We estimate the sums and where denotes the von Mangoldt function (and the Möbius function) whenever and are two coprime bases, (resp., ) is a strongly -multiplicative (resp., strongly -multiplicative) function of modulus , and is a real number. The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation, and combinatorial arguments. We deduce from these estimates a prime number theorem (and Möbius orthogonality) for sequences of integers with digit properties in two coprime bases.
"Prime numbers in two bases." Duke Math. J. 169 (10) 1809 - 1876, 15 July 2020. https://doi.org/10.1215/00127094-2019-0083