D. A. Goldston

Primes in tuples I

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, lim inf n→∞ Pn+1-Pn/log Pn/log = 0. We will quantify this result further in a later paper.

Pair Correlation of Zeros and Primes in Short Intervals

In 1943, A. Selberg [15] Deduced From The Riemann Hypothesis (Rh) that

SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES

\int\limits_{\rm{1}}^{\rm{X}} {{{\left( {\psi \left( {\left( {{\rm{1 + }}\delta } \right){\rm{x}}} \right){\rm{ - }}\psi \left( {\rm{x}} \right){\rm{ - }}\delta {\rm{x}}} \right)}^2}{{\rm{x}}^{{\rm{ - 2}}}}{\rm{dx}} \ll \delta {{\left( {{\mathop{\rm l}\nolimits} {\rm{ogX}}} \right)}^2}}

On the function S(T) in the theory of the Riemann zeta-function

(1) for X–1 ≤ δ ≤ X–1/4, X ≥ 2. Selberg was concerned with small values of δ and the constraint δ ≤ X–1/4 was imposed more for convenience than out of necessity. For Larger δ we have the following result.

Simple zeros of the Riemann zeta-function

Let qn denote the nth number that is a product of exactly two distinct primes. We prove that qn+1 - qnle; 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if ? is any positive integer, then (qn+1 - qn) ≤ e-γ(1 + o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms.

A differential delay equation arising from the sieve of Eratosthenes

Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim inf(qn+1―q n ) n→∞ 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.

Sieving the positive integers by large primes

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 importantly, he conjectured on number-theoretic grounds an asymptotic formula for the pair correlation of zeros and found that the form of this correlation exactly agreed with the Gaussian Unitary Ensemble (GUE) model for random Hermitian matrices which had been studied earlier by physicists. He was therefore able to formulate a general n-correlation conjecture for zeros. During the 1980s Odlyzko [23, 24] performed extensive numerical calculations of the correlations for zeros in ranges up to the 10th zero and found excellent agreement with the GUE model. More recently Hejhal [17] was able to prove the same partial result for triple correlation as Montgomery proved for pair correlation, and Rudnick and Sarnak [25] have done the same for n-correlation. Rudnick and Sarnak extended their results to a large class of L-functions, and also showed that the Riemann Hypothesis (RH) is not needed for smoother forms of the asymptotic result. Very recently Bogomolny and Keating [1, 2] have used a prime-twin type conjecture to derive heuristically the n-correlation conjecture beyond the range where the results of Rudnick and Sarnak apply. The conclusion of all this work is to ®rmly establish (but not prove) the GUE distribution for zeros of many zeta-functions. There is a dual relationship between zeros of the Riemann zeta-function and prime numbers. Following Montgomerys work, it was realized that information on pair correlation of zeros could be used to obtain information on primes. This connection was developed by Gallagher and Mueller [10] and Heath-Brown [15]. Later Goldston and Montgomery [14] found an equivalence under the Riemann Hypothesis between the pair correlation of zeros and the variance for the number of primes in short intervals. This equivalence arises out of the explicit formula via a Parseval relation, together with a Tauberian theorem. From this work one sees that to obtain new information on primes from zeros will require some new insight on the zeros and, while the connections to statistical physics mentioned above are a possible source of this insight, so far no progress has been made on this fundamental problem. The motivation for this paper is the observation that there are additional tools

Positive proportion of small gaps between consecutive primes

Abstract The function S(T) is the error term in the formula for the number of zeros of the Riemann zeta-function above the real axis and up to height T in the complex plane. We assume the Riemann hypothesis, and examine how well S(T) can be approximated by a Dirichlet polynomial in the L2 norm.