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Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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prove theorem

Definition 2.10. Given an arithmetical function f and FS. we define new function foF=GES as follows: foF(x) = Σ f(n) F (*). 1<n<x We note that fo F(x) = 0 for x € (0.1). hence indeed fo FS. If support of F is subset of natural numbers, we see that fo F is also supported on N and ƒ o F(m) = Σ f(n)F(") = [ƒ(n)F (7) = ƒ ⋆ F(m), 1<n<x nm as F(m/n) = 0 if n m because SuppF C N. We have following associative property. Theorem 2.15. Let f and g be two multiplicative functions and S. We have fo(goF) = (f*g) o F.
Transcribed Image Text:Definition 2.10. Given an arithmetical function f and FS. we define new function foF=GES as follows: foF(x) = Σ f(n) F (*). 1<n<x We note that fo F(x) = 0 for x € (0.1). hence indeed fo FS. If support of F is subset of natural numbers, we see that fo F is also supported on N and ƒ o F(m) = Σ f(n)F(") = [ƒ(n)F (7) = ƒ ⋆ F(m), 1<n<x nm as F(m/n) = 0 if n m because SuppF C N. We have following associative property. Theorem 2.15. Let f and g be two multiplicative functions and S. We have fo(goF) = (f*g) o F.
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