The aim of this work is to estimate exponential sums of the form $∑_{n≤x} Λ(n) exp(2iπ(f(n)+β n))$, 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.
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In a recent work we gave some estimations for exponential sums of the form $∑_{n≤x} Λ(n) exp(2iπ(f(n) + βn))$, where Λ denotes the von Mangoldt function, f a digital function, and β a real parameter. The aim of this work is to show how these results can be used to study the statistical properties of digital functions along prime numbers.
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