Fast algorithms for multiple evaluations of the Riemann zeta function

Abstract

The best previously known algorithm for evaluating the Riemann zeta function, ζ ( σ + i t ) \zeta (\sigma + it) , with σ \sigma bounded and t t large to moderate accuracy (within ± t − c \pm {t^{ - c}} for some c > 0 c > 0 , say) was based on the Riemann-Siegel formula and required on the order of t 1 / 2 {t^{1/2}} operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of ζ ( σ + i t ) \zeta (\sigma + it) with σ \sigma fixed and T ⩽ t ⩽ T + T 1 / 2 T \leqslant t \leqslant T + {T^{1/2}} to within ± t − c \pm {t^{ - c}} in O ( t ε ) O({t^\varepsilon }) operations on numbers of O ( log ⁡ t ) O(\log t) bits for any ε > 0 \varepsilon > 0 , for example, provided a precomputation involving O ( T 1 / 2 + ε ) O({T^{1/2 + \varepsilon }}) operations and O ( T 1 / 2 + ε ) O({T^{1/2 + \varepsilon }}) bits of storage is carried out beforehand. These algorithms lead to methods for numerically verifying the Riemann hypothesis for the first n n zeros in what is expected to be O ( n 1 + ε ) O({n^{1 + \varepsilon }}) operations (as opposed to about n 3 / 2 {n^{3/2}} operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as π ( x ) \pi (x) . The new zeta function algorithms use the fast Fourier transform and a new method for the evaluation of certain rational functions. They can also be applied to the evaluation of L L -functions, Epstein zeta functions, and other Dirichlet series.