs (p, p + 2r) withp ≤ x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r> 0. There seems to be no good conjecture for the remainders ω2r(x) =π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x)−li(x). However, there is a heuristic approximate formula for averages of the remainders ω2r(x) which is supported by numerical results. 1.

Abstract. Let X be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type (p, p + h], where p ≤ X is a prime number and h = o(X). Then we will apply this to prove that for every λ>1/2 there exists a positive proportion of primes p ≤ X such that the interval (p, p+λ log X] contains at least a prime number. As a consequence we improve Cheer and Goldston’s result on the size of real numbers λ> 1 with the property that there is a positive proportion of integers m ≤ X such that the interval (m, m + λ log X] contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers m ≤ X such that the interval (m, m + λ log X] contains at least a prime number. The last applications of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes p ≤ X such that the interval (p, p + λ log X] containsnoprimes. 1.