Halász’s theorem gives an upper bound for the mean value of a multiplicative function $f$ . The bound is sharp for general such $f$ , and, in particular, it implies that a multiplicative function with $|f(n)|\leqslant 1$ has either mean value $0$ , or is ‘close to’ $n^{it}$ for some fixed $t$ . The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s theorem).

Let $f$ and $g$ be 1-bounded multiplicative functions for which $f\ast g=1_{.=1}$ . The Bombieri–Vinogradov theorem holds for both $f$ and $g$ if and only if the Siegel–Walfisz criterion holds for both $f$ and $g$ , and the Bombieri–Vinogradov theorem holds for $f$ restricted to the primes.

We show that smooth-supported multiplicative functions $f$ are well distributed in arithmetic progressions $a_{1}a_{2}^{-1}\;(\text{mod}~q)$ on average over moduli $q\leqslant x^{3/5-\unicode[STIX]{x1D700}}$ with $(q,a_{1}a_{2})=1$ .

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$ , with $p,q$ both prime, satisfy $p\equiv q\equiv 3~(\text{mod}\,4)$ .

We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length $t{{R}^{1/3}}$ on a circle of radius $R$ , for any given $t\,>\,0$ . In particular we prove that any arc of length ${{\left( 40\,+\frac{40}{3}\sqrt{10} \right)}^{1/3}}\,{{R}^{1/3}}$ on a circle of radius $R$ , with $R\,>\,\sqrt{65}$ , contains at most three lattice points, whereas we give an explicit infinite family of 4-tuples of lattice points, $\left( {{v}_{1,n}},\,{{v}_{2,n}},\,{{v}_{3,n}},\,{{v}_{4,n}} \right)$ , each of which lies on an arc of length ${{\left( 40+\frac{40}{3}\sqrt{10} \right)}^{1/3}}R_{n}^{1/3}\,+\,o\left( 1 \right)$ on a circle of radius ${{R}_{n}}$ .

For given multiplicative function $f$ , with $\left| f\left( n \right) \right|\le 1$ for all $n$ , we are interested in how fast its mean value $\left( 1/x \right)\sum{_{n\le x}f\left( n \right)}$ converges. Halász showed that this depends on the minimum $M\,\left( \text{over}\,y\,\in \,\mathbb{R} \right)\,\text{of}\,\sum{_{p\le x}\left( 1-\operatorname{Re}\left( f\left( p \right){{p}^{-iy}} \right) \right)}/p$ , and subsequent authors gave the upper bound $\ll \left( 1+M \right){{e}^{-M}}$ . For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Halász-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of $y$ that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the $k$ -th powers mod $p$ .

The $abc$ -conjecture is applied to various questions involving the number of distinct fields $\mathbb{Q}\left( \sqrt{f(n)} \right)$ , as we vary over integers $n$ .

Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, $h_1(p)$ , is further examined. Whereas Kummer conjectured that $h_1(p) \sim G(p) := 2p(p/4\pi^2)^{(p-1)/4}$ it is shown, under certain plausible assumptions, that there exist constants $a_\alpha, b_\alpha$ such that $h_1(p) \sim \alpha G(p)$ for $\sim a_\alpha x/\log^{b_\alpha} x$ primes $p \le x$ whenever $\log \alpha$ is rational. On the other hand, there are $\ll_A x/\log^A x$ such primes when $\log \alpha$ is irrational. Under a weak assumption it is shown that there are roughly the conjectured number of prime pairs $p, mp\pm 1$ if and only if there are $\gg_m x/\log^2 x$ primes $p \le x$ for which $h_1(p) \sim e^{\pm 1/2m} G(p)$ .

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 $p\,<\,\sqrt{D}/2$ such that the Kronecker symbol $(D/P)\,=\,-1$ .

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.

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. Sarközy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

Under the assumption of the prime /c-tuplets conjecture we show that it is possible to construct an infinite sequence of integers, such that the average of any two is prime.

The aim of this paper is to establish a result on a family of congruences arising from the Fermat quotient: this result has an interesting application to the Fermat problem. For over 150 years the Fermat problem has been divided into two cases; the First Case being the assertion that for each odd prime p,

has no solution in non-zero integers x, y, z. Kummer's work on ideal numbers led, in 1847, to the complete solution of the Fermat problem for regular primes.