CECM Colloquium

Wednesday November 21, 2001 in K9509, SFU

3:30 - 4:30
Will Galway

Title: "Computing the Riemann Zeta Function by Numerical Quadrature."

Abstract: The Riemann zeta function, $\zeta(s)$, where $s=\sigma + i t$, may be computed to moderate accuracy, using $O(t^{1/2+\epsilon})$ arithmetic operations, by use of the Riemann-Siegel formula, which for fixed $\sigma$ gives an asymptotic expansion of $\zeta(s)$ as $t \to \infty$. In contrast, the Euler-Maclaurin summation formula may be used to compute $\zeta(s)$ to arbitrarily high accuracy, requiring $O(t^{1+\epsilon} + d^{1+\epsilon})$ operations to compute $d$ digits~\cite{CohenOlivierZeta}.

We analyze an integral underlying the Riemann-Siegel formula and show that numerical quadrature may be used to compute $\zeta(s)$ to arbitrarily high accuracy. This method requires $O(t^{1/2+\epsilon})$ operations to find $d$ digits of $\zeta(s)$, for fixed $d$ as $t \to \infty$. For large $t$, this quadrature method should allow computation of $\zeta(\sigma+i t)$ to high precision in much less time than by the Euler-Maclaurin formula. Further, precise error bounds for the Riemann-Siegel formula are known only in the case $\sigma=1/2$, while the quadrature method should allow computation of $\zeta(\sigma + i t)$ to a prescribed accuracy, for arbitrary $\sigma$, with efficiency comparable to that of the Riemann-Siegel formula. Although we do not find precise error bounds, we give heuristics for choosing parameters, such as step size, used in this method.