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Riemann's Zeta Function H. M. Edwards
Publisher: Academic Press Inc

\displaystyle \zeta(s) = \sum_{n=1}^. If we look at the Taylor expansion. For the Dirichlet series associated to f . $$\xi(s) = (s-1) \pi^{-s/2} \Gamma\left(1+\tfrac{1}{2} s\right) \zeta(s),$$. Observe at once that the Riemann zeta function is given by. So-defined because it puts the functional equation of the Riemann zeta function into the neat form $\xi(1-s) = \xi(s)$. These estimates resulted in the prime number conjecture, which is what Riemann was trying to prove when he invented his zeta function. In 1972, the number theorist Hugh Montgomery observed it in the zeros of the Riemann zeta function, a mathematical object closely related to the distribution of prime numbers. Riemann zeta function on critical line. Riemann zeta function on the critical line as a Fourier transform of exponential sawtooth function plus minus one.