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The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series), published by St Petersburg Academy in 1737.
In doing so, he discovered a connection between Riemann zeta function and prime numbers, known as the Euler product formula for the Riemann zeta function. Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem.
The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series = = = + + +Leonhard Euler considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem.
In doing so, he discovered the connection between the Riemann zeta function and prime numbers; this is known as the Euler product formula for the Riemann zeta function. [85] Euler invented the totient function φ(n), the number of positive integers less than or equal to the integer n that are coprime to n.
The Riemann zeta function ζ(s) is one of the most significant functions in mathematics because of its relationship to the distribution of the prime numbers. The zeta function is defined for any complex number s with real part greater than 1 by the following formula: ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty ...
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler .