zetaregularization

Zeta regularization is a technique used in theoretical physics and mathematics to handle divergent sums and integrals. It was introduced by Edward Witten in 1982 as a way to regularize and renormalize quantum field theories. The method involves analytically continuing the Riemann zeta function to non-integer values, which allows for the summation of divergent series.

The Riemann zeta function, denoted by ζ(s), is defined for complex numbers s with a real part greater than 1 as the sum of the reciprocals of the natural numbers raised to the power of s. By analytically continuing this function, it can be evaluated for all complex numbers except s = 1, where it has a simple pole.

In zeta regularization, the divergent sum is replaced by an expression involving the zeta function evaluated at a non-integer value. This value is chosen such that the resulting expression is finite and well-defined. The regularized sum is then defined as the value of this expression.

Zeta regularization has been applied to various areas of physics, including string theory, quantum gravity, and conformal field theory. It has also found applications in number theory and combinatorics. One of its key advantages is that it preserves the analytic properties of the original sum, making it a powerful tool for studying the behavior of physical systems.