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 denote a real or complex-valued function defined on the positive real axis such that for .
is any arithmetical function.
Generalized convolution is then defined by
“If for all nonintegral values of , the restriction of to integers is an arithmetical function and we find that .”
The question that sometimes arises in some form or other in relation to this is: why does have to be for all nonintegral values of for this to work? Is the point that we want to be when is not an integer?
The answer is yes. For example, to isolate , take to be the unit function . Then the generalized convolution for integer is
If whenever is not an integer, the above sum reduces to
I think Apostol could have simply said “If for all nonintegral values of , we find that for all integers …”