On the divisor function in short intervals

Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formula

$begin{array}{lll}sum limits_{x < n leq x+h(x)}d(n)sim h(x)log xend{array}$

for ({x to + infty,}) assuming a hypothetical estimate on the mean

$begin{array}{lll} int limits_X^{X+Y}(Delta(x+h(x))-Delta (x))^2,{d}x, end{array}$

which is a weakened form of a conjecture of M. Jutila.