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
[2]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[3]	K. Prachar, "Primzahlverteilung" , Springer (1957)

Comments

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.

References

• Odlyzko, A.M.; te Riele, Herman J.J. "Disproof of the Mertens conjecture" J. Reine Angew. Math. 357 (1985) 138-160 DOI 10.1515/crll.1985.357.138 Zbl 544.10047