“Riemann Hypothesis”的概念、定义、习题解题方法及翻译-数学辞典

单词

Riemann Hypothesis

释义

Riemann Hypothesis

First published in Riemann (1859), the Riemann hypothesis states that thenontrivial Roots of the Riemann Zeta Function

(1)

where

(the Complex Numbers), all lie on the ``Critical Line''

, where

denotes the Real Part of

. The Riemann hypothesis is also known as Artin'sConjecture.

In 1914, Hardy proved that an Infinite number of values for can be found for which and. However, it is not known if all nontrivial roots satisfy , so the conjecture remainsopen. André Weil proved the Riemann hypothesis to be true for field functions (Weil 1948, Eichler 1966, Ball andCoxeter 1987). In 1974, Levin showed that at least 1/3 of the Roots must lie on the Critical Line(Le Lionnais 1983), a result which has since been sharpened to 40% (Vardi 1991, p. 142). It is known that the zeros aresymmetrical placed about the line .

The Riemann hypothesis is equivalent to , where is the de Bruijn-Newman Constant (Csordaset al. 1994). It is also equivalent to the assertion that for some constant ,

(2)

where

is the Logarithmic Integral and

is the Prime Counting Function (Wagon 1991).

The hypothesis was computationally tested and found to be true for the first zeros by Brent et al. (1982).Brent's calculation covered zeros in the region .

There is also a finite analog of the Riemann hypothesis concerning the location of zeros for function fields defined byequations such as

(3)

This hypothesis, developed by Weil, is analogous to the usual Riemann hypothesis. The number of solutions for theparticular cases

, (3,3), (4,4), and (2,4) were known to Gauß.

See also Critical Line, Extended Riemann Hypothesis, Gronwall's Theorem, Mertens Conjecture,Mills' Constant, Riemann Zeta Function

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 75, 1987.

Brent, R. P. ``On the Zeros of the Riemann Zeta Function in the Critical Strip.'' Math. Comput. 33, 1361-1372, 1979.

Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. ``On the Zeros of the Riemann Zeta Function in the Critical Strip. II.'' Math. Comput. 39, 681-688, 1982. Abstract available at ftp://nimbus.anu.edu.au/pub/Brent/rpb070a.dvi.Z.

Csordas, G.; Smith, W.; and Varga, R. S. ``Lehmer Pairs of Zeros, the de Bruijn-Newman Constant and the Riemann Hypothesis.'' Constr. Approx. 10, 107-129, 1994.

Eichler, M. Introduction to the Theory of Algebraic Numbers and Functions. New York: Academic Press, 1966.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 25, 1983.

Odlyzko, A. ``The th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors.''

Riemann, B. ``Über die Anzahl der Primzahlen unter einer gegebenen Grösse,'' Mon. Not. Berlin Akad., pp. 671-680, Nov. 1859.

Sloane, N. J. A. SequenceA002410/M4924in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

van de Lune, J. and te Riele, H. J. J. ``On The Zeros of the Riemann Zeta-Function in the Critical Strip. III.'' Math. Comput. 41, 759-767, 1983.

van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. ``On the Zeros of the Riemann Zeta Function in the Critical Strip. IV.'' Math. Comput. 46, 667-681, 1986.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, p. 33, 1991.

Weil, A. Sur les courbes algébriques et les variétès qui s'en déduisent. Paris, 1948.

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