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Removed incorrect edit in first paragraph about arithmetic progression formula from elementary math. |
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<math>\psi_0(x) = \frac{1}{2} \lim_{h\to 0} (\psi(x+h) + \psi(x-h))</math> and the RHS is obtained from the [[residue theorem]], and then converting it into the formula that Riemann himself actually sketched.
This series is also conditionally convergent and the sum over zeroes should again be taken in increasing order of imaginary part :<ref name=Ing77>Ingham (1990) p.77</ref> <math>\sum_\rho\frac{x^\rho}{\rho} = \lim_{T \rightarrow \infty} S(x,T) \ </math> where <math>S(x,T) = \sum_{\rho:|\Im \rho| \le T} \frac{x^\rho}{\rho}</math>. The error involved in truncating the sum to ''S''(''x'',''T'') is always smaller than ln(''x'') in absolute value, and when divided by the [[natural logarithm]] of ''x'', has absolute value smaller than ''x''/''T'' divided by the distance from ''x'' to the nearest prime power.
==Weil's explicit formula ==
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