Z function

In mathematics, the Z-function is a function used for studying the Riemann zeta-function along the critical line where the argument is one-half. It is also called the Riemann-Siegel Z-function, the Riemann-Siegel zeta-function, the Hardy function, the Hardy Z-function and the Hardy zeta-function. It can be defined in terms of the Riemann-Siegel theta-function and the Riemann zeta-function by

It follows from the functional equation of the Riemann zeta-function that the Z-function is real for real values of t. It is an even function, and real analytic for real values. It follows from the fact that the Riemann-Siegel theta-function and the Riemann zeta-function are both holomorphic in the critical strip, where the imaginary part of t is between -1/2 and 1/2, that the Z-function is holomorphic in the critical strip also. Moreover, the real zeros of Z(t) are precisely the zeros of the zeta-function along the critical line, and complex zeros in the Z-function critical strip correspond to zeros off the critical line of the Riemann zeta-function in its critical strip.

Calculation of the value of Z(t) for real t, and hence of the zeta-function along the critical line, is greatly expedited by the Riemann–Siegel formula. This formula tells us

where the error term R(t) has a complex asymptotic expression in terms of the function

and its derivatives. If ${\displaystyle u=({\frac {t}{2\pi }})^{1/4}}$,${\displaystyle N=\lfloor u^{2}\rfloor }$ and ${\displaystyle p=u^{2}-N}$ then

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