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22
On the distribution of spacings between zeros of the zeta function
 MATH. COMP
, 1987
"... A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first 10 5 zeros and for zeros number 10 12 + 1 to 10 12 + 10 5 that are accurate to within ± 10 − 8, and which were calculated on the Cray1 and Cray XMP compute ..."
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Cited by 86 (9 self)
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A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first 10 5 zeros and for zeros number 10 12 + 1 to 10 12 + 10 5 that are accurate to within ± 10 − 8, and which were calculated on the Cray1 and Cray XMP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave similarly to eigenvalues of random hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by
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.
Small gaps between primes
"... ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between ..."
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Cited by 7 (3 self)
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ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.
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 ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. E ..."
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Cited by 6 (1 self)
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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 ElliottHalberstam 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, pn+1 − pn lim inf =0. n→ ∞ log pn We will quantify this result further in a later paper (see (1.9) below).
SMALL GAPS BETWEEN PRIMES II (PRELIMINARY)
"... Abstract. We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (pn+1 − pn) (1.1) lim inf n→ ∞ log pn(log log pn) −1 < ∞. log log log log pn Further we show that supposing the ..."
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Cited by 2 (2 self)
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Abstract. We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (pn+1 − pn) (1.1) lim inf n→ ∞ log pn(log log pn) −1 < ∞. log log log log pn Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ Xϑ with any level ϑ>1/2 we have bounded differences between consecutive primes infinitely often: (1.2) lim inf n→ ∞ (pn+1 − pn) ≤ C(ϑ) with a constant C(ϑ) depending only on ϑ. If the Bombieri–Vinogradov theorem holds with a level ϑ>20/21, in particular if the Elliott–Halberstam conjecture holds, then we obtain (1.3) lim inf n→ ∞ (pn+1 − pn) ≤ 20, that is pn+1 − pn ≤ 20 for infinitely many n. Inequalities (1.2)–(1.3) will follow from the even stronger following result Theorem A. Suppose the Bombieri–Vinogradov theorem is true for Q ≤ Xϑ with some ϑ>1/2. Then there exists a constant C ′ (ϑ) such that any admissible ktuple contains at least two primes for any (1.4) k ≥ C ′ (ϑ) if ϑ>1/2, where C ′ (ϑ) is an explicitly calculable constant depending only on ϑ. Further we have at least two primes for (1.5) k =7 if ϑ>20/21. Remark. For the definition of admissibility see (2.2) below. We will show some more general results for the quantity (ν is a given positive integer) (1.6) Eν = lim inf n→∞ pn+ν − pn log pn
Iterated absolute values of differences of consecutive primes
 Mathematics of Computation
, 1993
"... Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that verified this ..."
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Cited by 2 (0 self)
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Abstract. Let dç,(n) = p „ , the nth prime, for n> 1, and let dk+x(n) = \dk(n) dk(n + 1)  for k> 0, n> 1. A wellknown conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that dk(\) = 1 for all k> 1. This paper reports on a computation that verified this conjecture for k < tt(1013) » 3 x 10 ". It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences. 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.
PRIME CHAINS AND PRATT TREES
"... ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains wit ..."
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Cited by 1 (0 self)
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ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x. 1.