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
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