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Let us consider the function
and integrate it along a quadratic counterclockwise contour with vertices where is a big odd natural number. Hence,
We note that and that .
It’s also easy to see that
Hence, as we have that
By Residue theorem we have that
It is straightforward to show that
in the limit . The result follows.
Let denote the greatest prime factor of . For example , . Define . Examine if the sum
Let be the -th prime number.
If , then with , and . It folows that
From Merten’s theorem
and the original series has the same character as
which is convergent.
Let denote the – th prime. Evaluate the sum
Let be a real sequence such that
If is a real sequence that is square summable; i.e the sequence converges.
Prove that is also square summable.
Let be defined as
where . We note that
Equality holds when . Hence, . From the hypothesis, it follows that is pointwise bounded. It follows from the Uniform boundedness principle ( Banach – Steinhaus ) that are bounded. Hence, is square summable.
Let . Prove that
First of all we note that
Letting we get the requested value.