Euler-Riemann's zeta and Dirichlet's eta functions are defined for real negative numbers as analytic continuation. In the present book the author defines new series for zeta and eta functions, for real negative and imaginary numbers without analytic continuation. The new zeta and eta functions maintain the character of harmonic series and alternating harmonic series respectively for real negative numbers, as for real positive numbers. Zeta and eta functions have also been defined for multiple- and fractional harmonic series.