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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 Dumptyesque 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 (18321898) 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....
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This note was uploaded on 10/26/2011 for the course MATH 4400 taught by Professor Staff during the Spring '11 term at University of Georgia Athens.
 Spring '11
 Staff
 Algebra, Number Theory, Addition

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