We consider the Renyi alpha entropies for Luttinger liquids (LL). For large block lengths l, these are known to grow like lnl. We show that there are subleading terms that oscillate with frequency 2k(F) (the Fermi wave number of the LL) and exhibit a universal power-law decay with l. The new critical exponent is equal to K/(2 alpha), where K is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.
