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 ,

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