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Exercise 4. For a multiplicative function f, define the Dirichlet series for f by L(s, f) = f(n) We assume that s is chosen so that the series converges absolutely. (a) Prove that L(s, f) = p prime j=0 (b) Prove that if f is totally multiplicative, then L(s, f) = II p prime f(p³) pjs 1 f(p)

Exercise 4. For a multiplicative function f, define the Dirichlet series for f by L(s, f) = f(n) We assume that s is chosen so that the series converges absolutely. (a) Prove that L(s, f) = p prime j=0 (b) Prove that if f is totally multiplicative, then L(s, f) = II p prime f(p³) pjs 1 f(p)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.1: Inverse Functions
Problem 55E
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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 4. For a multiplicative function f, define the Dirichlet series for f by f(n) Σ n8 n=1 L(s, f) = We assume that s is chosen so that the series converges absolutely. (a) Prove that L(s, f) = ΠΣ) p prime j=0 pis (b) Prove that if f is totally multiplicative, then (s, f) = II p prime 1 1-f(p) p²
Transcribed Image Text:Exercise 4. For a multiplicative function f, define the Dirichlet series for f by f(n) Σ n8 n=1 L(s, f) = We assume that s is chosen so that the series converges absolutely. (a) Prove that L(s, f) = ΠΣ) p prime j=0 pis (b) Prove that if f is totally multiplicative, then (s, f) = II p prime 1 1-f(p) p²
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