. 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 DMS-9502952. 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 S-T E X Mauldin and Urba'nski Page 1 x1. Introduction: Setting and Notation Let I be a nonempty subset of N , the se...

We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. grant 1. Introduction For any x ? x 0 and for any positive valued function g(x) define R(x) = e fl log x log 2 x log 4 x=(log 3 x) 2 ; L(x) = exp(log x log 3 x= log 2 x) and E g (x) = exp \Gamma log x=(log 2 x) g(x) \Delta : Here log k x is the k-fold iterated logarithm, fl is Euler's constant, and x 0 is chosen large enough so that log 4 x 0 ? 1. The usual method used to find large gaps between successive prime numbers is to construct a long sequence S of consecutive integers, each of which has a "small" prime factor (so that they cannot be prime); then, the gap between the largest prime before S and the next, is at least as long as S. Similarly if one wishes to find an arithm...

. Call a set of integers fb1 ; b2 ; : : : ; bk g admissible if for any prime p, at least one congruence class modulo p does not contain any of the b i . Let ae (x) be the size of the largest admissible set in [1; x]. The Prime k-tuples Conjecture states that any for any admissible set, there are infinitely many n such that n+b1 ; n+b2 ; : : : n+bk are simultaneously prime. In 1974, Hensley and Richards [3] showed that ae (x) ? ß(x) for x sufficiently large, which shows that the Prime k-tuples Conjecture is inconsistent with a conjecture of Hardy and Littlewood that for all integers x; y 2, ß(x + y) ß(x) + ß(y): In this paper we examine the behavior of ae (x), in particular, the point at which ae (x) first exceeds ß(x), and its asymptotic growth.

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decades, one of the world’s foremost experts in modular forms, died on January 27, 2001 in Glasgow at the age of 85. He was one of the founding editors of The Ramanujan Journal. For this and the next issue of the The Ramanujan Journal, many well-known mathematicians have prepared articles in Rankin’s memory. In this opening paper, we provide a short biography of Rankin and discuss some of his major contributions to mathematics. At the conclusion of this article, we provide a complete list of Rankin’s doctoral students and a complete bibliography of all of Rankin’s writings divided into

One can apply Bonse’s inequality to points on the real line to find an upper bound on any prime gap, including both for first occurrences and for maximal prime gaps. However, such a result is neither as fine as the upper bound found by Mozzochi, nor as fine as the lower bound obtained by Rankin for maximal prime gaps. Without deep sieve methods, such as those used by Maier and Pomerance to compute a lower bound for maximal prime gaps, we show one can use Bonse’s inequality to arrive at an upper bound for any given prime gap without intricate derivations for any real constants. 1 Introduction and a new upper bound Γ(pk). Iwaniec, Pintz, and later Mozzochi [5] found good upper bounds on the difference between two consecutive primes, namely, pk+1 − pk ≪ p θ k, where either θ = 11 1 11 1

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Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s conjecture is still widely open. In this paper we prove that Koblitz’s conjecture is true on average over a two-parameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of Barban-Davenport-Halberstam,