Riemann–Siegel Formula

The Riemann–Siegel formula is an asymptotic expansion used to efficiently compute values of the Riemann zeta function ζ(s) when s lies on the critical line and t is large.

Mathematical Setup

• s = 1/2 + it, where t is a large real number
• Z(t) = eiθ(t) ζ(1/2 + it), a real-valued function
• θ(t) is a phase factor defined by:
  
  θ(t) = arg(Γ(1/4 + it/2)) − (t/2) log π

The Formula

The Riemann–Siegel formula expresses Z(t) as:

Z(t) = 2 ∑_n=1^N (1/√n) · cos(θ(t) − t · log n) + R(t)

• N = floor(√(t / 2π))
• R(t) is the remainder term, which becomes smaller as t increases

Why It’s Useful

• Efficiency: Reduces the number of terms needed for large t
• Accuracy: Remainder term can be bounded and estimated
• Applications: Used in verifying the Riemann Hypothesis numerically

Error Term and Refinements

The remainder R(t) has an asymptotic expansion involving powers of t−1/2. Researchers like Gabcke and Titchmarsh have refined bounds for this term. The Odlyzko–Schönhage algorithm further accelerates computations using FFT techniques.

Historical Note

Siegel discovered the formula in Riemann’s unpublished notes and published it in 1932. It remains a cornerstone in analytic number theory and computational mathematics.