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| viXra.org > Number Theory > viXra:2604.0097 |
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Number Theory |
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Authors: Yuhua Li
In this paper,we proved that for classic integral representation of Riemann $xi$-function $xi(s)=frac{1}{2}+frac{s(s-1)}{2}int_1^inftypsi(x)(x^{s/2-1}+x^{(1-s)/2-1})dx=-4psi'(1)+int_1^inftypsi'(x)((1-s)x^{s/2}+sx^{(1-)/2})dx=2int_1^infty(frac{3}{2}psi'(x)+xpsi''(x))(x^{s/2}+x^{(1-s)/2})dx$ , the common lower limitation 1 of the three divergent series equals each other (including $frac{-1}{2}$ for the first and $(-4)psi'(1)$ for the second).This provides a new approach to prove the three integral representation without using partial integration.
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[v1] 2026-04-26 18:30:17
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