The system under consideration is a two-dimensional one-component plasma influid regime, at density n and at arbitrary coupling Gamma=beta e^2 (e=unitcharge, beta = inverse temperature). The Helmholtz free energy of the model, asthe generating functional for the direct pair correlation c, is treated interms of a convergent renormalized Mayer diagrammatic expansion in density.Using specific topological transformations within the bond-renormalized Mayerexpansion we prove that the nonzero contributions to the regular part of theFourier component of c up to the k^2-term originate exclusively from the ringdiagrams (unable to undertake the bond-renormalization procedure) of theHelmholtz free energy. In particular, c(k)=-Gamma/k^2 + Gamma/(8 pi n) -k^2/[96(pi n)^2] + O(k^4). This result fixes via the Ornstein-Zernike relation,besides the well-known zeroth-, second- and fourth- moment sum rules, the newsix-momnt condition for the truncated pair correlation h, n(pi Gamma n/2)^3Integral r^6 h(r) d^2 r = 3(Gamma-6)(8-3 Gamma)/4.
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