By P. T. Bateman, Harold G. Diamond
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Extra info for Analytic number theory: an introductory course
If F is defined on R and F ( x ) = 0 for all x < 1, then we define F,, the total variation function of F , as an extended real valued function on R by setting F,(x) equal to the total variation of F on [O,x]. Familiar facts about F, include the following: (1) F, is monotone nondecreasing (write briefly: ‘Y,’ or “increasing”), (2) IF(b) - F(a)I 5 IF,(b) - F,(a)I for all a, b E Iw, (3) if F f, then F, = F . A function F supported in [l,oo) is loc. V. iff F,(x) < oo for each real z. For example, if F ( z ) = [XI - x 1 for x 2 1 and F ( z ) = 0 for x < 1, then F,(x)= + x - 1 for x 2 1.
Integration by parts yields the following integral inequality, which will be very useful in the sequel. 11 (Comparison of integrators). Let g be a nonnegative, left continuous, decreasing function o n (1 - E, oo). Let F and cp be functions o n R which are supported in [l,oo), are right continuous, and are locally of bounded variation. Assume F ( z ) 5 p(x) for all z. T h e n Proof. We have PX rx The right hand side is nonnegative since g J. and g is nonnegative. 11 By iterating the method of the preceding lemma (or otherwise) show that the series converges and is nonnegative for any real positive s.
8 Let f , g E A, f # 0 , g # 0 , and ( L f ) * g = f * ( L g ) . Show that g = cf for some constant c. Hint. First show that fnz(f) = fnz(g). 6 Let f E A. We say that f is invertible if there exists a g E A such that f * g = e. In this case we call g an inverse of f . Inverses are unique, when they exist: If g and g/ are both inverses of f , then If f * g = e, we shall write g = f * - l and refer to g as the inverse of f . 9 Let S = (2" : 0 5 n < oo}, 1s be the indicator function of S, and let p2 = e - e2.