§25.18 Methods of Computation

§25.18(i) Function Values and Derivatives

The principal tools for computing $\mathop{\zeta\/}\nolimits\!\left(s\right)$ are the expansion (25.2.9) for general values of $s$, and the Riemann–Siegel formula (25.10.3) (extended to higher terms) for $\mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right)$. Details are provided in Haselgrove and Miller (1960). See also Allasia and Besenghi (1989), Butzer and Hauss (1992), Kerimov (1980), and Yeremin et al. (1985). Calculations relating to derivatives of $\mathop{\zeta\/}\nolimits\!\left(s\right)$ and/or $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988).

For the Hurwitz zeta function $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ see Spanier and Oldham (1987, p. 653) and Coffey (2009).

For dilogarithms and polylogarithms see Jacobs and Lambert (1972), Osácar et al. (1995), Spanier and Oldham (1987, pp. 231–232), and Zudilin (2007).

For Fermi–Dirac and Bose–Einstein integrals see Cloutman (1989), Gautschi (1993), Mohankumar and Natarajan (1997), Natarajan and Mohankumar (1993), Paszkowski (1988, 1991), Pichon (1989), and Sagar (1991a, b).

§25.18(ii) Zeros

Most numerical calculations of the Riemann zeta function are concerned with locating zeros of $\mathop{\zeta\/}\nolimits\!\left(\frac{1}{2}+it\right)$ in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of $\mathop{\zeta\/}\nolimits\!\left(s\right)$ lie on the critical line $\Re{s}=\frac{1}{2}$. Calculations to date (2008) have found no nontrivial zeros off the critical line. For recent investigations see, for example, van de Lune et al. (1986) and Odlyzko (1987). For earlier work see Haselgrove and Miller (1960).