M. Drmota, C. Mauduit, J. Rivat:
"Prime numbers in two bases";
Duke Mathematical Journal, 169 (2020), 10; S. 1809 - 1876.

Kurzfassung englisch:
We estimate the sums ?n=x?(n)f(n)g(n)exp(2ip?n)
and ?n=xµ(n)f(n)g(n)exp(2ip?n),
where ? denotes the von Mangoldt function (and µ the Möbius function) whenever q1 and q2 are two coprime bases, f (resp., g) is a strongly q1-multiplicative (resp., strongly q2-multiplicative) function of modulus 1, 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.

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