Logarithmic accuracy Does anyone know how the method of logarithmic accuracy works and what do I have to know about it (as far as applied Mathematics is concerned)? Any references, examples or guidelines would be appreciated.
Thank you
 A: A quantity is said to be calculated “to logarithmic accuracy” when in an expansion like
$$
F(\varepsilon) = c \sum_{n=2}^{\infty} A^n \sum_{m=2}^{n} a_{n,m} \ln^{n-m}(\varepsilon/E) ~,
$$
only the coefficients $a_{n,m}$ associated with the most divergent terms are calculated exactly while the coefficients for $m>2$ are determined by an approximation procedure.
ref: Duke, C. B., and S. D. Silverstein,
“Does Logarithmic Accuracy Uniquely Define the Low‐Temperature Properties of Dilute Magnetic Alloy Systems?”, J. Appl. Phys. 39 (1968) 708, http://dx.doi.org/10.1063/1.2163592 .
As a side remark, infra-red divergence, i.e., divergence of calculated physical quantities at low energy/momentum/temperature scales, is a widely known (and perhaps, ubiquitous) issue in current-day condensed matter physics, and has many important consequences.

In models of condensed matter physics, ultraviolet divergences present
  not a problem but a nuisance. Their presence indicates that the
  continuum description is incomplete, i.e. the behaviour of
  long-wavelength excitations depends on shorter length scales. ... .
  Infrared divergences are more interesting. Their appearance is always
  an indication of an incorrectly chosen reference ground state. For
  example, if electrons in a metal attract, it is wrong to approximate
  them as free particles; the real ground state is a superconducting
  condensate of electron pairs. This ground state is orthogonal to the
  ground state of the non-interacting electron gas and therefore is
  unreachable by a perturbation expansion. There is no universal recipe
  for how to choose a correct ground state. Sometimes it is relatively
  easy (as for ordinary superconductors), sometimes it is not easy at
  all.

[Tsvelik, A. M. Quantum field theory in condensed matter physics. Cambridge University Press (2006), p. 48]
