Problem 5. Let f be a multiplicative function and g be defined on N by g(n) = Ef(d) (n E N). dn Prove that g is also multiplicative. Show that if f is completely multiplicative, it does not necessarily follow that g is completely multiplicative.

Problem 5. Let f be a multiplicative function and g be defined on N by g(n) = Ef(d) (n E N). dn Prove that g is also multiplicative. Show that if f is completely multiplicative, it does not necessarily follow that g is completely multiplicative.

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

Problem 5. Let f be a multiplicative function and g be defined on N by
g(n) = Ef(d)
(n e N).
din
Prove that g is also multiplicative. Show that if f is completely multiplicative, it
does not necessarily follow that g is completely multiplicative.
Problem 6. Use Problem 5 together with the fact that powers of multiplicative func-
tions are multiplicative to show that for n eN we have
Ed(m) = (Ed(m)
(1)
m/n
m/n

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