Z Function

In mathematics, the Z function, also called the Riemann-Siegel Z function or the Hardy function among other names, is used to study the Riemann zeta function along the critical line, where the real part of the argument equals one-half. It is defined in terms of the Riemann-Siegel theta function and the Riemann zeta function itself, and the functional equation of the zeta function makes the Z function real-valued, even and real analytic for real inputs; because both the theta function and the zeta function are holomorphic within the critical strip, the Z function is holomorphic there too. Its real zeros are exactly the zeta function's zeros along the critical line, and any complex zero of the Z function within the strip corresponds to a zero of the zeta function that lies off the critical line. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/