A long-standing conjecture claims that the Diophantine equation

has finitely many solutions, and, maybe, only those given by

It is well known that, if p is prime, the multiplicative group (ℤ/pℤ)* of reduced residues is cyclic. A generator is called a primitive root; there are φ(p − 1) of them, where φ is Euler's function. Thus, (φ(p − 1)/(p−1) is the proportion of primitive roots modulo p in (ℤ/pℤ)*. Elliott has proved that φp − 1)/(p − 1) has a limiting distribution function [2], in the sense that

exists for all real numbers u.

Assuming the abc-conjecture, it is shown that there are only finitely many powerful binomial coefficients with 3≤k≤n/2 in fact, if q2 divides , then . Unconditionally, it is shown that there are N1/2+σ(1) powerful binomial coefficients in the top N rows of Pascal's Triangle.

Let φ(n) be the Euler function (i.e., φ(n) denotes the number of integers less than n which are relatively prime to n), and define

These functions were extensively studied by several mathematicians. One of the problems investigated concerns their sign changes. We say that a function fx) has a sign change at x = x0 if f(x0 −) f(x0 +) < 0, and f(x) has a sign change on the integer n if (n)f(n+1) < 0. The numbers of sign changes and sign changes on integers of f(x) in the interval [1, T] are denoted by Xf(T) and Nf(T), respectively.

We are interested in the distribution of those zeros of the Riemann zeta-function which lie on the critical line ℜs = ½, and the maxima of the function between successive zeros. Our results are to be independent of any unproved hypothesis. Put

In this paper it is proved that, for x sufficiently large, every integer m with

can be written as m = Σ1≤n≤xεn/n, where εi, = 0 or 1.

In 1984 Heath-Brown [2] proved the conjecture of Erdős and Mirsky [1], and obtained the following result.

Theorem. There are infinitely many integers n for which d(n+ 1)= d(n). Indeed, for large x, the number of such n≤x is at least of order x ln-7 x.

A new version of Erdös-Turán's inequality is described. The purpose of the present paper is to show that the inequality provides better upper bounds for the discrepancies of some sequences than usual Erdös-Turán's inequality.

Investigations concerning the gaps between consecutive prime numbers have long occupied an important position on the interface between additive and multiplicative number theory. Perhaps the most famous problem concerning these gaps, the Twin Prime Conjecture, asserts that the aforementioned gaps are infinitely often as small as 2. Although a proof of this conjecture seems presently far beyond our reach (but see [5] and [10] for related results), weak evidence in its favour comes from studying unusually short gaps between prime numbers. Thus, while it follows from the Prime Number Theorem that the average gap between consecutive primes of size about x is around log x, it is now known that such gaps can be infinitely often smaller than 0–249 log x (this is a celebrated result of Maier [12], building on earlier work of a number of authors; see in particular [7], [13], [3] and [11]). A conjecture weaker than the Twin Prime Conjecture asserts that there are infinitely many gaps between prime numbers which are powers of 2, but unfortunately this conjecture also seems well beyond our grasp. Extending this line of thought, Kent D. Boklan has posed the problem of establishing that the gaps between prime numbers infinitely often have only small prime divisors, and here the latter divisors should be small relative to the size of the small gaps established by Maier [12]. In this paper we show that the gaps between consecutive prime numbers infinitely often have only small prime divisors, thereby solving Boklan's problem. It transpires that the methods which we develop to treat Boklan's problem are capable also of detecting multiplicative properties of more general type in the differences between consecutive primes, and this theme we also explore herein.

Let Tβ be the β-transformation on [0, 1). When β is an integer Tβ is ergodic with respect to Lebesgue measure and almost all orbits {} are uniformly distributed. Here we consider the non-integer case, determine when Tα, Tβ have the same invariant measure and when (appropriately normalised) orbits are uniformly distributed.

In the paper [5] bounds are found for the sum,

for a suitable Dirichlet character χ mod r, and real functionf(x). The proofs in that paper use Bombieri and Iwaniec's method [1], one formulation of which has as part of its first step the estimation of S in terms of a sum of many shorter sums of the form,

where e(x) = exp (2πix), mi∈ [M, 2M], and each mi, lies in its own interval, of length N ≥ M/4, that is disjoint from those of the others. This paper addresses a problem springing from above: to bound the numbers of ‘similar’ pairs, Si+, Si+, satisfying both

and

where ‖x‖ = min{|x − n|: n ∈ ℤ}. Lemma 5.2.1 of [3] (partial summation) shows that each sum in a similar pair is a bounded multiple of the other.

Let k be a positive integer and g(k) be the least integer n > k + 1 such that all prime factors of are >k. We prove that

where c is an absolute positive constant. We also establish a new theorem on the distribution of fractional parts of a smooth function.

A general analytic scheme for Poisson approximation to discrete distributions is studied in which the asymptotic behaviours of the generalized total variation, Fortet-Mourier (or Wasserstein), Kolmogorov and Matusita (or Hellinger) distances are explicitly characterized. Applications of this result include many number-theoretic functions and combinatorial structures. Our approach differs from most of the existing ones in the literature and is easily amended for other discrete approximations; arithmetic and combinatorial examples for Bessel approximation are also presented. A unified approach is developed for deriving uniform estimates for probability generating functions of the number of components in general decomposable combinatorial structures, with or without analytic continuation outside their circles of convergence.

Let p(n) be the usual partition function. Let l be an odd prime, and let r (mod t) be any arithmetic progression. If there exists an integer n ≡ r (mod t) such that p(n) ≢ 0 (mod l), then, for large X,

For a point set in the multidimensional unit torus we introduce an Lk-measure of uniformity of distribution, which for k=2 reduces to diaphony (and thus in this case essentially coincides with Weyl L2-discrepancy). For k ∈ [1, 2] we establish a sharp asymptotic for this new measure as the number of points of the set tends to infinity. Upper and lower-bound estimates are given also for k >2.

In 1918 Pólya and Vinogradov established the estimate for Dirichlet character sums that currently carries their names. It was forty years until Burgess gave an improvement of their bound [1], and it is forty years since that improvement.

The question as to which “natural” sequences contain infinitely many primes is of considerable fascination to the number-theorist. One such “natura” sequence is [nc[ where [·] denotes integer part. Piatetski-Shapiro [10] showed that there are infinitely many primes in this sequence for 1 < c < 12/11, obtaining the expected asymptotic formula for the number of such primes. The exponent 12/11 has been increased gradually by a number of authors to the present record 45/38 held by Kumchev [9]. It is expected that there are infinitely many primes of the form [nc[ for all cεε[1, ∞)/ℤ. Deshouillers [3] showed that this is almost always true, in the sense of Lebesgue measure on [1, ∞). Balog [2] improved and generalized this result to show that, for almost all c > 1,

where

As usual in Waring's problem, we let G(k) be the least number s such that all sufficiently large natural numbers can be written as the sum of s or fewer k-th powers of positive integers. Hardy and Littlewood gave the first general upper bound, G(k)≤(k − 2)2k − 1 + 5. Later [4], they reduced this to order k2 by first constructing an auxiliary set of natural numbers below x which are sums of sk-th powers, of cardinality with for large k. The argument, which is brief and elementary (see [15, Chapter 6]), is now referred to by R. C. Vaughan's term “diminishing ranges”, since the k-th powers are taken from intervals of decreasing lengths. This idea of choosing the variables from restricted ranges was refined by Vinogradov, whose application to exponential sum estimates gave for large k (see [18]). The recent iterative method of Vaughan and Wooley [16, 19, 21], which halves Vinogradov's bound, may be viewed as an evolved diminishing ranges argument, producing an auxiliary set with .

It is known that a system of r additive equations of degree k with greater than 2rk variables has a non-trivial p-adic solution for all p > k2r + 2. In this paper we consider the same system with more than crk variables, c > 2, and show the existence of a non-trivial solution for all p > r2k2+(2/(c − 2)) if r ≠ 1 and p > k2+(2/(c − 1)) if r = 1.

Recent advances in the theory of exponential sums (see, for example, [6], [7], [8], [12]) have contributed to corresponding progress in our understanding of the solubility of systems of simultaneous additive equations (see, in particular, [1], [2], [3], [4]). In a previous memoir [11] we developed a version of Vaughan's iterative method (see Vaughan [8]) suitable for the analysis of simultaneous additive equations of differing degrees, discussing in detail the solubility of simultaneous cubic and quadratic equations. The mean value estimates derived in [11] are, unfortunately, weaker than might be hoped, owing to the presence of undesirable singular solutions in certain auxiliary systems of congruences. The methods of [12] provide a flexible alternative to Vaughan's iterative method, and, as was apparent even at the time of their initial development at the opening of the present decade, such ideas provide a means of avoiding altogether the aforementioned problematic singular solutions. The systematic development of such an approach having been described recently in [15], in this paper we apply such methods to investigate the solubility of pairs of additive equations, one cubic and one quadratic, thereby improving the main conclusion of [11].