  The system under consideration is a two-dimensional one-component plasma in
fluid regime, at density n and at arbitrary coupling Gamma=beta e^2 (e=unit
charge, beta = inverse temperature). The Helmholtz free energy of the model, as
the generating functional for the direct pair correlation c, is treated in
terms of a convergent renormalized Mayer diagrammatic expansion in density.
Using specific topological transformations within the bond-renormalized Mayer
expansion we prove that the nonzero contributions to the regular part of the
Fourier component of c up to the k^2-term originate exclusively from the ring
diagrams (unable to undertake the bond-renormalization procedure) of the
Helmholtz 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 new
six-momnt condition for the truncated pair correlation h, n(pi Gamma n/2)^3
Integral r^6 h(r) d^2 r = 3(Gamma-6)(8-3 Gamma)/4.
