Friday, 29 April 2016

real analysis - About the differentiability of sumin=1nftyfracmu(n)sin(sqrt2n3x)sqrt2n3eisqrtn3x

Combining the identity (1) from [1], I am saying the specialization an=2n3 and bn=in3 (here i denotes the imaginary unit, thus i2=1) for integers n1, and the explanation of the criterion of Fubini's theorem, see [2] if you need it, I can prove that 1ζ(3)=0n=1μ(n)sin(2n3x)2n3ein3xdx,
where μ(n) is the Möbius function.





Question. I was wondering about questions involving this function f(x):=n=1μ(n)sin(2n3x)2n3ein3x 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 0x<.



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+ex1ex)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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