The Nyman–Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in space of square integrable functions on (0,?), involving dilations fractional part function by factors ? k ?(0,1), k?1. Randomizing generates new structures and criteria. One them a sufficient condition for RH that splits into

Abstract In this paper, we investigate the regularity properties and determine almost sure multifractal spectrum of a class random functions constructed as sums pulses with dilations translations. addition, continuity moduli sample paths these stochastic processes are investigated.