Integer division has been known to lie in P-uniform TC 0 since the mid-1980's, and recently this was improved to L- uniform TC 0 . At the time that the results in this paper were proved and submitted for conference presentation, it was unknown whether division lay in DLOGTIME-uniform TC 0 (also known as FOM). We obtain tight bounds on the uniformity required for division, by showing that division is complete for the complexity class FOM + POW obtained by augmenting FOM with a predicate for powering modulo small primes. We also show that, under a well-known number-theoretic conjecture (that there are many "smooth" primes), POW (and hence division) lies in FOM. Building on this work, Hesse has shown recently that division is in FOM [17].

The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups. 1.

We begin by explaining the basic idea of this paper in a simple case. We write np for the order of 2 modulo the prime p, so that np is the number of powers of 2 which are distinct mod p. We have the elementary bounds logp < £ np ^ p-1.

Using an elementary approach based on careful handlings of Cauchy integrals, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Moreover, we adapt an old result of Rademacher to get a convergent series expansion of these Fourier coecients and we show that this expansion allows to nd again these estimates. Our results improve previous ones by K. Mahler and O. Herrmann. In particular, we show that the Fourier coefficients of j are smaller than their asymptotically equivalent given by Petersson and Rademacher.

We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that “on average” these sums are close to the expected value. We also give applications of our result to sums of the divisor function twisted with characters (both additive and multiplicative) taken on the values of various functions, such as rational and exponential functions; in particular, we obtain upper bounds for such twisted sums. 1

We obtain rigorous upper bounds on the number of primes x for which p-1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications.

Introduction Let r be a positive integer. We say that r non-zero integers a 1 , ..., a r are multiplicatively independent if whenever there exist m 1 , ..., m r # Z such that r =1, it follows that m 1 =}}}=m r =0. We assume that none of a 1 , ..., a r is a perfect square or \1; let 1 denote the subgroup of Q* generated by a 1 , ..., a r and let |1 p | denote the order of such a group 1 (mod p). In the case r=1, 1=(a), let ord p (a) denote the order of a (mod p). The famous Artin Conjecture for primitive roots (see [1]) states that ord p (a)=p&1 for infinitely many primes p. Artin's Conjecture has been proved under the assumption of the Generalized Riemann Hypothesis by C. Hooley (See [13]). In his paper it is implicitly shown (unconditionally) that ord p (a)>-p#log p (1.1) for all but O(x#log x) primes p#x. article no. 0044 207 0022-314X#96 #18.00 Copyright # 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * Supported in part by C.I.C.M.A.

possible generalization of

with arithmetic weights attached to one of the variables