By P. T. Bateman, Harold G. Diamond

ISBN-10: 9812389385

ISBN-13: 9789812389381

ISBN-10: 9812560807

ISBN-13: 9789812560803

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CtT and note that in the exponential series all terms except the Pth vanish at n. 11). 20. 11) by induction on n. 4. For each p we can similarly evaluate exp ep, then express expu as (exp e2) * (exp e3) * (exp e5) * - .. 11). 16. The technique of resolving a function into a convolution product is effective for certain problems, and we shall study it systematically in the next section. 5 Multiplicative functions Let M := {f E A : f # 0, f ( m n )= f ( m ) f ( n if ) (rn,n)= I}. The elements of M are called multiplicative arithmetic functions.

In this section we define an exponential map on & and show that it does what we expect of an exponential. We define exp by setting expu = e u*u +u + 2! u*u*u +-+a**. 3! It is clear that exp v E A1. We shall generally use Greek letters to denote generic elements of the domain of exp and Latin letters for generic elements of the range. , if n = 2" for some QI 2 0, if n # 2" for all Q! 2 0. 0, By the chain rule of calculus, if f = exp 9, then f' = f -9'. Recalling that L is a derivation, in the following theorem we establish a similar formula for Calculus of Arithmetic Functions 26 arithmetic functions.

Thus * 1 = T1. 10 Given a sequence {fv}p=l with each fv E A, we say that {fv} converges (in A) if lim f v ( n )exists and is a finite number for Y+OO If the limit function is denoted by f , we shall write fv -+ f . each n E Z+. This is a pointwise notion of convergence. If we are given a sequence of complex numbers {an}r=o and f E A, define an f *n as the limit of the sequence of partial sums of the arithmetic functions a n f * n . We say that the infinite product f l * f2 * - . e. there exist at most a finite number of indices i for which fi(1) = 0 and with these factors omitted, the remaining product converges to a finite nonzero limit, and * f~ converges as N --+ 00.

### Analytic number theory: an introductory course by P. T. Bateman, Harold G. Diamond

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