Problems of the Millennium: the Riemann Hypothesis
I. The problem. The Riemann zeta function is the function 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 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 an analytic 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 2 imaginary 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 those limits 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 can state Riemann hypothesis. The nontrivial zeros of ζ(s) have real part equal to
In the opinion of many mathematicians the Riemann hypothesis, and its extension to general classes of L-functions, is probably today the most important open problem in pure mathematics. II. History and signiﬁcance of the Riemann hypothesis. For references pertaining to the early history of zeta functions and the theory of prime numbers, we refer to Landau [La] and Edwards [Ed].
1 We denote by (s) and (s) the real and imaginary part of the complex variable s . The use of the variable s is already inDirichlet’s famous work of 1837 on primes in arithmetic progression.
The connection between prime numbers and the zeta function, by means of the celebrated Euler product ζ(s) = (1 − p−s )−1
valid for (s) > 1 , appears for the ﬁrst time in Euler’s book Introductio in Analysin Inﬁnitorum, published in 1748. Euler also studied the values of ζ(s) at the even positive andthe negative integers, and he divined a functional equation, equivalent to Riemann’s functional equation, for the closely related function (−1)n−1 /ns (see the interesting account of Euler’s work in Hardy’s book [Hard]). The problem of the distribution of prime numbers received attention for the ﬁrst time with Gauss and Legendre, at the end of the eighteenth century. Gauss, in a letter to theastronomer Hencke in 1849, stated that he had found in his early years that the number π(x) of primes up to x is well approximated by the function2
dt . log t
In 1837, Dirichlet proved his famous theorem of the existence of inﬁnitely many primes in any arithmetic progression qn+a with q and a positive coprime integers. On May 24, 1848, Tchebychev read at the Academy of St.Petersburg his ﬁrst memoir on the distribution of prime numbers, later published in 1850. It contains the ﬁrst study of the function π(x) by analytic methods. Tchebychev begins by taking the logarithm of the Euler product, obtaining3 −
1 ) + log(s − 1) = log (s − 1)ζ(s) , ps
which is his starting point. Next, he proves the integral formula ζ(s) − 1 − 1 1 = s−1 Γ(s)
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