Riemann zeta function: Difference between revisions

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imported>Barry R. Smith
(Added a section on the history, and section headings coresponding to the functional equation, zeroes, special values, and generalizations.)
imported>Barry R. Smith
(Add a sentence giving prerequisites for understanding the definition.)
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==Definition==
==Definition==
To understand this definition, you must already understand the concepts of [[complex exponents]] and [[infinite series]] of complex numbers.


 
The Riemann zeta function is a [[meromorphic function]] defined for [[complex number]]s with [[real part]] <math>\scriptstyle \Re(s) > 1</math> by the infinite series
The Riemann zeta function is a [[meromorphic function]] defined for [[complex number]]s with [[real part]] <math>\scriptstyle \Re(s) > 1</math> by the [[infinite series]]


: <math> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} </math>
: <math> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} </math>
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The celebrated [[Riemann hypothesis]] is the conjecture that all non-real values of ''s'' for which &zeta;(''s'')&nbsp;=&nbsp;0 have real part 1/2.  The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics.
The celebrated [[Riemann hypothesis]] is the conjecture that all non-real values of ''s'' for which &zeta;(''s'')&nbsp;=&nbsp;0 have real part 1/2.  The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics.


==History==
==History==

Revision as of 21:18, 27 March 2008

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In mathematics, the Riemann zeta function, named after Bernhard Riemann, is one of the most important special functions. Its generalizations have important applications to number theory, arithmetic geometry, graph theory, and dynamical systems, to name a few examples. The Riemann zeta function in particular gained prominence when it was shown to have a connection with the distribution of the prime numbers. The most important result related to the Riemann zeta function is the Riemann hypothesis, which was the 8th of Hilbert's Problems, and is one of the seven Millenium Prize Problems presented by the Clay Institute of Mathematics. As such, anyone who determines its truth or falsity is entitled to $1 million (U.S.)

Definition

To understand this definition, you must already understand the concepts of complex exponents and infinite series of complex numbers.

The Riemann zeta function is a meromorphic function defined for complex numbers with real part by the infinite series

and then extended to all other complex values of s except s = 1 by analytic continuation. The function is holomorophic everywhere except for a simple pole at s = 1.

Euler's product formula for the zeta function is

(the index p running through the set of prime numbers).

The celebrated Riemann hypothesis is the conjecture that all non-real values of s for which ζ(s) = 0 have real part 1/2. The problem of proving the Riemann hypothesis is the most well-known unsolved problem in mathematics.

History

The origin of the Riemann zeta function can be traced to the Basel Problem. The solution to this problem states that

In deriving this identity, Leonard Euler also found the sums of the series for [1] These computations contain implicitly the germ of the idea of the zeta function. According to Andre Weil, these and related results due to Euler remained as "mere curiosities, and virtually unknown, until they received new life at the hands of Riemann in 1859".[2]

In an eight page paper, Riemann catapulted both himself and his namesake to worldwide renown. The paper includes the general definition of the zeta function (including the first use of the symbol to denote it) and the proof of its analytic continuation, the functional equation (see below), as well as results relating the function to the distribution of prime numbers and of course, the Riemann hypothesis. Since then, the function and its relatives have found applications in myriad research papers in a vast array of fields.

The functional equation

Zeroes

Special Values

Generalizations

  1. See William Dunham, Journey Through Genius.
  2. Andre Weil, Basic Number Theory, p.185.