An arithmetical function is a real- or complex-valued function defined on the
integers. Some examples of arithmetical functions are 
, the sum of
positive divisors of n, 
, the number of positive
divisors of n, and
, Euler's Totient function. Properties of these and
other functions are
widely studied in elementary number theory and recreational mathematics. The
study of is related to the study of perfect numbers, amicable
numbers, sociable numbers, and other interesting classes of integers. The
Euler- 
function has numerous applications and is fundamental to
the study
of congruences.
We develop the basic properties of the arithmetical functions such as the
convolution product. The Dirichlet series are introduced to explore the
multiplicative properties of these functions. Along the way, we prove the
Möbius inversion formula and develop properties of the Möbius function
. The function is related to the
distribution of prime numbers
through the connection between its Dirichlet series and the Riemann zeta
function.
The study of Dirichlet series leads naturally to a proof of Dirichlet's
theorem on the infinitude of primes in arithmetic progressions.
Dirichlet
proved that there are infinitely many primes in every arithmetic progression
of the form 
where a and
b are coprime. The text
includes a complete proof for the case a=5, a proof that is
representative
of the general case.