Zeta function regularization

In mathematics and theoretical physics, zeta function regularization is a summability method that allows one to give some meaningful values to seemingly meaningless expressions using the zeta function.

For example

${\displaystyle 1+2+3+\cdots =-{\frac {1}{12}}(\Re )}$

This infinite sum appears in the calculation of the Casimir force, and the regularized value of

${\displaystyle -{\frac {1}{12}}(\Re )}$

gives the correct physically measurable result.

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In theoretical physics, zeta regularization is a method to assign finite values to superficially divergent sums. The method is based on generalizing the sum to the case of more general values of an exponent and treating the sum as an analytic function of the exponent -- namely a function related to the Riemann zeta function. An analytic continuation gives the desired result.

The most well-known example is the sum

1+2+3+4+5+ ... = -1/12

whose value may be calculated as zeta(-1). Such a regularization and the corresponding result are guaranteed to preserve various symmetries of the physical system such as conformal symmetry. This particular result also occurs in the Casimir effect and the critical spacetime dimension of string theory.