# Riemann–Siegel formula

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##### 1: 25.10 Zeros
###### §25.10(ii) Riemann–SiegelFormula
Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp\left(i\vartheta(t)\right)$ to obtain the RiemannSiegel formula:
25.10.3 $Z(t)=2\sum_{n=1}^{m}\frac{\cos\left(\vartheta(t)-t\ln n\right)}{n^{1/2}}+R(t),$ $m=\left\lfloor\sqrt{t/(2\pi)}\right\rfloor$,
Calculations based on the RiemannSiegel formula reveal that the first ten billion zeros of $\zeta\left(s\right)$ in the critical strip are on the critical line (van de Lune et al. (1986)). …
##### 2: 25.19 Tables
• Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

• Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the RiemannSiegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

• ##### 3: 25.18 Methods of Computation
The principal tools for computing $\zeta\left(s\right)$ are the expansion (25.2.9) for general values of $s$, and the RiemannSiegel formula (25.10.3) (extended to higher terms) for $\zeta\left(\frac{1}{2}+it\right)$. …
##### 4: Bibliography B
• A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
• W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
• M. V. Berry and J. P. Keating (1998) $H=xp$ and the Riemann Zeros. In Supersymmetry and Trace Formulae: Chaos and Disorder, I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii (Eds.), pp. 355–367.
• M. V. Berry and J. P. Keating (1999) The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41 (2), pp. 236–266.
• M. V. Berry (1995) The Riemann-Siegel expansion for the zeta function: High orders and remainders. Proc. Roy. Soc. London Ser. A 450, pp. 439–462.