In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann who introduced it in 1859, is a prominent function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.
The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, i...
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In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann who introduced it in 1859, is a prominent function of great significance in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.
The Riemann hypothesis, a conjecture about the distribution of the zeros of the Riemann zeta function, is considered by many mathematicians to be the most important unsolved problem in pure mathematics.
The Riemann zeta function ζ(s) is a function of a complex variable s. The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:
The Riemann zeta function is defined as the analytic continuation of the function defined for Re(s) > 1 by the sum of the preceding series.
Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev...
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