Andrew Granville

DEFECT ZERO P-BLOCKS FOR FINITE SIMPLE GROUPS

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p-blocks remained unclassified were the alternating groups An. Here we show that these all have a p-block with defect 0 for every prime p ≥ 5. This follows from proving the same result for every symmetric group Sn, which in turn follows as a consequence of the t-core partition conjecture, that every non-negative integer possesses at least one t-core partition, for any t ≥ 4. For t ≥ 17, we reduce this problem to Lagranges Theorem that every non-negative integer can be written as the sum of four squares. The only case with t < 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for t-core partitions in terms of modular forms, and then controlling the size of the coefficients using Delignes Theorem (nee the Weil Conjectures). We also consider congruences for the number of p-blocks of Sn, proving a conjecture of Garvan, that establishes certain multiplicative congruences when 5 < p < 23. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime p and positive integer m, the number of p-blocks with defect 0 in Sn is a multiple of m for almost all n. We also establish that any given prime p divides the number of p-modularly irreducible representations of Sn, for almost all n.

Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters

Large character sums

Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n infinity and q -> infinity (q is the size of the finite field).

Limitations to the Equi-Distribution of Primes I

In an earlier paper FG] we showed that the expected asymptotic formula (x; q; a) (x)==(q) does not hold uniformly in the range q < x= log N x, for any xed N > 0. There are several reasons to suspect that the expected asymptotic formula might hold, for large values of q, when a is kept xed. However, by a new construction, we show herein that this fails in the same ranges, for a xed and, indeed, for almost all a satisfying 0 < jaj < x= log N x.

Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients

The distribution of squarefree binomial coefficients . For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they appear. It is evident that, if k is not too small, then must be highly composite in that it contains many prime factors and often to high powers. It is therefore of interest to enquire as to how infrequently is squarefree. One well-known conjecture, due to Erdős, is that is not squarefree once n > 4. Sarkozy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

It is easy to determine whether a given integer is prime

“The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers ... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated ... It is in the nature of the problem that any method will become more complicated as the numbers get larger. Nevertheless, in the following methods the difficulties increase rather slowly ... The techniques that were previously known would require intolerable labor even for the most indefatigable calculator.” —from article 329 of Disquisitiones Arithmeticae (1801) by C. F. Gauss There are few better known or more easily understood problems in pure mathematics than the question of rapidly determining whether a given integer is prime. As we read above, the young Gauss in his first book Disquisitiones Arithmeticae regarded this as a problem that needs to be explored for “the dignity” of our subject. However it was not until the modern era, when questions about primality testing and factoring became a central part of applied mathematics, that there was a large group of researchers endeavoring to solve these questions. As we shall see, most of the key ideas in recent work can be traced back to Gauss, Fermat and other mathematicians from times long gone by, and yet there is also a modern spin: With the growth of computer science and a need to understand the true difficulty of a computation, Gauss’s vague assessment “intolerable labor” was only recently Received by the editors January 27, 2004, and, in revised form, August 19, 2004. 2000 Mathematics Subject Classification. Primary 11A51, 11Y11; Secondary 11A07, 11A41, 11B50, 11N25, 11T06. L’auteur est partiellement soutenu par une bourse du Conseil de recherches en sciences naturelles et en genie du Canada. Because of their use in the data encryption employed by public key cryptographic schemes;

Unexpected Irregularities in the Distribution of Prime Numbers

In 1849 the Swiss mathematican ENCKE wrote to GAUSS, asking whether he had ever considered trying to estimate Π(x), the number of primes up to x, by some sort of “smooth” function. On Christmas Eve 1849, GAUSS replied that “he had pondered this problem as a boy” and had come to the conclusion that “at around x, the primes occur with density 1/log x.” Thus, he concluded, π(ϰ) could be approximated by

Decay of Mean Values of Multiplicative Functions

Li(x): = \int_2^x {\frac{{dt}}{{\log t}} = \frac{x}{{\log x}} + \frac{x}{{{{\log }^2}x}} + O(\frac{x}{{{{\log }^3}x}})}