## Zeta Regularization Techniques with ApplicationsThis book is the result of several years of work by the authors on different aspects of zeta functions and related topics. The aim is twofold. On one hand, a considerable number of useful formulas, essential for dealing with the different aspects of zeta-function regularization (analytic continuation, asymptotic expansions), many of which appear here, in book format, for the first time are presented. On the other hand, the authors show explicitly how to make use of such formulas and techniques in practical applications to physical problems of very different nature. Virtually all types of zeta functions are dealt with in the book. |

### Contents

ZetaRegularization of | 3 |

Riemann and Hurwitz Zeta Functions | 4 |

References | 20 |

Regularization of Double Series of the Type EpsteinHurwitz | 33 |

Multiple Zeta Functions with Arbitrary Exponents | 43 |

Some Remarks on the Use of the ZetaFunction Regularization | 49 |

ZETA FUNCTION REGULARIZATION GENERALIZED | 57 |

Applications to Effective Lagrangians for Constant Background Fields | 63 |

Oneloop Effective Action for Scalar Fields on KaluzaKlein Space | 162 |

References | 169 |

Path Integral Approach for a Class of Quantum pBrane Models | 183 |

The Casimir Energy for pBranes Compactified on Constant Curva | 197 |

References | 206 |

FiniteTemperature Effects with NonVanishing Chemical Potential | 220 |

Applications to KaluzaKlein Models | 226 |

Free Energy for the Open Superstring Theory and Laurent Series | 235 |

Zeta Functions vs HeatKernel Regularization | 70 |

References | 78 |

Definition of the Casimir Energy and its Relation to the Vacuum | 84 |

ZetaRegularization of Lapla | 107 |

Casimir Energy for Spherical and Cylindrical Universes | 119 |

Path Integral Techniques and ZetaFunction Regularization of | 143 |

Thermodynamics of the Open Bosonic String in an External Mag | 248 |

References | 257 |

282 | |

References | 316 |

### Common terms and phrases

a₁ analytic continuation asymptotic expansion behaviour boundary conditions calculation Casimir effect Casimir energy Chap coefficients compact computation consider constant contour contribution convergence corresponding covariant curvature cutoff defined derivative dimensions Dirichlet divergences effective action eigenvalues Elizalde energy density equation evaluation expression finite free energy gauge given heat kernel Hurwitz zeta function hyperbolic infinite infinity integral invariant Laplacian Lett log˛ manifold massless Math meromorphic method metric Nucl obtained one-loop operator p-brane parameter Phys pole potential procedure quantization quantum field renormalization representation residue result Riemann spheres Riemann zeta function scalar field Selberg Selberg trace formula space spacetime spinor summation supermembrane temperature theory topology transform vacuum energy vanish zero modes zeta function regularization zeta-function regularization