On the normal density of primes in short intervals

Selberg has shown on the basis of the Riemann hypothesis that for every ε > 0 most intervals |x,x+xϵ| of length xϵ contain approximately xϵlogx primes. Here by “most” we mean “for a set of values of x of asymptotic density one.” Prachar has extended Selberg's result to primes in arithmetic progressions. Both authors noted that if we assume the quasi Riemann hypothesis, that ζ(s) has no zeros in the domain {σ>12+δ} for some δ 2 δ. Here we give a simple proof of these theorems in a general context, where an arbitrary signed measure takes the place of d[ψ(x)−x]. Then we show by a counterexample that this general theorem is the best of its kind: the condition ε > 2δ cannot be replaced by ε = 2δ. In our example, the associated Dirichlet integral is an entire function which remains bounded on the domain {σ≥12+δ}. Thus its growth and regularity properties are better than those of ζ′(s)ζ(s). Nevertheless the corresponding signed measure behaves badly.