Many consider it to be the most important unsolved problem in pure mathematics Bombieri It is of great interest in number theory because it implies results about the distribution of prime numbers. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture , comprise Hilbert's eighth problem in David Hilbert 's list of 23 unsolved problems ; it is also one of the Clay Mathematics Institute's Millennium Prize Problems. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
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Zeros of the Riemann zeta function come in two different types. So-called "trivial zeros" occur at all negative even integers , , , In general, a nontrivial zero of is denoted , and the th nontrivial zero with is commonly denoted Brent ; Edwards , p. Wiener showed that the prime number theorem is literally equivalent to the assertion that has no zeros on Hardy , p.
The Riemann hypothesis asserts that the nontrivial zeros of all have real part , a line called the " critical line. An attractive poster plotting zeros of the Riemann zeta function on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research The plots above show the real and imaginary parts of plotted in the complex plane together with the complex modulus of.
As can be seen, in right half-plane, the function is fairly flat, but with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros of lie. The position of the complex zeros can be seen slightly more easily by plotting the contours of zero real red and imaginary blue parts, as illustrated above. The zeros indicated as black dots occur where the curves intersect.
The figures above highlight the zeros in the complex plane by plotting where the zeros are dips and where the zeros are peaks. The above plot shows for between 0 and As can be seen, the first few nontrivial zeros occur at the values given in the following table Wagon , pp. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, OEIS A The numbers of nontrivial zeros less than 10, , , The so-called xi-function defined by Riemann has precisely the same zeros as the nontrivial zeros of with the additional benefit that is entire and is purely real and so are simpler to locate.
ZetaGrid is a distributed computing project attempting to calculate as many zeros as possible. It had reached The following table lists historical benchmarks in the number of computed zeros Gourdon Numerical evidence suggests that all values of corresponding to nontrivial zeros are irrational e.
No known zeros with order greater than one are known. While the existence of such zeros would not disprove the Riemann hypothesis, it would cause serious problems for many current computational techniques Derbyshire , p. Some nontrivial zeros lie extremely close together, a property known as Lehmer's phenomenon. The Riemann zeta function can be factored over its nontrivial zeros as the Hadamard product. Let denote the th nontrivial zero of , and write the sums of the negative integer powers of such zeros as.
Lehmer , Keiper , Finch , p. But by the functional equation, the nontrivial zeros are paired as and , so if the zeros with positive imaginary part are written as , then the sums become. These values can also be written in terms of the Li constants Bombieri and Lagarias It is also equal to the constant from Li's criterion. Bombieri, E.
Number Th. Brent, R. Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, Derbyshire, J. New York: Penguin, Edwards, H. Riemann's Zeta Function. New York: Dover, Farmer, D. Combinatorics 2 , No. Finch, S. Mathematical Constants. Cambridge, England: Cambridge University Press, p. Gourdon, X. Gram, J. Hardy, G. New York: Chelsea, Havil, J. Hayes, B. Hutchinson, J. Keiper, J. Landau, E. Lehmer, D. Odlyzko, A. Pegg, E. Sabbagh, K. Atlantic Books, Sloane, N.
Titchmarsh, E. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, Voros, A. Wagon, S. Mathematica in Action. New York: W. Freeman, Wiener, N. Wolfram Research. Weisstein, Eric W.
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Riemann Zeta Function Zeros
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