Problems of the Millennium: the Riemann Hypothesis
E. Bombieri

I. The problem. The Riemann zeta function is the function of the complex variable s, defined in the half-plane1 (s) > 1 by theabsolutely convergent series ζ(s) := 1 , ns n=1


and in the whole complex plane C by analytic continuation. As shown by Riemann, ζ(s) extends to C as a meromorphic function with only a simple pole at s= 1 , with residue 1 , and satisfies the functional equation s 1−s π −s/2 Γ( ) ζ(s) = π −(1−s)/2 Γ( ) ζ(1 − s). 2 2 (1)

In an epoch-making memoir published in 1859, Riemann [Ri] obtained ananalytic formula for the number of primes up to a preassigned limit. This formula is expressed in terms of the zeros of the zeta function, namely the solutions ρ ∈ C of the equation ζ(ρ) = 0 . In this paper,Riemann introduces the function of the complex variable t defined by 1 s ξ(t) = s(s − 1) π −s/2 Γ( ) ζ(s) 2 2 with s = 1 +it, and shows that ξ(t) is an even entire function of t whose zeros have 2imaginary part between −i/2 and i/2 . He further states, sketching a proof, that in the range between 0 and T the function ξ(t) has about (T /2π) log(T /2π) − T /2π zeros. Riemann then continues: “Manfindet nun in der That etwa so viel reelle Wurzeln innerhalb dieser Grenzen, und es ist sehr wahrscheinlich, dass alle Wurzeln reell sind.”, which can be translated as “Indeed, one finds between thoselimits about that many real zeros, and it is very likely that all zeros are real.” The statement that all zeros of the function ξ(t) are real is the Riemann hypothesis. The function ζ(s) has zeros at thenegative even integers −2, −4, . . . and one refers to them as the trivial zeros. The other zeros are the complex numbers 1 + iα 2 where α is a zero of ξ(t). Thus, in terms of the function ζ(s), we canstate Riemann hypothesis. The nontrivial zeros of ζ(s) have real part equal to
1 2

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In the opinion of many mathematicians the Riemann hypothesis, and its extension to general classes of L... [continua]

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