This paper presents a new probabilistic primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high prob-ability or dependent on an unproven assumption. Thc test terminates in expected polynomial time on all but at most an exponentially vanishing fraction of the inputs of length k, for every k. This result implies: • There exist an infinite set of primes which can be recognized in expected polynomial time. • Large certified primes can be generated in expected polynomial time. Under a very plausible condition on the distribution of primes in "small" intervals, the proposed algorithm can be shown'to run in expected polynomial time on every input. This

“It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random ’ means. ” — R. C. Vaughan (February 1990). After the first world war, Cramér began studying the distribution of prime numbers, guided by Riesz and Mittag-Leffler. His works then, and later in the midthirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramér’s ideas have directed and motivated research ever since. One can only fully appreciate the significance of Cramér’s contributions by viewing his work in the appropriate historical context. We shall begin our discussion with the ideas of the ancient Greeks, Euclid and Eratosthenes. Then we leap in time to the nineteenth century, to the computations and heuristics of Legendre and Gauss, the extraordinarily analytic insights of Dirichlet and Riemann, and the crowning glory of these ideas, the proof the “Prime Number Theorem ” by Hadamard and de la Vallée Poussin in 1896. We pick up again in the 1920’s with the questions asked by Hardy and Littlewood,

Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN ∈P = ⇒ P =NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication HN ̸∈P = ⇒ P ̸=NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence on the underlying field, while the second can be interpreted as a quantitative statement on the higher moments of the zeroes of Dedekind zeta functions. In particular, both assumptions can still hold even if GRH is false. We thus obtain a new application of Dedekind zero estimates to computational algebraic geometry. Along the way, we also apply recent explicit algebraic and analytic estimates, some due to Silberman and Sombra, which may be of independent interest.

In this paper we prove that inf max n∑ z ν k ∣ = √ n + O ( n 0.2625+ǫ). (ǫ> 0) |zk|≥1 ν=1,...,n2 k=1 This improves on the bound inf max n∑ |zk|≥1 ν=1,...,n2 z k=1 ν k ∣ ≤ √ 6n log(1 + n2) of Erdős and Renyi. In the special case of n + 1 being a prime we have previously obtained the much sharper result n ≤ inf max n∑ z ν k

The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d−1 is a prime power. In this paper, we consider the case when d − 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of “almost ” Ramanujan graphs for almost every value of d. More precisely, for any fixed ǫ> 0 and for almost every value of d (in the sense of natural density), there are infinitely many d-regular graphs such that all the non-trivial eigenvalues of the adjacency matrices of these graphs have absolute value less than (2 + ǫ) √ d − 1. 1

Abstract not found

: In an earlier paper [FG] we showed that the expected asymptotic formula ß(x; q; a) ¸ ß(x)=OE(q) does not hold uniformly in the range q ! x= log N x, for any fixed N ? 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept fixed. However, by a new construction, we show herein that this fails in the same ranges, for a fixed and, indeed, for almost all a satisfying 0 ! jaj ! x= log N x. 1. Introduction. For any positive integer q and integer a coprime to q, we have the asymptotic formula (1:1) ß(x; q; a) ¸ ß(x) OE(q) as x ! 1, for the number ß(x; q; a) of primes p x with p j a (mod q), where ß(x) is the number of primes x, and OE is Euler's function. In fact (1.1) is known to hold uniformly for (1:2) q ! log N x and all (a; q) = 1, for every fixed N ? 0 (the Siegel--Walfisz Theorem), for almost all q ! x 1=2 = log 2+" x and all (a; q) = 1 (the Bombieri--Vinogradov Theorem) and for almost all q !...

Several results involving d(n!) are obtained, where d(m) denotes the number of positive divisors of m. These include estimates for d(n!)/d((n - 1)!), d(n!) - d((n - 1)!), as well as the least number K with d((n +K)!)/d(n!) # 2. 1

We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)-#(x-y) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available zero-free regions for the Riemann zeta-function, and also on the strength of density bounds for the zeros themselves. We also study implications in the opposite direction: assuming that an asymptotic formula like the above is valid for almost all x in a given range of values for y, we find zero-free regions or density bounds.

In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zeta-function.