Zeta Function and Paris Prize

These two justly famous papers arose under special circumstances, which partly explains why they were never expounded in complete form. During his trip to Berlin in 1859, Riemann and Kronecker discussed a famous problem that both Legendre and Gauss had attempted to solve: can one find an analytic formula that asymptotically estimates the number of primes below a given large number? Riemann's contribution to this question, his short paper on the zeta function, was published soon after that visit, whereas his incomplete solution to a question posed by the Paris Academy was submitted at the very last moment before the deadline in 1861. Since it failed to win the prize, this paper vanished until Dedekind and Weber placed it in Riemann's Collected Works, first published in 1876. Both papers are technically demanding, so as with Riemann's 1857 paper on Abelian functions, discussed in Chapter 7, they are only summarized here. Riemann's zeta function was defined via complexification of Euler's version, which already revealed, via Euler's product formula, its surprising connection with the prime numbers. Riemann found an ingenious way to extend it to the entire complex plane. The later famous Riemann conjecture, stating that all the nontrivial zeros of the zeta function lie on the line s = 1/2, was only one of the questions he left unresolved. In fact, Riemann's paper attracted rather little attention until the 1890s when Jacques Hadamard proved the prime number theorem along with several still open questions raised in Riemann's paper. In the case of the Paris prize paper, this later attracted considerable attention in the wake of Einstein's theory of general relativity. The second part of Riemann's paper presents some of the technical machinery omitted from the Habilitation lecture, including what came to be known as Riemann's curvature tensor. Even in this century, experts have continued to scrutinize the difficult problem of discerning in what ways these two texts may have been related in Riemann's mind.