The function field Sathe--Selberg formula in arithmetic progressions and `short intervals'

Arithmetic Progressions of Primes in Short Intervals

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: If N is sufficiently large and M is not too small compared with N , then the primes in the interval [N, N + M ] contains many arithmetic progressions of length k.

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

For any > 0 and any non-exceptional modulus q ≥ 3, we prove that, for x large enough (x ≥ α log q), the interval [ ex, ex+ ] contains a prime p in any of the arithmetic progressions modulo q. We apply this result to establish that every integer n larger than exp(71 000) is a sum of seven cubes.

The partition function in arithmetic progressions

In celebration of G.E. Andrews' 60 th birthday.

Integers without large prime factors in short intervals and arithmetic progressions

Primes in Short Segments of Arithmetic Progressions

Λ is the von Mangoldt function, and ∑ * a(q) denotes a sum over a set of reduced residues modulo q. We shall assume throughout x ≥ 2, 1 ≤ q ≤ x, 1 ≤ h ≤ x, (1.3) the other ranges being without interest. As far as we are aware the only known result concerning the general function I(x, h, q) is due to Prachar [11], who showed that, assuming the Generalized Riemann Hypothesis (GRH) I(x, h, q) ≪ hx...