Hardy function | Hardy Z-function | Z-function

For a real positive t, the Riemann-Siegel Z function is defined by Z(t) congruent e^(i ϑ(t)) ζ(1/2 + i t). This function is sometimes also called the Hardy function or Hardy Z-function (Karatsuba and Voronin 1992, Borwein et al. 1999). The top plot superposes Z(t) (thick line) on left bracketing bar ζ(1/2 + i t) right bracketing bar , where ζ(z) is the Riemann zeta function. For real t, the Riemann-Siegel theta function ϑ(t) is defined as ϑ(t) | = | ℑ[ln Γ(1/4 + 1/2 i t)] - 1/2 t ln π | = | arg[Γ(1/4 + 1/2 i t)] - 1/2 t ln π.

Gram point | Gram's law | Riemann-Siegel formula | Riemann zeta function | xi-function

Bernhard Riemann | Carl Ludwig Siegel