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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. [1] [2]
In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L -functions or zeta functions .
The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +:
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.
This book concerns the Riemann zeta function and the Riemann hypothesis on the location of the zeros of this function. It includes a translation of Riemann's original paper on these subjects, and analyzes this paper in depth; it also covers methods of computing the function such as Euler–Maclaurin summation and the Riemann–Siegel formula.
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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.