By P. T. Bateman, Harold G. Diamond
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Additional info for Analytic number theory: an introductory course
18 Then Let g E A, g(l) = 0 and let j=O cj”=, ajg*j be a power series. j=l Proof. Fix n 2 1. We evaluate each side of the formula at n. The summation on each side extends over at most 1 [logn / log 21 indices j . 1). 0 + 25 Exponential mapping An example of the last lemma is L( x g * j / j ! ) = ( x g * j / j ! ) j=O *~ g . 5) is an analogue of the familiar exponential series. In the next section we shall investigate some properties of this series and show its use in multiplicative number theory.
The - ( x + 00). Now we investigate the rate of convergence in the last expression. 4 lpl = 1 * f , where f ( n ) := p ( f i ) i f n is a square and f (n) := O otherwise. Proof. Let g := p * IpI. Since p and lpJ are multiplicative, so is 9. An easy calculation shows that g(1) = 1 and that for any p , g ( p 2 ) = -1 and g ( p a ) = 0 if a # 0, 2. Also, f is multiplicative. Thus g = f , and we have 0 (by Mobius inversion) 1pl = 1 * p * 1pl = 1 * f. 5 + x = (1 - z2)/(1- x). rr21 < 3z1j2. Proof. Let f be as in the preceding lemma.
23 Let f be any completely multiplicative function except e. Show that f * f and f * - l are not completely multiplicative. 24 Let f E CAI. Show that f is completely multiplicative if and only if f * f = f - T . 25 Let f E d1. Show that f is completely multiplicative if and only if for each g and h E A we have f - ( g * h ) = (f - 9 )* (f - h ) . 26 Show that Ipl is multiplicative but not completely multiplicative. 26. Describe the functions fi. Find log IpI. 27 Liouville's X function is defined by X(n) = (-l)n(n).
Analytic number theory: an introductory course by P. T. Bateman, Harold G. Diamond