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Concept
Z function
Summary
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 , and then
where the ellipsis indicates we may continue on to higher and increasingly complex terms.
Other efficient series for Z(t) are known, in particular several using the incomplete gamma function. If
then an especially nice example is
From the critical line theorem, it follows that the density of the real zeros of the Z function is
for some constant c > 2/5. Hence, the number of zeros in an interval of a given size slowly increases. If the Riemann hypothesis is true, all of the zeros in the critical strip are real zeros, and the constant c is one. It is also postulated that all of these zeros are simple zeros.
Because of the zeros of the Z function, it exhibits oscillatory behavior.
Official source
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