Number Theory- arithmetic functions

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SUMMARY

The discussion centers on proving that the arithmetic function σk(n) = Σd|n d^k is multiplicative for each integer k. Participants reference a key lemma stating that if g is a multiplicative function and f(n) = Σd|n g(d), then f is also multiplicative. The challenge lies in identifying the appropriate function g to apply this lemma effectively to the problem at hand.

PREREQUISITES
  • Understanding of multiplicative functions in number theory
  • Familiarity with arithmetic functions, specifically σk(n)
  • Knowledge of summation notation and divisor functions
  • Basic comprehension of mathematical lemmas and their applications
NEXT STEPS
  • Research the properties of multiplicative functions in number theory
  • Study the specific arithmetic function σk(n) and its applications
  • Learn how to apply mathematical lemmas in proofs
  • Explore examples of functions g that are multiplicative
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This discussion is beneficial for mathematicians, students studying number theory, and anyone interested in the properties of arithmetic functions and their applications in mathematical proofs.

roca
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Problem: Show that for each k, the function σk(n)=Ʃd|n dk is multiplicative.



The attempt at a solution:

What I know is that I am supposed to use the Lemma which states that if g is a multiplicative function and f(n)=Ʃd|n g(d) for all n, then f is multiplicative. I am just very confused on how to apply this theorem to my problem.
 
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What would g be in this case?
 
just a function
 

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