= Prime-character sum near one
{title2=$F_\chi(s)=\sum_{p\nmid N}\chi(p)p^{-s}$}
For real $s>1$, the logarithm defined by the <Euler product> of a <Dirichlet L-function> differs from $F_\chi(s)$ by a quantity of absolute value at most $\sum_{n\geq2}1/(n(n-1))=1$. If $L(\chi,1)\ne0$, a local <holomorphic logarithm> differs from this continuous logarithm by one fixed multiple of $2\pi i$ on a short real interval, so $F_\chi(s)$ stays bounded as $s\downarrow1$. For the <principal Dirichlet character>, the pole of the <Riemann zeta function> instead gives $F_{\chi_0}(s)=\log(1/(s-1))+O(1)$. <Orthogonality of Dirichlet characters> then makes the sum over any reduced residue class diverge, proving infinitude of its <primes>.
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