Let denote the least common multiple. Prove that
for all .
Solution
For each of the form the number of solutions of is given by . It follows that the series is equal to
and since the function is multiplicative we have successively:
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Let denote the least common multiple. Prove that
for all .
Solution
For each of the form the number of solutions of is given by . It follows that the series is equal to
and since the function is multiplicative we have successively:
Let denote the – th prime. Evaluate the sum
Let denote the Möbius function. Prove that
Solution
Summing only over primes , where , we have that
Let . Prove that