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

We prove a central limit theorem for $\log|\zeta(1/2+it)|$ with respect to the measure $|\zeta^{(m)}(1/2+it)|^{2k}dt$ ($k,m\in\mathbb N$), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure $|\zeta(1/2+it+i\alpha)|^{2k}dt$, with $\alpha\in(-1,1)$. Finally we prove unconditionally the analogue result in the random matrix theory context.


Introduction and statement of the main results
In 1946 Selberg [20] 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(1/2+ it)| is approximately Gaussian with mean 0 and variance 1 2 log log T , i.e.
for any fixed V ∈ R, as T goes to infinity. The author [9] studied the distribution of log |ζ(1/2 + it)| with respect to the weighted measure |ζ(1/2 + it)| 2 dt and, assuming the Riemann Hypothesis (RH), proved that it is asymptotically Gaussian with mean log log T and variance 1 2 log log T . In this paper we investigate the value distribution of log |ζ(1/2 + it)| with respect to the measure (1.2) |ζ (m) (1/2 + it)| 2k dt for any fixed m, k non negative integers. 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 [21], Soundararajan speculates that an upper bound like holds in a large range for V , in particular we expect (see [19,Conjecture 2], see also [12]) 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 2T T e iu(log |ζ(1/2+it)|−k log log T ) |ζ(1/2 + it)| 2k dt du 2k + iu 1 and is then related to the distribution of log |ζ(1/2 + it)| 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(1/2 + it)| with respect to the measure |ζ (m) (1/2 + it)| 2k dt, assuming RH only. Corollary 1. Assume the Riemann Hypothesis, let m be a non negative integer and k = 1 or k = 2. As t varies in T ≤ t ≤ 2T , the distribution of log |ζ(1/2 + it)| is asymptotically Gaussian with mean k log log T and variance 1 2 log log T , with respect to the weighted measure |ζ (m) (1/2 + it)| 2k dt.
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 2k-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 Hughes and Young conjecture and we assume the following statement.
With this assumption we can prove the main theorem.
Theorem 1. Let k, m ∈ N and assume the Riemann Hypothesis and Conjecture 1 for k. As t varies in T ≤ t ≤ 2T , the distribution of log |ζ(1/2 + it)| is asymptotically Gaussian with mean k log log T and variance 1 2 log log T , with respect to the weighted measure |ζ (m) (1/2 + it)| 2k dt.
In particular, being Conjecture 1 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] and [4]), we notice that Corollary 1 trivially follows from Theorem 1.
We notice that Theorem 1 shows that the m-th derivative has no effect in the weighted distribution of log |ζ(1/2 + it)|. This is consistent with the conjecture (see [14,Conjecture 6.1] and [8]) which indicates that |ζ ′ (1/2 + it)| 2h amplifies the contribution coming from the large values in the same way as |ζ(1/2 + it)| 2h would do (up to a normalization of (log T ) 2h ). 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 log m T , in accordance with Theorem 1. 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, tilting with |ζ(1/2 + it)| 2k the mean of log |ζ(1/2 + it)| moves to the right as k grows and this reflects the fact that the measure |ζ(1/2 + it)| 2k dt gives more and more weight to the large values of the Riemann zeta function.
Moreover, we look at the shifted weighted measure |ζ(1/2 + it + iα)| 2k dt with α a real number such that |α| < 1. As we will see, the distribution of log |ζ(1/2 + it)| 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 (1.5) start decorrelating, 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).
Theorem 2. Let k ∈ N and assume the Riemann Hypothesis and Conjecture 1 for k. As t varies in T ≤ t ≤ 2T , for any fixed and real α such that |α| < 1, the distribution of log |ζ(1/2 + it)| is asymptotically Gaussian with mean k µ α and variance 1 2 log log T , with respect to the measure |ζ(1/2 + it + iα)| 2k dt.
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 In this shifted case too, if k ≤ 2 then Theorem 2 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 θ n of matrices in the CUE in the limit as N → ∞ (see e.g. [16]). Then we consider a tilted version of the Haar measure and we have the following theorem. Theorem 3. As N → ∞, the value distribution of log |Z| is asymptotically Gaussian with mean k log N and variance 1 2 log N with respect to the measure |Z| 2k d Haar . As usual, the correspondence with Theorem 1 holds if we identify the mean density of the eigenangles θ n , N/2π, with the mean density of the Riemann zeros at a height T up the critical line, 1 2π log T 2π , i.e. if N = log T 2π .

Proof of Theorems 1 and 2
To prove both the theorems, we introduce a set of shifts α 1 , . . . , α k , β 1 , . . . , β k and we denote for the sake of brevity The general strategy of the proof is similar to the one of Theorem 1 in [9], 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's denote where x := T ε , with ε := (log log log T ) −1 . Now we show that log |ζ(1/2 + it)| has the same distribution as ℜ P (t) with respect to the measure ζ α,β (t)dt. This is achieved in the following proposition by bounding the second moment of the difference.
Proposition 2.1. For k a non negative integer, assume Conjecture 1 and RH. Let T be a large parameter and α 1 , . . . , α k , Proof. From Tsang's work [22, equation (5.15)] we know that where the sum in the definition of L(t) is over all the non-trivial zeros of ζ, then we have to bound the second moments of the terms on the right hand side with respect to weighted measure ζ α,β (t)dt, by using Conjecture 1 (note that we are allowed to apply the conjecture, since the shifts are small up to a change of variable, being |α i − β j | ≪ (log T ) −1 ).
Let's start with where 1 x (·) is the indicator function of primes up to x. If we denote Ω(n) the function which counts the number of prime factors of n with multiplicity, then we have with g x the multiplicative function defined by g x (p α ) = p −4α/ log x − 1 and We now analyze the first term I 0 α,β (z, w; s) and then we will see how to apply the method to deal with all the others. Since the sum in the definition (2.6) is multiplicative we have p a 2 + b 2 +m 1 ( 1 2 +α 1 +s)+···+m k ( 1 2 +α k +s)+n 1 ( 1 2 +β 1 +s)+···+n k ( 1 2 +β k +s) · p>x m 1 +···+m k = =n 1 +···+n k 1 p m 1 ( 1 2 +α 1 +s)+···+m k ( 1 2 +α k +s)+m 1 ( 1 2 +β 1 +s)+···+n k ( 1 2 +β k +s) (2.8) and by putting in evidence the first terms in the Euler products, this is where A α,β (z, w; s) and A * α,β (z, w; s) are arithmetical factors (Euler products) converging absolutely in a half-plane ℜ(s) > −δ for some δ > 0, uniformly for |z|, |w| ≤ 1, such that their derivatives with respect to z and w at 0 also converge in the same half plane. We now have extracted the polar part, hence we are ready to shift the integral over s in (2.5) to the left of zero. To do so, it is convenient to prescribe the same conditions as in [4, Remarks after Lemma 2.1], assuming that G(s) vanishes at s = − α i +β j 2 for all i, j, so that 6 the only pole we pick in the contour shift is at s = 0. Moreover we assume that the shifts are such that |α i + β j | ≫ (log T ) −1 for every 1 ≤ i, j ≤ k, so that Hence by (2.8), (2.9) and (2.10), we get being |ℜ(α i )|, |ℜ(β j )| ≤ (log T ) −1 and p≤x |p −4/ log x − 1|/p ≪ 1; this last bound can be obtained by using the Taylor's approximation e −z = 1 + O(z) for z ≪ 1, which yields by Mertens' first theorem. All the other terms I j,S,T α,β (z, w) can be treated exactly in the same way as I 0 α,β (z, w) by assuming that |α i ± β j | ≫ (log T ) −1 for every i, j, since they only differ from the first case by permutations and changes of signs of the shifts. Therefore we get Plugging this into (2.5), we prove that Moreover, since the left hand side in (2.12) is holomorphic in terms of the shifts, although we have proved the above for α i , β j such that (2.11) holds, the maximum modulus principle can be applied to obtain the bound to the enlarged domain we need.
We will treat similarly also the other pieces. As regards the second one, we have that As before, we analyze the first term only since all the others are completely analogous. The term for j = 0 is · · · n 1 2 +β k +s k ds and this time, because of the condition r 1 , r 2 ≥ 2, when we estimate the sum via the first terms of its Euler product we just get that the above is and, applying the same machinery as before, this yields We use the same approach in order to bound the second moment of S 3 as well, which is since the sum x<p≤x 3 p −1 is bounded.
We deal with R in the same way and we get that where the extra factor with the triple log comes from the second term in the definition of R, while the first one can be treated analogously to S 3 .
Finally we have to bound the second moment of ℜL(t). To do so, in view of equations (2.8) and (2.9) in [9], it suffices to study where η t := min ρ |t − γ| and log + t := max(log t, 0), with the aim of proving that this is ≪ k T (log T ) k 2 (log log log T ) 5/2 . By applying the Cauchy-Schwarz inequality, the above is 16) and the first term can be treated as R above and it is ≪ k T (log T ) k 2 (log log log T ) 4 . We now conclude the proof, bounding the second term in (2.16). If we denote τ : by Hölder inequality. The remaining sum can be bounded under RH in view of Kirila's [17, Theorem 1.2] and Milinovich's [18] works, which generalize the well known result due to Gonek about sum over the non-trivial zeros of zeta [10, Corollary 1]. Indeed, since the shifts α j +i t log x in the sum over zeros in (2.17) have modulus ≤ 1 and real part ≤ (log T ) −1 in absolute value, then we have for every j = 1, . . . , k and putting this into (2.17) we get (2.18) Plugging (2.18) into (2.16) we prove that concluding the proof of the proposition. 9 The second step is getting rid of the small primes, showing that their contribution does not affect the distribution asymptotically. This simple fact will simplify the third and last step of the proof, as we will see in the following. Let's define (2.20) P (t) : where X := (log T, x] (we recall that x = T ε and ε = (log log log T ) −1 ).

Proposition 2.2.
For k a non negative integer, assume Conjecture 1 and RH. Let T be a large parameter and α 1 , . . . , α k , β 1 , . . . , β k ∈ C such that |α i |, |β j | < 1, |ℜ(α i )|, |ℜ(β j )| ≤ (log T ) −1 and |α i − β j | ≪ (log T ) −1 for all 1 ≤ i, j ≤ k. Then we have: Proof. This can be proved with the same method used in Proposition 2.1. We recall that 1 log T (·) denotes the indicator function of primes up to log T . We start by studying ab m 1 2 +α 1 +s 1 · · · n 1 2 +β k +s k ds and, as usual, we estimate the sum with the first terms of its Euler product, extract the polar part, shift the integral over s to the left, getting that (2.21) is and by the same argument as in the proof of Proposition 2.1 the above is and this concludes the proof.
Finally we investigate the distribution of the polynomial ℜP (t), which has the same distribution as log |ζ(1/2 + it)| thanks to Propositions 2.1 and 2.2. The most natural method to do so is studying the moments and this is achieved in the following result.
− zkµ α vanishes for all S, T for this choice of the shifts. The thesis follows as in the proof of Theorem 1.

Proof of Theorem 3
In the usual notations we set in the introduction, let us define the moment generating function where the mean has to be considered over the group U(N) with respect to the Haar measure) and the cumulants Q j = Q j (N) by In [16], among other things, Keating and Snaith studied the cumulants showing that and deduced a central limit theorem proving that the limiting distribution of log |Z| is Gaussian with mean 0 and variance 1 2 log N. Here, for any k ∈ N, we study the distribution of random variable log |Z| with respect to the tilted measure |Z| 2k d Haar . Before starting with our analysis, we recall that the moments of |Z| are known also for non integer k (see [16] equation (6) and (16)): where k ∈ R and G denotes the Barnes G-function. We denote M 2k := N j=1 Γ(j)Γ(j+2k) Now we are ready to consider the first moment by (3.1) and (3.4). We compute the derivative by Leibniz's rule, writing log Γ(j) + log Γ(j + 2k + x) − 2 log Γ(j + k + x/2) and we get Moreover an application of Stirling's formula yields hence, by (3.5) and (3.6), the weighted mean of the random variable log |Z| is Then we study the weighted n-th moment of the random variable log |Z|: . (3.7) If we denote f j (x) = f N,k,j (x) := log Γ(j) + log Γ(j + 2k + x) − 2 log Γ(j + k + x 2 ), then we can carry on the computation in (3.7) by computing the derivative, getting where the sums in (3.8) are over the n-uple (m 1 , . . . , m n ) such that m 1 + 2m 2 + · · · + nm n = n.
Using Stirling's approximation formula, one can easily estimate the derivatives of f j (x) and prove