Z function

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.

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.

Because of the zeros of the Z function, it exhibits oscillatory behavior.

It also slowly grows both on average and in peak value.

For instance, we have, even without the Riemann hypothesis, the Omega theorem that where the notation means that

Higher even powers have been much studied, but less is known about the corresponding average value.

It is conjectured, and follows from the Riemann hypothesis, that for every positive ε.

Here the little "o" notation means that the left hand side divided by the right hand side does converge to zero; in other words little o is the negation of Ω.

The best known bound on this rate of growth is not strong, telling us that any

It would be astonishing to find that the Z function grew anywhere close to as fast as this.