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Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
List of topics named after Leonhard Euler
Euler hypergeometric integral Euler–Riemann zeta function Euler's identity e iπ + 1 = 0. Euler's four-square identity, which shows that the product of
Leibniz formula for π
to be converted to an infinite product with one term for each prime number. Such a product is called an Euler product. It is: π 4 = ( ∏ p ≡ 1 ( mod
Euler's totient function
\ln(x)} or log e ( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle
Proof of the Euler product formula for the Riemann zeta function
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations
Euler characteristic
algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant
Riemann hypothesis
his solution to the Basel problem. He also proved that it equals the Euler product ζ ( s ) = ∏ p prime 1 1 − p − s = 1 1 − 2 − s ⋅ 1 1 − 3 − s ⋅ 1 1 −
Dirichlet beta function
( s ) {\displaystyle \zeta (s)} which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact
