Möbius functions of higher rank and Dirichlet series

2021/08/04 by Kobayashi, Masato
#FOS: Mathematics #Number Theory (math.NT) #Primary:11A25 #Secondary:11M32

paper · doi:10.48550/arxiv.2108.01822

Abstract

We introduce Möbius functions of higher rank, a new class of arithmetic functions so that the classical Möbius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all r-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role.

Related

• On the series expansion of a square-free zeta series
• The Dirichlet series that generates the Möbius function is the inverse of the Riemann zeta function in the right half of the critical strip
• Multiplicative functions resembling the Möbius funciton
• Moebius functions Irreducible Polynomials and Dirichlet Series
• Partial sums of the Möbius function in arithmetic progressions assuming GRH