This reminds me: I would like to see a plot of the complex plane, with the locus of Re(Zeta(z)) = 0 in blue and the Im(Zeta(z)) = 0 in red. The red and blue lines would cross many times along the critical line, but would avoid each other elsewhere. On Fri, Sep 14, 2018 at 11:45 AM Allan Wechsler <acwacw@gmail.com> wrote:
I don't see why it has to converge. The zeta function slowly gets "wrinklier" as you go up the critical line. and (as you have observed) the derivatives at the zeroes tend to cluster around the real line, so I am guessing that the derivatives gradually crawl rightward. If you did a million zeroes, you might see values around 4. (I think the tendency is logarithmic, that is, you have to double the distance out the critical line to see an increment in average derivative.)
On Fri, Sep 14, 2018 at 11:26 AM Wouter Meeussen < wouter.meeussen@telenet.be> wrote:
what is the limit of the sum of derivatives of the zeta function taken at all its zero’s? Shouldn’t it be a real number because of symmetry?
for the first 1000 zero’s I get 2.86778 -0.00254555 i for the next 1001-2000 it’s 3.43164 +0.00472815 i for 2001-3000 3.64089 -0.00544207 i for 3001-4000 3.80755 -0.00334517 i
and so the running average up to n goes like
n=1000 : 2.86778 -0.00254555 i 2000 : 3.14971 +0.0010913 i 3000 : 3.31343 -0.00108649 i 4000 : 3.43696 -0.00165116 i
no sweet convergence sofar
just curious, Wouter _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun