Series sigman 1 n-z converges pointwise in U = Re(z) > 1 to the Riemann zeta fun

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The Riemann zeta function ?(s) is a function of a complex variable s = ? + it (here, s, ? and t are traditional notations associated with the study of the ?-function). The following infinite series converges for all complex numbers s with real part greater than 1, and defines ?(s) in this case: The Riemann zeta function is defined as the analytic continuation of the function defined for ? > 1 by the sum of the preceding series. Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1.[2] The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that ? > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ? 1. For s = 1 the series is the harmonic series which diverges to +?, and Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1. so not uniform.

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