Andrew Granville's Home Page

1992 Publications

Limitations to the equi-distribution of primes III (with John Friedlander)
Compositio Mathematicae, 81 (1992), 19-32.

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.


Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle,
American Mathematical Monthly , 99 (1992), 318-331; (Corrigendum) 104 (1997), 848-851.

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 \( f(x_1^k,\ldots, x_n^k)=0\)


Squares in arithmetic progressions (with Enrico Bombieri and Janos Pintz)
Duke Mathematical Journal, 66 (1992), 165-204.

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 \( 1, 25, 49,\ldots, 24N-23\).


Computation on the first factor of the class number of cyclotomic fields (with Gilbert Fung and Hugh C. Williams)
Journal of Number Theory, 42 (1992), 297-312.

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


On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters,
Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno,Salerno, Italy, 1992, 157-194.

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.


Relevance of the residue class to the abundance of primes (with John Friedlander)
Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Univ. Salerno, Salerno, Italy, 1992, 95-104.

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.


Primality testing and Carmichael numbers,
Notices of the American Mathematical Society, 39 (1992), 696-700.

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