Note on Generalized Convolution in Number Theory

In Tom Apostol’s famous Introduction to Analytic Number Theory there is a section about generalized convolutions which sometimes causes confusion (see Apostol, Chapter 2, p.39 and also my Advanced Number Theory Note #3). In this short note I want to quickly try to clarify this bit.

Let F denote a real or complex-valued function defined on the positive real axis such that F(x) = 0 for 0 < x < 1.

\alpha is any arithmetical function.

Generalized convolution is then defined by

(\alpha \circ F)(x)= \sum_{n \leq x} \alpha(n)F(\frac{x}{n})

Apostol says:

“If F(x) = 0 for all nonintegral values of x, the restriction of F to integers is an arithmetical function and we find that (\alpha \circ F)(m) =(\alpha \ast F)(m).”

Here (\alpha \ast \beta)(n) = \sum_{d | n} \alpha(n)\beta(\frac{n}{d}).

The question that sometimes arises in some form or other in relation to this is: why does F(x) have to be 0 for all nonintegral values of x for this to work? Is the point that we want F(\frac{x}{n}) to be 0 when \frac{x}{n} is not an integer?

The answer is yes. For example, to isolate F, take \alpha to be the unit function u. Then the generalized convolution for integer m \geq 1 is

(u \circ F)(m) = \sum_{n \leq m} F(\frac{m}{n})

If F(\frac{m}{n}) = 0 whenever \frac{m}{n} is not an integer, the above sum reduces to

\sum_{n | m} F(\frac{m}{n}) = (u \ast F)(m)

I think Apostol could have simply said “If F(x) = 0 for all nonintegral values of x, we find that (\alpha \circ F)(m) = (\alpha \ast F)(m) for all integers m \geq 1…”