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Exercise 6. Prove that the following functions are multiplicative. (a) d(n) = #{de N: dn} (b) 2w(n), where w(n) = #{p/n: p prime} (-1)w(n) if n is squarefree, otherwise (c) μ(n) = = {(-1)-(

Exercise 6. Prove that the following functions are multiplicative. (a) d(n) = #{de N: dn} (b) 2w(n), where w(n) = #{p/n: p prime} (-1)w(n) if n is squarefree, otherwise (c) μ(n) = = {(-1)-(

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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Definition 0.1. A function f: N→ C is multiplicative if f(mn) = f(m)f(n) whenever (m, n) 1. A multiplicative function f is totally multiplicative (or completely multi- plicative) if f(mn) = f(m) f(n) for all m, n € N.
Transcribed Image Text:Definition 0.1. A function f: N→ C is multiplicative if f(mn) = f(m)f(n) whenever (m, n) 1. A multiplicative function f is totally multiplicative (or completely multi- plicative) if f(mn) = f(m) f(n) for all m, n € N.
Exercise 6. Prove that the following functions are multiplicative. (a) d(n) = #{de N: d\n} (b) 2w (n), where w(n) = #{p/n: p prime} (-1)(n) if n is squarefree, (c) μ(n) = otherwise
Transcribed Image Text:Exercise 6. Prove that the following functions are multiplicative. (a) d(n) = #{de N: d\n} (b) 2w (n), where w(n) = #{p/n: p prime} (-1)(n) if n is squarefree, (c) μ(n) = otherwise
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