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. In this paper, we establish Möbius inversion formulae for the Fourier expansions of the Apostol-Bernoulli, Apostol-Euler and ApostolGenocchi polynomials. As an application, by specializing our formulae at some special values we obtain interesting number-theoritical relations. We derive explicit formulae for Apostol-Bernoulli numbers. These formulae involve Stirling numbers of the second kind and powers of cotangent. Our proofs are very simple.

Möbius inversion formulae for Apostol-Bernoulli type polynomials and numbers