By Hildebrand A.J.

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01 Chapter 2 Arithmetic functions II: Asymptotic estimates The values of most common arithmetic functions f (n), such as the divisor function d(n) or the Moebius function µ(n), depend heavily on the arithmetic nature of the argument n. As a result, such functions exhibit a seemingly chaotic behavior when plotted or tabulated as functions of n, and it does not make much sense to seek an “asymptotic formula” for f (n). However, it turns out that most natural arithmetic functions are very well behaved on average, in the sense that the arithmetic means Mf (x) = (1/x) n≤x f (n), or, equivalently, the “summatory functions” Sf (x) = n≤x f (n), behave smoothly as x → ∞ and can often be estimated very accurately.

I) If f and g are multiplicative, then so is f ∗ g. (ii) If f is multiplicative, then so is the Dirichlet inverse f −1 . (iii) If f ∗ g = h and if f and h are multiplicative, then so is g. (iv) (Distributivity with pointwise multiplication) If h is completely multiplicative, then h(f ∗ g) = (hf ) ∗ (hg) for any functions f and g. Remarks. (i) The product of two completely multiplicative functions is multiplicative (by the theorem), but not necessarily completely multiplicative. , with the coprimality condition).

Since d = 1 ∗ 1, and the function 1 is multiplicative, the function d is multiplicative as well. Similarly, since σ = id ∗1, and 1 and id are multiplicative, σ is multiplicative. 1 Evaluate the function f (n) = d2 |n µ(d) (where the summation runs over all positive integers d such that d2 |n), in the sense of expressing it in terms of familiar arithmetic functions. , d∗ (n) = {(a, b) ∈ N2 : ab = n, (a, b) = 1}. Show that d∗ is multiplicative, and find its values on prime powers. 3 Determine an arithmetic function f such that 1 = φ(n) d|n 1 n f d d (n ∈ N).

### Introduction to Analytic Number Theory by Hildebrand A.J.

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