One loop partition functions in a compactified open bosonic string theory

We calculate the one loop partition functions of a toroidally compactified open bosonic string (fromD=26 toD=10) in the path integral formalism. By a certain combination of the compactified sectors it is shown that the partition functions of a compactified open bosonic string vanish in both annulus and Möbius strip topologies as in the open superstring case. We also discuss the consistency requirement for the equivalence between the compactified open bosonic string and the open superstring with the gauge groups (SO(8192) andSO(32), respectively.