CHAPTER 1

Panorama of Arithmetic Functions

Throughout these notes we denote by Z, Q, R, C the sets of integers, rationals,

real and complex numbers, respectively. The positive integers are called natural

numbers. The set

N = {1, 2, 3, 4, 5, . . . }

of natural numbers contains the subset of prime numbers

P = {2, 3, 5, 7, 11, . . . }.

We will often denote a prime number by the letter p.

A function f : N → C is called an arithmetic function. Sometimes an arithmetic

function is extended to all Z. If f has the property

(1.1) f(mn) = f(m) + f(n)

for all m, n relatively prime, then f is called an additive function. Moreover, if (1.1)

holds for all m, n, then f is called completely additive; for example, f(n) = log n is

completely additive. If f has the property

(1.2) f(mn) = f(m)f(n)

for all m, n relatively prime, then f is called a multiplicative function. Moreover,

if (1.2) holds for all m, n, then f is called completely multiplicative; for example,

f(n) = n−s for a fixed s ∈ C, is completely multiplicative.

If f : N → C has at most a polynomial growth, then we can associate with f

the Dirichlet series

Df (s) =

∞

1

f(n)n−s

which converges absolutely for s = σ + it with σ suﬃciently large. The product of

Dirichlet series is a Dirichlet series

Df (s)Dg(s) = Dh(s),

with h = f ∗ g defined by

(1.3) h(l) =

mn=l

f(m)g(n) =

d|l

f(d)g(l/d),

which is called the Dirichlet convolution.

The constant function f(n) = 1 for all n ∈ N has the Dirichlet series

(1.4) ζ(s) =

∞

1

n−s

3

http://dx.doi.org/10.1090/ulect/062/01