Combining the identity (1) from [1], I am saying the specialization an=√2n3 and bn=i√n3 (here i denotes the imaginary unit, thus i2=−1) for integers n≥1, and the explanation of the criterion of Fubini's theorem, see [2] if you need it, I can prove that 1ζ(3)=∫∞0∞∑n=1μ(n)sin(√2n3x)√2n3e−i√n3xdx,
where μ(n) is the Möbius function.
Question. I was wondering about questions involving this function f(x):=∞∑n=1μ(n)sin(√2n3x)√2n3e−i√n3x defined on [0,∞) that I know how solve or I don't know how solve those.
I know that the function f(x) is uniformly and absolutely convergent on [0,∞), but what about the differentiability? I am asking what relevant facts we can deduce about the differentiability of our function f(x) for real numbers 0≤x<∞.
Many thanks.
Feel free, if you prefer, add hints for some of previous question, instead of a full answer.
References, both from this Mathematics
[1] See the answer by D'Aurizio for Cantarini's lemma, identity (1) from: Find the closed form for ∫∞0cosxln(1+e−x1−e−x)dx=∑∞n=01n2+(n+1)2.
[2] See the second paragraph of the answer by Eldredge: When can a sum and integral be interchanged?
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