Expressions for Sudakov form factors for heavy quarks are presented. They are used to construct resummed jet rates for up to four jets in e + e − annihilation. The coefficients of leading and nextto-leading logarithmic corrections, mandatory for a combination with higher order matrix elements, are evaluated up to second order in αs.PACS numbers: 12.38. Bx, 12.38.Cy, 13.66.Bc, 14.65.Fy The formation of jets is the most prominent feature of perturbative QCD in e + e − annihilation into hadrons. Jets can be visualized as large portions of hadronic energy or, equivalently, as a set of hadrons confined to an angular region in the detector. In the past, this qualitative definition was replaced by quantitatively precise schemes to define and measure jets, such as the cone algorithms of the Weinberg-Sterman [1] type or clustering algorithms, e.g. the Jade [2,3] or the Durham scheme (k ⊥ scheme) [4]. A refinement of the latter one is provided by the Cambridge algorithm [5]. Within the context of this paper, however, the Durham and the Cambridge algorithms are equivalent and they will be referred to as k ⊥ algorithm. A clustering according to the relative transverse momenta has a number of properties that minimize the effect of hadronization corrections and allow an exponentiation of leading (LL) and next-to-leading logarithms (NLL) [4,6] stemming from soft and collinear emission of secondary partons.Equipped with a precise jet definition the determination of jet production cross sections and their intrinsic properties is one of the traditional tools to investigate the structure of the strong interaction and to deduce its fundamental parameters. In the past decade, precision measurements, especially in e + e − annihilation, have established both the gauge group structure underlying QCD [7,8,9,10,11] and the running of its coupling constant α s over a wide range of scales [12]. In a similar way, also the quark masses should vary with the scale.A typical strategy to determine the mass of, say, the bottom-quark at the centre-of-mass (c.m.) energy of the collider is to compare the ratio of three-jet production cross sections for heavy and light quarks [13,14,15,16,17]. At jet resolution scales below the mass of the quark, i.e. for gluons emitted by the quark with a relative transverse momentum k ⊥ smaller than the mass, the collinear divergences are regularized by the quark mass. In this region mass effects are enhanced by logarithms ln(m b /k ⊥ ), increasing the significance of the measurement. Indeed, this leads to a multiscale problem since in this kinematical region also large logarithms ln( √ s/k ⊥ ) appear such that both logarithms need to be resummed simultaneously. A solution to a somewhat similar two-scale problem, namely for the average subjet multiplicities in two-and three-jet events in e + e − annihilation was given in [35]. These large logarithms can be deduced by inspection of the corresponding splitting functions for massive quarks [18,19,20,21,22] and of the boundaries for their integration over the energy...