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Let denote the Bessel function of the first kind. Prove that
We recall that
Using the fact that the looks like an ‘almost periodic’ function with decreasing amplitude. If we denote by the zeros of then as and furthermore
as for each . So the integral converges uniformly in this case justifying the interchange of limit and integral.
The result follows.
Let denote the Möbius function and the Euler’s totient function. Prove that
Since both functions are multiplicative it suffices to prove the identity for prime numbers. Hence for we have
Let denote the Möbius function. Prove that
Summing only over primes , where , we have that
For the values of for which the following series makes sense, prove that
We have successively:
due to the extended binomial theorem.
Evaluate the sum
Lemma: It holds that
Proof: Consider the function and let us it integrate over the following contour
By the residue theorem it follows that
For the residues we have
The integrals along the sides vanish; hence:
the result follows.
Back to the problem. We have successively: