I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I am wondering, then, is there any record of his attempts to prove RH?
Number Theory – Riemann’s Attempts to Prove the Riemann Hypothesis
cv.complex-variablesnt.number-theoryriemann-hypothesisriemann-zeta-function
Best Answer
The short answer is no. If anyone were aware of such a record, it would surely have been Carl Siegel, who undertook a careful study of Riemann’s unpublished notes. However, Siegel wrote:
Riemann himself, in his paper on the zeta function, said only that he made “some fleeting, vain attempts” (einigen flüchtigen vergeblichen Versuchen) to prove (what we now call) the Riemann hypothesis, and gave no indication that he recorded these attempts.
Having said that, I want to mention that there is some interesting speculation in Chapter 7 (on the Riemann–Siegel formula; see especially section 7.8) in H. M. Edwards's book Riemann’s Zeta Function about what Riemann’s train of thought might have been.