Weighted value distributions of the Riemann zeta function on the critical line

1 Introduction and statement of the main results

In 1946, Selberg [18] proved a central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line, showing that the distribution of log⁡|ζ⁢(12+i⁢t)| is approximately Gaussian with mean 0 and variance 12⁢log⁡log⁡T, i.e.

for any fixed V∈ℝ, as T goes to infinity. The author [9] studied the distribution of log⁡|ζ⁢(12+i⁢t)| with respect to the weighted measure |ζ⁢(12+i⁢t)|2⁢d⁢t and, assuming the Riemann hypothesis (RH), proved that it is asymptotically Gaussian with mean log⁡log⁡T and variance 12⁢log⁡log⁡T. In this paper, we investigate the value distribution of log⁡|ζ⁢(12+i⁢t)| with respect to the measure

for any fixed non-negative integers m,k. The motivation is due to the study of the large values of the Riemann zeta function, i.e. the uniformity in V in the central limit theorem (1.1). In [19], Soundararajan speculates that an upper bound like

holds in a large range for V. In particular, we expect (see [17, Conjecture 2] and [11]) that for any fixed k,

Expressing the characteristic function of positive reals as a Mellin transform, the left-hand side of (1.3) can be written as

and is then related to the distribution of log⁡|ζ⁢(12+i⁢t)| with respect to the weighted measure (1.2), with m=0.

In the case k=1,2, we can prove a central limit theorem for log⁡|ζ⁢(12+i⁢t)| with respect to the measure |ζ(m)⁢(12+i⁢t)|2⁢k⁢d⁢t, assuming RH only.

In the case k>2, since not even the moments of zeta are known, we cannot expect to prove unconditionally a central limit theorem. However, assuming the asymptotic formula [13, Conjecture 7.1] suggested by the recipe [6] for the twisted and shifted (2⁢k)-th moments of the Riemann zeta function

one can deal with the general case too. More precisely, we use the strategy of [4, Theorem 1.2] in order to re-write the conjecture of Hughes and Young and we assume the following statement.

With this assumption we can prove the main theorem.

In particular, since Conjecture 1.2 is known in the cases k=1 (see [2, 5] and, e.g., [3] for the easy modifications needed to account for the shifts) and k=2 (see [13, 4]), we notice that Corollary 1.1 trivially follows from Theorem 1.3.

We notice that Theorem 1.3 shows that the m-th derivative has no effect in the weighted distribution of log⁡|ζ⁢(12+i⁢t)|. This is consistent with the conjecture (see [12, Conjecture 6.1] and [8])

which indicates that |ζ′⁢(12+i⁢t)|2⁢h amplifies the contribution coming from the large values in the same way as |ζ⁢(12+i⁢t)|2⁢h would do (up to a normalization of (log⁡T)2⁢h). More generally, as far as moments are concerned, the m-th derivative of zeta should behave like zeta itself (see, e.g., [7]), up to a normalization of logm⁡T, in accordance with Theorem 1.3. We also note that, while in Selberg’s classical case the mean is 0 because the contribution of the small values and that of the large values of zeta balance out, by tilting with |ζ⁢(12+i⁢t)|2⁢k, the mean of log⁡|ζ⁢(12+i⁢t)| moves to the right as k grows, and this reflects the fact that the measure |ζ⁢(12+i⁢t)|2⁢k⁢d⁢t gives more and more weight to the large values of the Riemann zeta function.

Moreover, we look at the shifted weighted measure |ζ⁢(12+i⁢t+i⁢α)|2⁢k⁢d⁢t with a real number α such that |α|<1. As we will see, the distribution of log⁡|ζ⁢(12+i⁢t)| is quite sensitive to the parameter α. Indeed, in computing the integral

one expects the same magnitude as in the unshifted case if |α| is smaller than (log⁡T)-1, which is the typical scale for the Riemann zeta function. On the other hand, if |α| is larger, then the two factors in the integral (1.4) start decorrelating, and thus the size of the integral decreases. This phenomenon is shown in the following result, in which we use the notation

for any α∈(-1,1).

This theorem shows that the shift has no effect if it is smaller than (log⁡T)-1. On the contrary, for larger values of the shift the mean gets smaller; for instance, if

with δ∈(0,1), then the mean is ∼(1-δ)⁢k⁢log⁡log⁡T. In this shifted case too, if k≤2, then Theorem 1.4 holds assuming RH only.

Lastly, we show that in the random matrix theory setting, an analogous weighted central limit theorem can be proved unconditionally. We consider the characteristic polynomials

of N×N unitary matrices U and we investigate their distribution of values with respect to the circular unitary ensemble (CUE). It has been conjectured that the limiting distribution of the non-trivial zeros of the Riemann zeta function, on the scale of their mean spacing, is the same as that of the eigenphases θn of matrices in the CUE in the limit as N→∞ (see, e.g., [14]). Then we consider a tilted version of the Haar measure and we have the following theorem.

As usual, the correspondence with Theorem 1.3 holds if we identify the mean density of the eigenangles θn, N/2⁢π, with the mean density of the Riemann zeros at a height T up the critical line 12⁢π⁢log⁡T2⁢π, i.e. if

2 Proof of Theorems 1.3 and 1.4

To prove both the theorems, we introduce a set of shifts α1,…,αk,β1,…,βk and we denote for the sake of brevity

The general strategy of the proof is similar to the one of [9, Theorem 1], but here we avoid the detailed combinatorial analysis we performed in [9], by working with Euler products instead of Dirichlet series, inspired by [1, Proposition 5.1]. The first step is then approximating the logarithm of the Riemann zeta function with a suitable Dirichlet polynomial. Let us set

where x:=Tε, with ε:=(log⁡log⁡log⁡T)-1. Now we show that log⁡|ζ⁢(12+i⁢t)| has the same distribution as ℜ⁡P~⁢(t) with respect to the measure ζα,β⁢(t)⁢d⁢t. This is achieved in the following proposition by bounding the second moment of the difference.