Bounded mean oscillation and the distribution of primes

Abstract

As usual, let π(x) denote the number of prime numbers [les ]x and ψ(x) the well known Chebyshev's function. Let E(x) denote either (ψ(x)−x)/√x or (π(x)−li x)/(√x/log x), x[ges ]2. The study of E occupies a central place in the theory of primes. A classical result of Littlewood [7] states that E(x)=Ω_±(log log log x) as x tends to infinity, showing in particular that E is unbounded. We expect rather erratic behaviour of E, but still one can wonder if it belongs to one of the classic function spaces [Xscr ], necessarily containing some unbounded functions. Let us extend definition of E(x) for x<2 by putting E(x)=0. A natural question is if it belongs to BMO, the space of functions with bounded mean oscillation, see, e.g. [2]. A locally integrable function f on the real line belongs to BMO if there exists a constant C such that for every bounded interval I⊂R we have

formula here

with a suitable constant α_I∈R. [mid ]I[mid ] denotes here the length of I. Without any loss in generality one can take

formula here

the average of f over I (cf. [2], chapter VI). BMO is important and intensely studied in the complex analysis. It is obvious that BMO is larger than the space of bounded (measurable) functions and thus it seems a natural candidate for [Xscr ]. E∈BMO would mean that E behaves in a certain predictable way. Otherwise, we obtain another confirmation of the big irregularity in the distribution of primes.