Kelin Zhu

The classic version states that if g and f are arithmetic functions satisfying g ( n ) = ∑ d ∣ n f ( d ) for every integer n ≥ 1 {\displaystyle g(n)=\sum _{d\mid n}f(d)\quad {\text{for every integer }}n\geq 1} {\displaystyle g(n)=\sum _{d\mid n}f(d)\quad {\text{for every integer }}n\geq 1} then f ( n ) = ∑ d ∣ n μ ( d ) g ( n d ) for every integer n ≥ 1 {\displaystyle f(n)=\sum _{d\mid n}\mu (d)g\left({\frac {n}{d}}\right)\quad {\text{for every integer }}n\geq 1} {\displaystyle f(n)=\sum _{d\mid n}\mu (d)g\left({\frac {n}{d}}\right)\quad {\text{for every integer }}n\geq 1} where μ is the Mörbius function and the sums extend over all positive divisors d of n (indicated by d ∣ n {\displaystyle d\mid n} {\displaystyle d\mid n} in the above formulae). In effect, the original f(n) can be determined given g(n) by using the inversion formula. The two sequences are said to be Mörbius transforms of each other.