Tuesday, 24 September 2019

complex analysis - Evaluating zetaleft(frac12right) as an integral zetaleft(frac12right)=frac12inti0nftyfrac[x]xx3/2,dx




I am reading the second chapter of Titchmarsh's book on the Riemann Zeta Function. I would have written:



ζ(12)=1+12+13+=



If you think about it for a moment. This doesn't decay nearly fast enough, and so the sequence diverges. Then I had to look up the actual definition of ζ(s) in the region s=σ+it and 0<σ<1. We have:



ζ(s)={1nsRe(s)>1s0[x]xxs+1dx1>Re(s)>0



Then if I evaluate at s=32=32+0i we use the second formula:



ζ(12)=120[x]xx3/2dx=n=0[2n2n+11n+1]?<0



Is this thing negative? Is ζ(12)<0. The book overs several "analytic continuations" and I'm only looking at this first one, to make sure I understand.



Could someone help me evaluate the integral? I didn't use any fancy changes of variables. The main step is:




0f(x)dx=n+1nf(x)dx=10f(x)dx+21f(x)dx+


Answer






You can take advantage of the
following identity:





ζ(12)=Nk=11k2N12Nxxx3/2dx,N=1,2,3,




Note that
0<|12Nxxx3/2dx|<1N such that you don't need to evaluate the integral. Namely,





\bbx{\zeta\pars{1 \over 2} = \lim_{N \to \infty}\pars{\sum_{k = 1}^{N}{1 \over \root{k}} - 2\root{N}}}


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