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The Riemann zeta function can also be defined in terms of by

Proof of the Riemann hypothesis is number 8 of and number 1 of .

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The Riemann zeta function satisfies the

Zeros of come in (at least) two different types. So-called "trivial zeros" occur at negative even integers , , , ..., and "nontrivial zeros" at certain

(Ayoub 1974), which was proved by Riemann for all complex  (Riemann 1859).

The Riemann hypothesis was computationally tested and found to be true for the first zeros by Brent (1982), covering zeros in the region ). S. Wedeniwski used ZetaGrid () to prove that the first trillion () nontrivial zeros lie on the . Gourdon (2004) then used a faster method by Odlyzko and Schönhage to verify that the first ten trillion () nontrivial zeros of the function lie on the . This computation verifies that the Riemann hypothesis is true at least for all less than 2.4 trillion. These results are summarized in the following table, where indicates a .

The Riemann zeta function can be split up into

As defined above, the zeta function with a is defined for . However, has a unique to the entire , excluding the point , which corresponds to a with 1 (Krantz 1999, p. 160). In particular, as , obeys

In 1914, Hardy proved that an number of values for can be found for which and (Havil 2003, p. 213). However, it is not known if nontrivial roots satisfy . Selberg (1942) showed that a positive proportion of the nontrivial zeros lie on the , and Conrey (1989) proved the fraction to be at least 40% (Havil 2003, p. 213).

The Riemann zeta function is related to the and by

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André Weil proved the Riemann hypothesis to be true for field functions (Weil 1948, Eichler 1966, Ball and Coxeter 1987). In 1974, Levinson (1974ab) showed that at least 1/3 of the must lie on the (Le Lionnais 1983), a result which has since been sharpened to 40% (Vardi 1991, p. 142). It is known that the zeros are symmetrically placed about the line . This follows from the fact that, for all complex numbers ,

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  • Clay Mathematics Institute. "The Riemann Hypothesis." .

    The Riemann hypothesis is equivalent to the statement that all the zeros of the (a.k.a. the alternating zeta function)

  • Conrey, J. B. "The Riemann Hypothesis." 50,341-353, 2003. .

    The Riemann zeta function can also be defined in the complex plane by the

  • Derbyshire, J. New York: Penguin, pp. 371-372, 2004.

    Balazard, M. and Saias, E. "The Nyman-Beurling Equivalent Form for the RiemannHypothesis." 18, 131-138, 2000.

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The of the Riemann zeta function for is defined by

In 2000, the Clay Mathematics Institute () offered a $1 million prize () for proof of the Riemann hypothesis. Interestingly, of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the ), does not earn the $1 million award.

de Branges, L. "Riemann Zeta Functions." May 24, 2004. .

Here, the sum on the right-hand side is exactly the (sometimes also called the alternating zeta function). While this formula defines for only the , equation (◇) can be used to analytically continue it to the rest of the . can also be performed using . A globally convergent series for the Riemann zeta function (which provides the of to the entire except ) is given by

and "Riemann Hypothesis." From --A Wolfram Web Resource.

It is also known that the nontrivial zeros are symmetrically placed about the , a result which follows from the functional equation and the symmetry about the line . For if is a nontrivial zero, then is also a zero (by the functional equation), and then is another zero. But and are symmetrically placed about the line , since , and if , then . The Riemann hypothesis is equivalent to , where is the (Csordas 1994). It is also equivalent to the assertion that for some constant ,

Conrey, J. B. "The Riemann Hypothesis." 50,341-353, 2003. .

This hypothesis, developed by Weil, is analogous to the usual Riemann hypothesis. The number of solutions for the particular cases , (3,3), (4,4), and (2,4) were known to Gauss.

Derbyshire, J. New York: Penguin, 2004.

discovering a formula for the area of a sphericaltriangle), but is most famous for publishingworks on his "principle of indivisibles" (calculus);these were very influentialand inspired further development by Huygens, Wallis and Barrow.

Keiper, J. "The Zeta Function of Riemann." 4,5-7, 1995.

Fellow geniuses are the best judges of genius, and BlaisePascal had this to say of Fermat:"For my part, I confess that [Fermat's researches about numbers]are far beyond me, and I am competent only to admire them."E.T.

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