1.7. Notes 31
The proof of Proposition 1.2 is modeled on the one given by Ramanujan
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
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
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
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
1 − xm