4. a) Show that if n = p₁¹.ps where p₁,...,Ps are s distinct primes and a₁,..., ag € Z with a; ≥2 for all i then 2s | (n). Show that for any n E Z b) $(n) = Σ µ(d) ². 0
4. a) Show that if n = p₁¹.ps where p₁,...,Ps are s distinct primes and a₁,..., ag € Z with a; ≥2 for all i then 2s | (n). Show that for any n E Z b) $(n) = Σ µ(d) ². 0
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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Question
Please solve all the subpart of the problem
![4. a)
Show that if n = p₁¹ ...ps where p₁, ... , ps are s distinct
primes and a₁,..., a¸ € Z with a¿ ≥ 2 for all i then 2³ | ø(n).
Show that for any n E Z
b)
o(n) = Σ µ(d) ².
d
0<d/n
[Hint: Use the fact that the summatory function of is the identity
function.]
c)
Check the formula in part b for n = 12.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1becd8c0-6409-4777-a3ed-b129ce4472d0%2Fca1e63b0-7e90-4394-94af-26347d0c210d%2Fqdez7lc_processed.png&w=3840&q=75)
Transcribed Image Text:4. a)
Show that if n = p₁¹ ...ps where p₁, ... , ps are s distinct
primes and a₁,..., a¸ € Z with a¿ ≥ 2 for all i then 2³ | ø(n).
Show that for any n E Z
b)
o(n) = Σ µ(d) ².
d
0<d/n
[Hint: Use the fact that the summatory function of is the identity
function.]
c)
Check the formula in part b for n = 12.
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