The “Riemann hypothesis” for the Hawkins random Sieve
We establish a general mean value result for arithmetic functions over short intervals with the Selberg-Delange method. As an application, we generalize the Deshouillers-Dress-Tenenbaum arcsine law on divisors to the short interval case.
The present paper deals with the summatory function of functions acting on the digits of an -ary expansion. In particular let be a positive integer, then we callits -ary expansion. We call a function strictly -additive, if for a given value, it acts only on the digits of its representation, i.e.,Let with , , and at least one . Then we call a pseudo-polynomial.The goal is to prove that for a -additive function there exists an such thatwhere is the mean of the values of ...
The aim of this work is to estimate exponential sums of the form , where Λ denotes von Mangoldt’s function, f a digital function, and β ∈ ℝ a parameter. This result can be interpreted as a Prime Number Theorem for rotations (i.e. a Vinogradov type theorem) twisted by digital functions.