zeta0

Zeta0 is not a standard mathematical constant; the term can appear in different contexts as a shorthand related to the Riemann zeta function. In many cases, zeta0 is used to denote the value of the zeta function at zero, i.e., zeta0 := zeta(0).

The Riemann zeta function zeta(s) is defined for complex numbers with real part greater than 1 by the Dirichlet series zeta(s) = sum_{n≥1} n^{-s} and is extended to the rest of the complex plane by analytic continuation. It satisfies zeta(0) = -1/2. The function also has trivial zeros at negative even integers, zeta(-2n) = 0 for n ≥ 1, and nontrivial zeros in the critical strip 0 < Re(s) < 1, many of which lie on the critical line Re(s) = 1/2 (the Riemann hypothesis).

Outside a fixed convention, zeta0 may also appear as a label in numerical experiments or software, representing the first element in a sequence of zeta evaluations or as a placeholder for a base value in zeta-regularization methods in physics. Because zeta0 is not standardized, its meaning depends on the source; always consult the surrounding definitions in the given text or code to determine its precise interpretation.

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