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 αI∈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.