2. Prove that|#(n)| _ S(s)S(2s)n=1if s > 1.

Riemann zeta function is a complex-valued function defined as ςs=∑n=1∞1ns. There is a relationship…

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2. Prove that |µ(n)| _ S(s) S(2s) n=1 if s > 1.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.2: Properties Of Group Elements

Problem 30E: 30. Prove statement of Theorem : for all integers .

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Question

2. Prove that
|#(n)| _ S(s)
S(2s)
n=1
if s > 1.

Expert Solution

Step 1

Riemann zeta function is a complex-valued function defined as $ς (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}}$ .

There is a relationship between the prime number and the zeta function. For a prime number $p$ ,

$\begin{array}{rcl} ς (s) & = & \sum_{n = 1}^{\infty} \frac{1}{n^{s}} \\ = & \prod_{p is prime} \frac{1}{1 - p^{- s}} \end{array}$

This relationship is very significant in calculating various properties of the zeta function.

In the given question, we show a relationship between the zeta function and the Mobius function. Precisely, we prove that

$\begin{array}{rcl} \sum_{n = 1}^{\infty} \frac{|μ (n)|}{n^{s}} & = & \frac{ς (s)}{ς (2 s)} \end{array}$

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