Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
4  
see store.doverpublications.com/0486417409.html especially reproduction on page 156 and discussion of Riemann's hand calculations from about 155-162 –  Will Jagy Sep 19 '13 at 4:32
    
Thanks Will. I found your reference very helpful. –  Mustafa Said Sep 19 '13 at 6:04

2 Answers 2

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.

share|improve this answer

"Know" is hard for those of us without a ouija board, but I think people believe that the Riemann-Siegel formula was used.

share|improve this answer
2  
I'm looking at Edward's book, this is discussed in pages 155-162 and on. He reproduces the page with the formula from the Gottingen Library. –  Will Jagy Sep 19 '13 at 3:06

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.