Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)} dz$ over appropriate rectangle(s) in the critical strip. This still seems difficult, however, without a computer.
[Math] How did Riemann calculate the first few non-trivial zeros of the zeta-function
analytic-number-theorycv.complex-variablesnt.number-theoryriemann-zeta-function
Best Answer
In searching through the Riemann Nachlass in Gottingen (including those folders not listed as connected with $\zeta(s)) $ there is no evidence -- at least that has been saved -- that Riemann computed anything more than the first few zeros (I think up to ordinate about 80).
The method he used was the expansion that is now called the Riemann-Siegel formula. I did not see any use, e.g., of an approach based on Euler-Maclaurin. The limited accuracy Riemann obtained reflects that of the error term in the R-S formula.