1.7. Notes 31

The proof of Proposition 1.2 is modeled on the one given by Ramanujan

[Ram19].

Landau gave suﬃcient conditions for the heuristic mentioned at the beginning

of Section 1.4 to hold true; see page 201 of his treatise [Lan74] on prime number

theory, or the paper [Lan00]. Partial summation goes back to a paper of N. H.

Abel [Abe26] on power series. The Euler-Maclaurin summation formula [Eul38,

Mac42] dates to the first half of the eighteenth century. The work of Mertens is

in [Mer74a, Mer74b].

The formula for d(n) dates to 1673 and is due to J. Kersey [Ker73]. Obscure

today, to his contemporaries he was known as an author of well-regarded textbooks.

It seems that R. Descartes [Des79] was the first to explicitly note an arithmetic

function multiplicative; in a posthumous manuscript he stated that the sum-of-

divisors function σ(n) has this property. From a letter [Des98] to M. Mersenne it

seems likely that he knew this by 1638.

The von Mangoldt function, though not the notation for it used today, and the

convolution identity 1 ∗ Λ = log, are due to N. W. Bugaev [Bug73] and E. C´esaro

[C´ es88].

The convolution identity 1∗φ = id was proved by Gauss in article 39 of the Dis-

quisitiones, and this may well be the first instance of a divisor sum of an arithmetic

function.

E. T. Bell [Bel15] and M. Cipolla [Cip15] independently in 1915 considered

the set of arithmetic functions as an algebraic structure and gave Proposition 1.12.

But a good many particular convolutions of multiplicative functions were known in

the nineteenth century, so this result may well have been appreciated earlier.

A. F. M¨ obius [M˝ ob31] introduced the function named after him in 1831. But

already Euler had considered infinite series whose terms involved values of the

M¨ obius function.

Proposition 1.13 is due to J. W. R. Dedekind [Ded57] and J. Liouville [Lio57]

independently in 1857, and Proposition 1.14 to M¨ obius [M˝ ob31].

The asymptotic estimate for Φ(x) is due to Mertens [Mer74b]. Dirichlet had

obtained a similar estimate with

xε

in place of log(x) in [Dir49].

The maximal order of log(d(n)) was found by S. Wigert [Wig07] around 1906

using the Prime Number Theorem. The dependence on the PNT was removed some

years later by Ramanujan [Ram15]. The proof of Proposition 1.15 is a modified

version of the one given by Wigert.

In two papers [Dir38a, Dir38b] of 1838 Dirichlet considers the question of how

one can study arithmetic functions that fluctuate irregularly. It is not unreasonable

to consider these papers as founding the theory of arithmetic functions, but Dirichlet

himself refers to earlier work, and in particular to remarks of Gauss in article 301 of

the Disquisitiones. What Dirichlet set out to do in the first of these papers was to

find ‘das asymptotische Gesetz’ in the sense of Gauss, of the divisor function d(n).

He indicated an argument, based on the Lambert series expansion

∞

n=1

d(n)xn

=

∞

m=1

xm

1 − xm

,