1.5. Sums over divisors 19

Proposition 1.14 (Second M¨ obius inversion formula). Suppose that F is a

function on the interval [1, ∞). If

G(x) =

n≤x

F

x

n

then

F (x) =

n≤x

μ(n)G

x

n

and conversely on this interval.

Proof. First

n≤x

μ(n)G

x

n

=

n≤x

μ(n)

m≤x/n

F

x/n

m

=

mn≤x

μ(n)F

x

mn

=

N≤x

F

x

N

n|N

μ(n) = F (x)

and then

n≤x

F

x

n

=

n≤x

m≤x/n

μ(m)G

x/n

m

=

mn≤x

μ(m)G

x

mn

=

N≤x

G

x

N

m|N

μ(m) = G(x)

since 1 ∗ μ = e.

The second M¨ obius inversion formula throws light on the method of

Chebyshev. The relation

ψ(x) =

n≤x

μ(n)T

x

n

holds by M¨ obius inversion. Since T(x) is quite precisely known, it might

seem possible to estimate ψ(x) fairly accurately by means of this formula.

The problem here is that there is a great deal of cancellation in the sum,

due to the oscillation of sign of μ(n). Too little is known about the behavior

of μ(n) for this approach to promise much success. But the estimates of

Chebyshev may be obtained by replacing μ(n) by an approximation of a

particular kind. The approximation associated with the proof of Proposition

1.1 is μ(n) ≈ e1(n)−2e2(n) where ek(n) is the arithmetic function that equals