We show that if \( a\) is fixed then there are values of \( q\leq x/(\log x)^N\), which are coprime to \( a\), such that the asymptotic \( \pi(x;q,a) \sim \pi(x)/\phi(q)\) fails to hold. Article

The number of odd entries in a row of Pascal's triangle is always a power of 2. These are either equally split between 1 and 3 mod 4, or are all 1 mod 4. Similarly, for every odd \( a\), the number of entries in a given row of Pascal's trinagle that are \( \equiv a \pmod 8\) is either 0 or a power of 2. We develop a theory to explain this. Article, Corrigendum

For a given polynomial \( f\) we use local methods to find exponents \( k\) for which there are no non-trivial integer solutions to the Diophantine equation \( f(x_1^k,\ldots, x_n^k)=0\)Article

We show that there are \( \ll N^{2/3+o(1)}\) squares in any given arithmetic progression of length \( N\). This was later improved by Bombieri and Zannier to \( \ll N^{3/5+o(1)}\); while the conjecture is that the maximum is \( \sqrt{8N/3}+O(1) \), given by
\( 1, 25, 49,\ldots, 24N-23\). Article

We compute the first factor of the class number of the \( p\)th cyclotomic field for each prime \( p\leq 3000\).Article

We show that Selberg's formula, by itself, leads to two possible behaviours for the prime number theorem in arithmetic progression. This allows us to deduce the behaviour of a possible Siegel zero using elementary methods.Article

We show strong upper bounds for the average number of primes
\( \equiv a \pmod q\) as one varies over \( q\) coprime to \( a\). Asymptotics were attained much later by Fiorilli. Article

A survey on Carmichael numbers just after it was proved that there are infinitely many Article