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PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS 3 and proves the identity4 ∑ n≤x ψ( x n ) = log [x]! . From this identity, he finally obtains numerical upper and lower bounds for ψ(x), ϑ(x) and π(x). Popular variants of Tchebychev’s method, based on the integrality of suitable ratios of factorials, originate much later and cannot be ascribed to Tchebychev. Riemann’s memoir on π(x) is really astonishing for the novelty of ideas intro- duced. He first writes ζ(s) using the integral formula, valid for (s) > 1 : ζ(s) = 1 Γ(s) ∫ ∞ 0 e−x 1 − e−x xs−1 dx, (4) and then deforms the contour of integration in the complex plane, so as to obtain a representation valid for any s. This gives the analytic continuation and the functional equation of ζ(s). Then he gives a second proof of the functional equation in the symmetric form (1), introduces the function ξ(t) and states some of its properties as a function of the complex variable t. Riemann continues by writing the logarithm of the Euler product as an integral transform, valid for (s) > 1 : 1 s log ζ(s) = ∫ ∞ 1 ∏ (x)x−s−1 dx (5) where ∏ (x) = π(x) + 1 2 π( 2 √x) + 1 3 π( 3 √x) + . . . . By Fourier inversion, he is able to express ∏ (x) as a complex integral, and compute it using the calculus of residues. The residues occur at the singularities of log ζ(s) at s = 1 and at the zeros of ζ(s). Finally an inversion formula expressing π(x) in terms of ∏ (x) yields Riemann’s formula. This was a remarkable achievement which immediately attracted much attention. Even if Riemann’s initial line of attack may have been influenced by Tchebychev (we find several explicit references to Tchebychev in Riemann’s unpublished Nach- lass5) his great contribution was to see how the distribution of prime numbers is determined by the complex zeros of the zeta function. At first sight, the Riemann hypothesis appears to be only a plausible interesting property of the special function ζ(s), and Riemann himself seems to take that view. He writes: “Hiervon w¨are allerdings ein strenger Beweis zu w¨unschen; ich habe indess die Aufsuchung desselben nach einigen fl¨uchtigen vergeblichen Versuchen vorl¨aufig bei Seite gelassen, da er f¨ur den n¨achsten Zweck meiner Untersuchung entbehrlich schien.”, which can be translated as “Without doubt it would be desir- able to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study.” 4 Here [x] denotes the integral part of x . 5 The Nachlass consists of Riemann’s unpublished notes and is preserved in the mathematical library of the University of G¨ottingen. The part regarding the zeta function was analyzed in depth by C.L. Siegel [Sie].

4 E. BOMBIERI On the other hand, one should not draw from this comment the conclusion that the Riemann hypothesis was for Riemann only a casual remark of minor interest. The validity of the Riemann hypothesis is equivalent to saying that the deviation of the number of primes from the mean Li(x) is π(x) = Li(x) + O(√x log x); the error term cannot be improved by much, since it is known to oscillate in both directions to order at least Li(√x) log log log x (Littlewood). In view of Riemann’s comments at the end of his memoir about the approximation of π(x) by Li(x), it is quite likely that he saw how his hypothesis was central to the question of how good an approximation to π(x) one may get from his formula. The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers. This fact alone singles out the Riemann hypothesis as the main open question of prime number theory. The Riemann hypothesis has become a central problem of pure mathematics, and not just because of its fundamental consequences for the law of distribution of prime numbers. One reason is that the Riemann zeta function is not an isolated object, rather is the prototype of a general class of functions, called L-functions, associated with algebraic (automorphic representations) or arithmetical objects (arithmetic varieties); we shall refer to them as global L-functions. They are Dirichlet series with a suitable Euler product, and are expected to satisfy an appropriate functional equation and a Riemann hypothesis. The factors of the Euler product may also be considered as some kind of zeta functions of a local nature, which also should satisfy an appropriate Riemann hypothesis (the so-called Ramanujan property). The most important properties of the algebraic or arithmetical objects underlying an L-function can or should be described in terms of the location of its zeros and poles, and values at special points. The consequences of a Riemann hypothesis for global L-functions are important and varied. We mention here, to indicate the variety of situations to which it can be applied, an extremely strong effective form of Tchebotarev’s density theorem for number fields, the non-trivial representability of 0 by a non-singular cubic form in 5 or more variables (provided it satisfies the appropriate necessary congruence conditions for solubility, Hooley), and Miller’s deterministic polynomial time pri- mality test. On the other hand, many deep results in number theory which are consequences of a general Riemann hypothesis can be shown to hold independently of it, thus adding considerable weight to the validity of the conjecture. It is outside the scope of this article even to outline the definition of global L- functions, referring instead to Iwaniec and Sarnak [IS] for a survey of the expected properties satisfied by them; it suffices here to say that the study of the analytic properties of these functions presents extraordinary difficulties. Already the analytic continuation of L-functions as meromorphic or entire func- tions is known only in special cases. For example, the functional equation for the L-function of an elliptic curve over Q and for its twists by Dirichlet characters is an easy consequence of, and is equivalent to, the existence of a parametrization of the curve by means of modular functions for a Hecke group Γ0(N ); the real difficulty lies in establishing this modularity. No one knows how to prove this functional equation by analytic methods. However the modularity of elliptic curves over Q has been established directly, first in the semistable case in the spectacular work