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DIRICHLET SERIES
PETE L. CLARK

1. Introduction
In considering the arithmetical functions f : N → C as a ring under pointwise
addition and “convolution”:
f ∗ g (n) =

f (d1 )g (d2 ),
d1 d2 =n

we employed that old dirty trick of abstract algebra. Namely, we introduced an
algebraic structure without any motivation and patiently explored its consequences
until we got to a result thatwe found useful (M¨bius Inversion), which gave a sort
o
of retroactive motivation for the definition of convolution.
This definition could have been given to an 18th or early 19th century mathematical audience, but it would not have been very popular: probably they would
not have been comfortable with the Humpty Dumpty-esque redefinition of multiplication.1 Mathematics at that time did havecommutative rings: rings of numbers,
of matrices, of functions, but not rings with a “funny” multiplication operation
defined for no better reason than mathematical pragmatism.
So despite the fact that we have shown that the convolution product is a useful operation on arithmetical functions, one can still ask what f ∗ g “really is.”
There are (at least) two possible kinds of answers to thisquestion: one would be to
create a general theory of convolution products of which this product is an example
and there are other familiar examples. Another would be to show how f ∗ g is
somehow a more familiar multiplication operation, albeit in disguise.
To try to take the first approach, consider a more general setup: let (M, •) be a
commutative monoid. Recall from the first homework assignment thatthis means
that M is a set endowed with a binary operation • which is associative, commutative, and has an identity element, say e: e • m = m • e = m for all m ∈ M . Now
consider the set of all functions f : M → C. We can add functions in the obvious
“pointwise” way:
(f + g )(m) := f (m) + g (m).
We could also multiply them pointwise, but we choose to do something else, defining
(f ∗ g )(m):=

f (d1 )g (d2 ).
d1 •d2 =m

With the assistance of Richard Francisco and Diana May.
1Recall that Lewis Carroll – or rather Charles L. Dodgson (1832-1898) – was a mathematician.
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PETE L. CLARK

But not so fast! For this definition to make sense, we either need some assurance
that for all m ∈ M the set of all pairs d1 , d2 such that d1 · d2 = m is finite (so
the sum is a finitesum), or else some analytical means of making sense of the sum
when it is infinite. But let us just give three examples:
Example 1: (M, •) = (Z+ , ·). This is the example we started with – and of course
the set of pairs of positive integers whose product is a given positive integer is finite.
Example 2: (M, •) = (N, +). This is the “additive” version of the previous example:
(f ∗ g )(n) =

f(i)g (j ).
i+j =n

Of course this sum is finite: indeed, for n ∈ N it has exactly n +1 terms. As we shall
see shortly, this “additive convolution” is closely related to the Cauchy product of
infinite series.
Example 3: (M, •) = (R, +). Here we have seem to have a problem, because
for functions f, g : R → C, we are defining
(f ∗ g )(x) =

f (x − y )g (y ),

f (d1 )g (d2 ) =
y ∈R

d1 +d2=x

and although it is possible to define a sum over all real numbers, it turns out never
to converge unless f and g are zero for the vast majority of their values.2 However,
there is a well-known replacement for a “sum over all real numbers”: the integral.
So one should probably define
∞

(f ∗ g )(x) =

f (x − y )g (y )dy.
−∞

Here still one needs some conditions on f and g to ensureconvergence of this
“improper” integral. It is a basic result of analysis that if
∞

∞

|f | < ∞,
−∞

|g | < ∞,
−∞

then the convolution product is well-defined. The convolution is an all-important
operation in harmonic analysis: roughly speaking, it provides a way of “mixing
together” two functions. Like any averaging process, it often happens that f ∗ g
has nicer properties than...