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20
Conformal Iterated Function Systems With Applications To The Geometry Of Continued Fractions
, 1998
"... . In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set o ..."
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Cited by 30 (9 self)
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. In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries. 1 Research supported by NSF Grant DMS9502952. AMS(MOS) subject classifications(1980). Primary 28A80; Secondary 58F08, 58F11, 28A78 Key words and phrases. Iterated function systems, continued fractions, Hausdorff dimension, Hausdorff and packing measures, arithmetic densities. Typeset by A M ST E X Mauldin and Urba'nski Page 1 x1. Introduction: Setting and Notation Let I be a nonempty subset of N , the se...
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.
Chains of large gaps between consecutive primes
 Adv. in Math
, 1981
"... ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx log log log 10gx(loglog logx)2, where c = eY and y is Euler's constant. Here, this result is shown with ..."
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Cited by 11 (3 self)
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ABSTRACT. Let G(x) denote the largest gap between consecutive grimes below x, In a series of papers from 1935 to 1963, Erdos, Rankin, and Schonhage showed that G(x):::: (c + o ( I)) logx loglogx log log log 10gx(loglog logx)2, where c = eY and y is Euler's constant. Here, this result is shown with c = coe Y where Co = 1.31256... is the solution of the equation 4 / Co e4/co = 3. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus. 1.
Some remarks on Number Theory
 Israel Journal of Mathematics
, 1962
"... This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1
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Cited by 9 (1 self)
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This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1<j<co be an infinite sequence of numbers on the unit circle. Put
Yıldırım, Small gaps between primes or almost primes
"... Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of ex ..."
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Cited by 8 (2 self)
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Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→ ∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of the ElliottHalberstam Conjecture is true, then the above bound can be improved to 6. 1.
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.
On the incompatibility of two conjectures concerning prime numbers
 Proc. Symp. Pure Math. (Analytic Number Theory
, 1972
"... Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a measure of good luck. Several instances of this will emerge in due course, but one example now may give the ide ..."
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Cited by 6 (0 self)
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Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a measure of good luck. Several instances of this will emerge in due course, but one example now may give the idea: The speaker and his coworker, Douglas Hensley,
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).