CECM Colloquium

**Wednesday November 21, 2001
in K9509, SFU
**

3:30 - 4:30

Will Galway

CECM

**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.