The Riemann zeta function, denoted by the Greek letter zeta, is a function of a complex variable defined for real part greater than 1 by the infinite series of reciprocal powers of the positive integers, and extended to the rest of the complex plane, apart from a single pole at 1, by analytic continuation. Leonhard Euler first studied the function for real arguments in the early 1700s, and Bernhard Riemann's 1859 paper extended it to complex variables and established the functional equation connecting its values on either side of the critical strip. The function can also be written as an infinite product over the prime numbers, the Euler product, which ties its behavior directly to the distribution of primes; the Riemann Hypothesis, that every non-trivial zero of the function has real part one half, remains one of the most consequential unsolved problems in mathematics.
Facts
Origin YearEuler had already studied the same series for real arguments in the early 1700s; 1859 marks Riemann's own paper extending it to the complex plane and establishing its functional equation, which the entity's name commemorates. FAQs
What are the trivial zeros of the Riemann zeta function?
They sit at the negative even integers; the unsolved question concerns the other zeros.
The trivial zeros are the zeros that occur at all the negative even integers. The nontrivial zeros lie in the critical strip, and the Riemann Hypothesis asserts that all of them have real part one half, a vertical line called the critical line.
Connections
Associated With
Riemann's 1859 paper extended the function to the complex plane and is the work the function's name commemorates.
Source Riemann Zeta Function (Wikipedia)
Attributed To
Source Riemann Zeta Function (Wikipedia)
In Branch
Source Riemann Zeta Function (Wikipedia)
Is Kind Of Object
Sources
1. Riemann Zeta Function (Wikipedia)
Wikimedia FoundationLead section
The Riemann zeta function or Euler-Riemann zeta function, denoted by the lowercase Greek letter ΞΆ (zeta), is a mathematical function of a complex variable.
- In Branch: Analytic Number Theory
- Associated With: Bernhard Riemann
Attributed To: Leonhard Euler, Lead
Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century.
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