Problems of the Millennium: the Riemann Hypothesis
E. Bombieri

I. The problem. The Riemann zeta function is thefunction of the complex variable s, defined in the half-plane1 (s) > 1 by the absolutely 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 ameromorphic 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 zerosof the zeta function, namely the solutions ρ ∈ C of the equation ζ(ρ) = 0 . In this paper, Riemann introduces thefunction 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 2 imaginary part between −i/2 and i/2 . He further states, sketchinga 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 c [continua]

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