This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a non-trivial problem to actually do so. In particular so that the constants involved do not render the explicit estimates useless in practical applications. We have used the practical bounds that are needed to prove Theorem 1 as motivation for our results here, though we hope that this work will be applicable to a variety of other problems which routinely apply these or related exponential sum estimates. In particular our results here can be used to say something about the questions of estimating the number of integers free of large prime factors in short intervals (see [FL]), and of the largest prime factor of an integer in an interval (see [J]). Our key result is

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: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fellow. Supported, in part, by the National Science Foundation Integers, without large prime factors, in arithmetic progressions, II Andrew Granville 1. Introduction. The study of the distribution of integers with only small prime factors arises naturally in many areas of number theory; for example, in the study of large gaps between prime numbers, of values of character sums, of Fermat's Last Theorem, of the multiplicative group of integers modulo m, of S--unit equations, of Waring's problem, and of primality testing and factoring algorithms. For over sixty years this subject has received quite a lot of attention from analytic number theorists and we have recently begun to attain a very pre...

this paper we shall study the complex zeros of f p (t). Using zero locating software one finds that, for primes p up to 1000, about half of the zeros lie on the unit circle; leading one to expect this to be the general phenomenon. It turns out to be fairly easy to prove that at least half of the zeros of f p (t) are on the unit circle (that is jtj = 1): First note that F p (z) := z

this paper we will apply the uniform abc-conjecture to the very large solutions of Diophantine equations that arise from modular functions and deduce a lower bound for the class number of imaginary quadratic fields. This extends an idea of Chowla [1,2] who indicated, via a conjecture of Hall, how unlikely it is that

Abstract. Seven widely spaced regions of integers with π4,3(x) <π4,1(x) have been discovered using conventional prime sieves. Assuming the generalized Riemann hypothesis, we modify a result of Davenport in a way suggested by the recent work of Rubinstein and Sarnak to prove a theorem which makes it possible to compute the entire distribution of π4,3(x) − π4,1(x) including the sign change (axis crossing) regions, in time linear in x, using zeroes of L(s, χ),χ the nonprincipal character modulo 4, generously provided to us by Robert Rumely. The accuracy with which the zeroes duplicate the distribution (Figure 1) is very satisfying. The program discovers all known axis crossing regions and finds probable regions up to 10 1000. Our result is applicable to a wide variety of problems in comparative prime number theory. For example, our theorem makes it possible in a few minutes of computer time to compute and plot a characteristic sample of the difference li(x) − π(x) with fine resolution out to and beyond the region in the vicinity of 6.658 × 10 370 discovered by te Riele. This region will be analyzed elsewhere in conjunction with a proof that there is an earlier sign change in the vicinity of 1.39822 × 10 316. 1.

: We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that `(x; q; a) ¸ 2x=OE(q) for each a 62 H and `(x; q; a) = o(x=OE(q)), otherwise. From here, we deduce that if the second case holds at all, then it holds only for the multiples of some fixed integer q 0 ? 1. Actually, even if the Prime Number Theorem for arithmetic progressions, modulo q, does hold, these methods allow us to deduce the behaviour of a possible `Siegel zero' from Selberg's formula. We also propose a new method for determining explicit upper and lower bounds on `(x; q; a), which uses only elementary number theoretic computations. 1. Introduction. Define `(x) = P px log p, where p only denot...

Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n> 1. We show that, for every real number 32/17 < η < (4 + 3 √ 2)/4, there exists a constant c(η)> 1 such that for every integer a � = 0, the set � p ∈ P: p = P(q − a) for some prime q with p η < q < c(η) p η � has relative asymptotic density one in the set of all prime numbers. Moreover, in the range 2 ≤ η < (4+3 √ 2)/4, one can take c(η) = 1+ε for any fixed ε> 0. In particular, our results imply that for every real number 0.486 ≤ ϑ ≤ 0.531, the relation P(q − a) ≍ q ϑ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisors of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ → P(q − a) for a> 0, and show that for infinitely many primes q, this map can be iterated at least (log log q) 1+o(1) times before it terminates. 1.

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant D ? 3705, there is always at least one prime p ! p D=2 such that the Kronecker symbol (D=p) = \Gamma1. 1991 Mathematics Subject Classification 11R11, 11Y40 The first author is a Presidential Faculty Fellow. His research is partiallly supported by the NSF. The research of the second two authors is partially supported by NSERC of Canada 1 1 Introduction Let D be the fundamental discriminant of a real quadratic field and let S = f5; 8; 12; 13; 17; 24; 28; 33; 40; 57; 60; 73; 76; 88; 97; 105; 124; 129; 136; 145; 156; 184; 204; 249; 280; 316; 345; 364; 385; 424; 456; 520; 609; 616; 924; 940; 984; 1065; 1596; 2044; 2244; 3705g: At the end of Chapter 6 of [5], the second author made the following conjecture. Conjecture. The values of D for which the least prime p such that the Kronecker symbol (D=p) = \Gamma1 satisfies p ? p D=2 are precisely those in S. He also veri...

this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is well-known (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n