# Möbius series

A class of functions of the form $$F_n(x) = \sum_{s=1}^\infty f(x^s) s^{-n} \ .$$ These series were investigated by A. Möbius [1], who found for a series (*) the inversion formula $$f(x) = \sum_{s=1}^\infty \mu(s) F_n(x^s) s^{-n} \ ,$$ where $\mu(s)$ is the Möbius function. Möbius considered also inversion formulas for finite sums running over the divisors of a natural number $n$: $$F(n) = \sum_{d | n} f(d) \ ,\ \ \ f(n) = \sum_{d | n} \mu(d) F(n/d) \ .$$

Another inversion formula: If $P(n)$ is a totally multiplicative function for which $P(1) = 1$, and $f(x)$ is a function defined for all real $x > 0$, then $$g(x) = \sum_{n \le x} P(n) f(x/n)$$ implies $$f(x) = \sum_{n \le x} \mu(n) P(n) g(x/n) \ .$$

#### References

 [1] A. Möbius, "Ueber eine besondere Art der Umkehrung der Reihen" J. Reine Angew. Math. , 9 (1832) pp. 105–123 DOI 10.1515/crll.1832.9.105 Zbl 009.0333cj [2] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) Zbl 0057.28201 [3] K. Prachar, "Primzahlverteilung" , Springer (1957) Zbl 0080.25901

All these (and many other) inversion formulas follow from the basic property of the Möbius function that it is the inverse of the unit arithmetic function $E(n) \equiv 1$ under Dirichlet convolution, cf. (the editorial comments to) Möbius function and Multiplicative arithmetic function.
The term "Möbius series" is also applied to the summatory function of the Möbius function $$M(x) = \sum_{n \le x} \mu(n) \ .$$ Mertens conjectured in 1897 that the bound $|M(x)| < \sqrt x$ holds: this would imply the Riemann hypothesis. Odlyzko and te Riele disproved the Mertens conjecture in 1985.