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

2. Prove that |µ(n)| _ S(s) S(2s) n=1 if s > 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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2. Prove that |#(n)| _ S(s) S(2s) n=1 if s > 1.
Transcribed Image Text: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 ςs=∑n=1∞1ns. 

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

ςs=∑n=1∞1ns=∏p is prime11-p-s

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 

∑n=1∞μnns=ςsς2s

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