Zeta function regularization is a mathematical technique used to assign values to certain divergent series or integrals that are typically undefined in the classical sense. This technique involves the use of the Riemann zeta function and related functions to provide a meaningful interpretation of these divergent expressions. ### Key Concepts 1. **Divergent Series**: Many series or integrals encountered in quantum field theory, number theory, or statistical mechanics can diverge.
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Zeta function regularization assigns a finite value to a divergent spectral sum by first forming a convergent complex-power series and then using its analytic continuation. For positive frequencies , continue to to define a regulated frequency sum, when that continuation is regular there. This does not turn the divergent positive-term sum into an ordinary convergent sum. It is a useful subtraction convention for the normal-ordering constant of a string and for vacuum energies.