Riemann–Siegel formula

The Riemann–Siegel formula expresses the Riemann zeta function on the critical line as a finite oscillatory sum plus a correction term. It enables efficient computation of the zeta function at large heights.

Riemann zeta function: The meromorphic continuation of the series ζ(s) = ∑ₙ₌₁∞ n⁻ˢ, central to the distribution of prime numbers. The formula computes this function specifically at points on its critical line.
Bernhard Riemann: A German mathematician whose work transformed analysis, geometry, and number theory. Riemann's zeta-function investigations supplied the problem the formula addresses.
Odlyzko–Schönhage algorithm: An algorithm for evaluating the Riemann zeta function at many points efficiently using fast Fourier transforms. It accelerates large-scale zeta computations that use Riemann–Siegel evaluations.
Dirichlet series: A series of the form ∑ₙ₌₁∞ aₙn⁻ˢ, including the defining series for the Riemann zeta function. The defining series converges too slowly on the critical line at large heights.
Critical line: The vertical line Re(s) = 1/2 in the complex plane, where the nontrivial zeros of the Riemann zeta function are conjectured to lie. The formula is tailored to evaluate zeta at s = 1/2 + it.
Carl Ludwig Siegel: A German mathematician known for major contributions to number theory and the theory of quadratic forms. Siegel uncovered and developed the formula from Riemann's unpublished papers.
Critical-line zero computation: The numerical search and verification of zeros of the Riemann zeta function on Re(s) = 1/2. The formula makes high zero searches feasible by rapidly evaluating zeta values.
Riemann–Siegel integral formula: An integral representation associated with Riemann's method for studying the zeta function. The finite-sum formula is a computationally useful outcome of Riemann's integral approach.
Riemann–Siegel theta function: The real-valued phase function that makes the Hardy Z-function real on the critical line. Its phase organizes the oscillatory terms in the formula's finite sum.
Riemann's Nachlass: The collection of Bernhard Riemann's unpublished mathematical papers and notes. Siegel found the formula's source in Riemann's mathematical estate.

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Riemann–von Mangoldt formulaRelated: It provides a computational route to evaluating the zero-counting function.