by the function within the Ω does not tend to zero with increasing t. Average growth The average growth of the Z function has also been much studied. We can find the root mean square (abbreviated RMS) average from or which tell us that the RMS size of Z(t) grows as . This estimate can be improved to If we increase the exponent, we get an average value which depends more on the peak values of Z. For fourth powers, we have from which we may conclude that the fourth root of the mean fourth power grows as The Lindelöf hypothesis 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. This conjecture is called the Lindelöf hypothesis, and is weaker than the Riemann hypothesis. It is normally stated in an important equivalent form, which is in either form it tells us the rate of growth of the peak values cannot be too high. The best known bound on this rate of growth is not strong, telling us that any is suitable. It would be astonishing to find that the Z function grew anywhere close to as fast as this. Littlewood proved that on the Riemann hypothesis, and this seems far more likely. References External links Wolfram Research – Riemann-Siegel function Z (includes function plotting and evaluation) Zeta and L-functions