Number of prime ideals in short intervals

Abstract

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramer on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun-Titchmarsh theorem.