Moments of the Derivative of the Riemann Zeta-function and of Characteristic Polynomials

Characteristic polynomials of unitary matrices are extremely useful models for the Riemann zeta-function ζ(s). The distribution of their eigenvalues give insight into the distribution of zeros of the Riemann zeta-function and the values of these characteristic polynomials give a model for the value distribution of ζ(s). See the works [KS] and [CFKRS] for detailed descriptions of how these models work. The important fact is that formulas for the moments of the Riemann zeta-function are modelled by the moments of the characteristic polynomials of unitary matrices. A question of interest in number theory is the horizontal distribution of zeros of ζ ′(s). Knowledge of this distribution is the key element in Levinson’s famous proof [Lev] that more than 1/3 of the zeros of ζ(s) have real part equal to 1/2. To elaborate, we recall that the Riemann Hypothesis asserts that all non-real zeros of ζ(s) have real part 1/2. Speiser proved that the Riemann Hypothesis is equivalent to the assertion that all non-real zeros of ζ ′(s) have real part greater than or equal to 1/2. It is not difficult to show that if ζ ′(1/2 + iγ) = 0 for a real number γ then ζ(1/2 + iγ) = 0; in words, the derivative of zeta vanishes on the 1/2-line only at a multiple zero of zeta. It is widely believed that all of the zeros of ζ(s) are simple. Consequently, it is expected that all of the non-real zeros of ζ ′(s) will lie strictly to the right of the 1/2-line. The point of departure for Levinson’s celebrated work was a theorem of Levinson and Montgomery [LM] asserting that up to a height T above the real axis, ζ(s) and ζ ′(s) have the same number of zeros strictly to the left of the 1/2-line, apart from a small number O(log T ) possible exceptions. Consequently, if the proportion of zeros of ζ ′(s) to the left of the 1/2-line is at most δ, then the proportion of zeros of ζ(s) to the left of the 1/2-line is also at most δ. The zeros of ζ(s) are symmetric about the 1/2-line. Hence, the proportion of zeros of ζ(s) to the right of the 1/2-line is also at most δ. Then the proportion of zeros of ζ(s) on the 1/2-line must be at least 1− 2δ. Levinson set out to find an upper bound for δ. Levinson proved the inequality 1 N(T ) ∑