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?



Riemann's attempts to prove RH
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anyone were aware of such a record, it would surely have been Carl Siegel, who undertook a careful study of Riemann’s unpublished notes.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](https://arxiv.org/abs/1810.05198):



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> Approaches to a proof of the so-called “Riemann hypothesis” or even to a proof of the existence of infinitely many zeros of the zeta function on the critical line are not included in Riemann’s papers.
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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.


