This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 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 ) = X d 1 d 2 = n f ( d 1 ) g ( d 2 ) , 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 that we found useful (M obius Inversion), which gave a sort of retroactive motivation for the definition of convolution. This definition could have been given to an 18th or early 19th century mathe- matical audience, but it would not have been very popular: probably they would not have been comfortable with the Humpty Dumpty-esque redefinition of multipli- cation. 1 Mathematics at that time did have commutative 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 use- ful operation on arithmetical functions, one can still ask what f * g really is. There are (at least) two possible kinds of answers to this question: 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 that this means that M is a set endowed with a binary operation which is associative, commuta- tive, 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 ) := X d 1 d 2 = m f ( d 1 ) g ( d 2 ) . With the assistance of Richard Francisco and Diana May. 1 Recall that Lewis Carroll or rather Charles L. Dodgson (1832-1898) was a mathematician. 1 2 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 d 1 ,d 2 such that d 1 d 2 = m is finite (so the sum is a finite sum), 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....
View Full Document