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

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