Youness Lamzouri
email: lamzouri@mathstat.yorku.edu

AND Stephen Lester
email: sjlester@gmail.com

Maksym Radziwi
email: maksym.radziwill@gmail.com

DISCREPANCY BOUNDS FOR THE DISTRIBUTION OF THE RIEMANN ZETA-FUNCTION AND APPLICATIONS

We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For 1 2 < σ ≤ 1 we obtain an upper bound on the discrepancy between the distribution of ζ(s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ(s) inside of the critical strip, strengthening a previous result of the first author.

As an application of these results we obtain the first effective error term for the number of solutions to ζ(s) = a in a strip 1 2 < σ 1 < σ 2 < 1. Previously in the strip 1 2 < σ < 1 only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg). In general our results are an improvement of the classical Bohr-Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function.

).

Introduction and statement of main results

Let {X(p)} p be a sequence of independent random variables uniformly distributed on the unit circle where p runs over the prime numbers. Consider the random Euler product

ζ(σ, X) = p 1 - X(p) p σ -1
, which converges almost surely for σ > 1 2 . Due to the unique factorization of the integers we intuitively expect that the functions p -it interact like the independent random variables X(p). This suggests that ζ(σ, X) should be a good model for the Riemann zeta-function, and one may ask: How well does the distribution of ζ(σ, X) approximate that of the Riemann zeta-function?

A probabilistic treatment of a theorem of Bohr and Jessen [START_REF] Bohr | Über die Werteverteilung der Riemannschen Zetafunktion, erste Mitteilung[END_REF][START_REF]Über die Werteverteilung der Riemannschen Zetafunktion[END_REF] due to Jessen and Wintner [START_REF] Jessen | Distribution functions and the Riemann zeta function[END_REF] asserts that log ζ(σ + it) has a continuous limiting distribution in the complex plane for σ > 1 2 . In fact, it can be seen from these works that log ζ(σ + it) converges in distribution to log ζ(σ, X) for σ > 1 2 . In this article we investigate the discrepancy between the distributions of the random variable log ζ(σ, X) and that of where the supremum is taken over all rectangles R with sides parallel to the coordinate axes and

P T f (t) ∈ R := 1 T • meas T ≤ t ≤ 2T : f (t) ∈ R ,
where here and throughout we denote by meas the Lebesgue measure on R. This quantity measures the extent to which the distribution function of the random variable log ζ(σ, X) approximates that of log ζ(σ + it). We prove Theorem 1.1. Let 1 2 < σ < 1 be fixed. Then

D σ (T ) 1 (log T ) σ .
Additionally, for σ = 1 we have

D 1 (T )
(log log T ) 2 log T .

Theorem 1.1 improves upon a previous discrepancy estimate due to Harman and Matsumoto [START_REF] Harman | Discrepancy estimates for the value-distribution of the Riemann zeta-function. IV[END_REF]. For fixed 1 2 < σ ≤ 1 they showed that the discrepancy satisfies the bound D σ (T ) 1 (log T ) (4σ-2)/(21+8σ)-ε for any ε > 0. One new feature of our estimate is that the power of the logarithm does not decay to zero as σ → 1 2 . We introduce a different technique to study this problem that relies upon careful analysis of large complex moments of the Riemann zeta-function inside of the critical strip. Some of the tools developed by Selberg to study the distribution of log ζ( 1 2 + it) are also used, such as Beurling-Selberg functions. We have not been able to determine conjecturally the correct size of the discrepancy D σ (T ). In this direction we have only the following result, whose proof we include in the Appendix. Proposition 1.2. Let 1/2 < σ < 1 be fixed, and ε > 0 be small. Then, we have

D σ (T ) = Ω(T 1-2σ-ε ).
Moreover, If D σ (T ) = O(T 1-2σ+ε ) then the Zero Density Hypothesis holds.

The zero density hypothesis states that for 1/2 < σ < 1, the number of zeros ρ = β + iγ of ζ(s) such that β ≥ σ and |γ| ≤ T is T 2(1-σ)+ε (see [START_REF] Iwaniec | Analytic number theory[END_REF] for the current developements towards this conjecture). There is an apparent difference between our upper and lower bounds for D σ (T ) in Theorem 1.1 and Proposition 1.2. It would be very interesting to work out a reliable heuristic to predict the correct size of this discrepancy.

An important problem in the theory of the Riemann zeta-function is to understand its maximal order within the critical strip. The Riemann hypothesis implies that for On the other hand, Montgomery [START_REF] Montgomery | Extreme values of the riemann zeta function[END_REF] proved that log |ζ(σ + it)| = Ω (log t) 1-σ+o (1) , and based on a probabilistic argument, he conjectured that this omega result is in fact optimal, namely that log |ζ(σ + it)| (log t) 1-σ+o(1) . This motivates the study of the extent to which the extreme values of ζ(σ + it) can be modeled by those of the random variable ζ(σ, X). For if the distribution of the extreme values of ζ(σ + it) matches that of ζ(σ, X) in the viable range, then Montgomery's conjecture follows.

To this end we would like to understand log P T (log |ζ(σ + it)| > τ ).

(1.1)

For fixed τ an upper bound for (1.1) was first given by Jessen and Wintner [START_REF] Jessen | Distribution functions and the Riemann zeta function[END_REF], and further improvements, including lower bounds, were later given in [START_REF]On the magnitude of asymptotic probability measures of Dedekind zeta-functions and other Euler products[END_REF][START_REF] Hattori | Large deviations of Montgomery type and its application to the theory of zeta-functions[END_REF][START_REF] Joyner | Distribution theorems of L-functions[END_REF]. More recently, an asymptotic formula has been derived by Hattori and Matsumoto [START_REF]A limit theorem for Bohr-Jessen's probability measures of the Riemann zeta-function[END_REF]. Improving on these results, the first author [START_REF]On the distribution of extreme values of zeta and L-functions in the strip 1 2 < σ < 1[END_REF] established a uniform asymptotic formula for (1.1) in nearly the full conjectured range of τ . More precisely, he showed that there is a positive constant A(σ), such that uniformly in the range τ (log T ) 1-σ+o (1) , we have log

P T log |ζ(σ + it)| > τ = (1 + o(1)) log P log |ζ(σ, X)| > τ = (-A(σ) + o(1)) τ 1 1-σ (log τ ) σ 1-σ . (1.2)
He also proved that the same estimate holds when log |ζ(σ + it)| and log |ζ(σ, X)| are replaced by arg ζ(σ + it) and arg ζ(σ, X) respectively. We strengthen this result, obtaining an asymptotic formula for P T (log |ζ(σ + it)| > τ ) (and P T (arg ζ(σ + it) > τ )) in the same range.

Theorem 1.3. Let 1 2 < σ < 1 be fixed. There exists a positive constant b(σ) such that for 3 ≤ τ ≤ b(σ)(log T ) 1-σ (log log T ) 1-1 σ we have

P T log |ζ(σ + it)| > τ = P log |ζ(σ, X)| > τ × 1 + O (τ log τ ) σ 1-σ • (log log T ) (log T ) σ .
Moreover, the same asymptotic estimate holds when log |ζ(σ + it)| and log |ζ(σ, X)| are replaced by arg ζ(σ + it) and arg ζ(σ, X) respectively.

The terms (log T ) σ appearing in the error term in Theorem 1.3 and in the upper bound of Theorem 1.1 are related. An improvement in our method would produce an improvement in both results. Since we do not believe that we will be able to extend significantly the range of Theorem 1.3, it seems that our bound for D σ (T ) is as well optimal given the method used.

We also apply Theorem 1.1 to study the roots, s, to the equation ζ(s) = a where a is a nonzero complex number. These points are known as a-points and the study of their distribution is a classical topic in the theory of the Riemann zeta-function.

Let N a (σ 1 , σ 2 ; T ) be the number of a-points in the strip 1 2 < σ 1 < σ < σ 2 < 1, T ≤ t ≤ 2T . In 1948 Borchsenius and Jessen [START_REF] Borchsenius | Mean motions and values of the Riemann zeta function[END_REF] proved that there exists a constant c(a, σ 1 , σ 2 ) > 0 such that as T → ∞ N a (σ 1 , σ 2 ; T ) ∼ c(a, σ 1 , σ 2 )T.

(1.

3)

The constant c(a, σ 1 , σ 2 ) can be explicitly given in terms of the random variable ζ(σ, X). Indeed, let

f a (σ) = E log |ζ(σ, X) -a| .
Then,

c(a, σ 1 , σ 2 ) = f a (σ 2 ) -f a (σ 1 ) 2π .
The differentiability of f a (σ) is not trivial, and was established by Borchsenius and Jessen.

Using Theorem 1.1 we obtain the first effective error term for N a (σ 1 , σ 2 ; T ) valid for

σ 1 < σ 2 in the critical strip. Theorem 1.4. Let 1 2 < σ 1 < σ 2 < 1.
For every nonzero complex number a there exists a constant c(a, σ 1 , σ 2 ) > 0 such that

N a (σ 1 , σ 2 ; T ) = c(a, σ 1 , σ 2 )T + O T • log log T (log T ) σ 1 /2 .
Inside the critical strip, an effective error term was known previously only slightly to the left of the half-line under the assumption of the Riemann hypothesis, thanks to unpublished work of Selberg (see [START_REF] Selberg | Old and new conjectures and results about a class of Dirichlet series[END_REF] and [START_REF] Tsang | The distribution of the values of the Riemann zeta-function[END_REF], Chapter 8). A related (but easier) problem studies the a-points of log ζ(s). In this setting, Borchsenius and Jessen [START_REF] Borchsenius | Mean motions and values of the Riemann zeta function[END_REF] proved an asymptotic formula for the number of these a-points in the region 1 2 < σ 1 < σ < σ 2 < 1, T ≤ t ≤ 2T . Moreover, an effective formula for the number of such a-points with an error term that has a power saving of (log log T ) 1/ log 4 T has been established by Matsumoto [START_REF] Matsumoto | Discrepancy estimates for the value-distribution of the Riemann zeta-function. II, Number theory and combinatorics[END_REF], [START_REF]Discrepancy estimates for the value-distribution of the Riemann zeta-function. III[END_REF], where here and throughout we let log k denote the k-th iterated logarithm. In the region of absolute convergence (σ > 1) Matsumoto's formula is slightly stronger and has an error term with a power saving of log log T . We have not determined the limits of our method for σ > 1, but it should give a formula for the number of apoints of ζ(s) with an error term with a saving of at least (log T ) 1/2 . Additionally, our method applies to counting the a-points of log ζ(s) and a modification of our argument should give an effective formula for the number of such points with a power saving of log T inside the critical strip.

It is likely that our ideas can be generalized to other situations where the Bohr-Jessen framework [START_REF] Bohr | Über die Werteverteilung der Riemannschen Zetafunktion, erste Mitteilung[END_REF] applies. For example, our methods were recently used by Gonek and Lee [START_REF] Gonek | Zero-density estimates for epstein zeta functions[END_REF] to obtain an effective error term for the number of zeros of the Epstein zeta-function of a quadratic form with class number n > 1, in a strip σ 1 < Re(s) < σ 2 , where 1/2 < σ 1 < σ 2 . This improves on a previous result of Lee [START_REF] Lee | On the zeros of Epstein zeta functions[END_REF], where only an asymptotic estimate for this quantity was obtained.

Key ideas and detailed results

In probability theory, the classical Berry-Esseen Theorem states that if the characteristic functions of two real valued random variables are close, then their corresponding probability distributions are close as well. In order to establish Theorem 1.1 the key ingredient is to show that the characteristic function of the joint distribution of

Then we prove

Theorem 2.1. Let 1 2 < σ < 1 and A ≥ 1 be fixed. There exists a positive constant

b 1 = b 1 (σ, A) such that for |u|, |v| ≤ b 1 (log T ) σ we have Φ σ,T (u, v) = Φ rand σ (u, v) + O 1 (log T ) A .
(2.1)

Remark 2.1. The analogous asymptotic estimate for Φ σ,T (u, v) when σ = 1 follows from Theorem 2 of [START_REF] Lamzouri | The two-dimensional distribution of values of ζ(1 + it)[END_REF], which states that for all complex numbers z 1 , z 2 with |z 1 |, |z 2 | ≤ log T /(50(log log T ) 2 ) we have

1 T 2T T ζ(1 + it) z 1 ζ(1 + it) z 2 dt = E ζ(1, X) z 1 ζ(1, X) z 2 + O exp - log T 2 log log T .
Indeed, by taking z 1 = i 2 (u -iv) and z 2 = i 2 (u + iv) we obtain that for all real numbers u, v with |u|, |v| ≤ log T /(50(log log T ) 2 ) we have

Φ 1,T (u, v) = Φ rand 1 (u, v) + O exp - log T 2 log log T . (2.2)
To deduce Theorem 1.1 from Theorem 2.1 and Remark 2.1, we use Beurling-Selberg functions (see Section 6 below) to relate the distribution function P T (log ζ(σ + it) ∈ R) to the characteristic function Φ σ,T (u, v). We should note that any improvement in the range of validity of Theorem 2.1 would lead to an improved bound for the discrepancy D σ (T ). Indeed, we can show that D σ (T ) 1/L if the asymptotic formula (2.1) holds in the range |u|, |v| L. In order to investigate the distribution of large values of log |ζ(σ+it)| (or arg ζ(σ+it)) and prove Theorem 1.3, we study large complex moments of ζ(σ +it) and compare them to the corresponding complex moments of ζ(σ, X). Define

M z (T ) := 1 T 2T T |ζ(σ + it)| z dt.
Assuming the Riemann hypothesis, the first author [START_REF]On the distribution of extreme values of zeta and L-functions in the strip 1 2 < σ < 1[END_REF] established an asymptotic formula for M z (T ) uniformly in the range |z| (log T ) 2σ-1 , and conjectured that such an asymptotic should hold in the extended range |z| (log T ) σ . The assumption of the Riemann hypothesis is necessary in this case, since |ζ(σ + it)| z is very large when σ + it is close to a zero of ζ(s) and z is a negative real number. Also note that, when Re(z) is large, the moment M z (T ) is heavily affected by the contribution of the points t where |ζ(σ + it)| is large. Thus, short of proving strong bounds for |ζ(σ + it)| and without assuming the Riemann hypothesis, we cannot hope for asymptotics of the moments M z (T ), except in a narrow range of values for z. To overcome this difficulty, we compute instead complex moments of ζ(σ + it) after first removing a small set of "bad" points t in [T, 2T ], namely those close to zeros of ζ(s) and those for which |ζ(σ + it)| is large. Using this method we obtain an asymptotic formula for these moments in the full conjectured range |z| (log T ) σ .

Theorem 2.2. Let 1 2 < σ < 1 and A ≥ 1 be fixed. There exist positive constants b 3 = b 3 (σ, A) and b 4 = b 4 (σ, A) and a set E(T ) ⊂ [T, 2T ] of measure meas(E(T )) ≤ T exp -b 3 log T / log log T , such that for all complex numbers z with |z| ≤ b 4 (log T ) σ we have

1 T [T,2T ]\E(T ) |ζ(σ + it)| z dt = E (|ζ(σ, X)| z ) + O E |ζ(σ, X)| Re(z) (log T ) A ,
R Y (σ + it) := p n ≤Y 1 np n(σ+it) and R Y (σ, X) := p n ≤Y X(p) n np σn .
We extract Theorems 2.1 and 2.2 from the following key proposition.

1 T A(T ) exp z 1 R Y (σ + it) + z 2 R Y (σ + it) dt = E exp z 1 R Y (σ, X) + z 2 R Y (σ, X) + O exp -b 6 log T log log T , where A(T ) is the set of those t ∈ [T, 2T ] such that |R Y (σ + it)| ≤ (log T ) 1-σ / log log T.
Compared to earlier treatments our main innovation consists in the introduction of the condition

|R Y (σ + it)| ≤ (log T ) 1-σ / log log T in A(T ). Without this constraint the range of |z 1 | and |z 2 | in Proposition 2.3 would be reduced to (log T ) 2σ-1 . Using Littlewood's Lemma (see equation (8.2) below), one can count the number of a-points of ζ(s) in the strip 1 2 < σ 1 < σ < σ 2 < 1, T ≤ t ≤ 2T , if one can estimate the integral 2T T log |ζ(σ + it) -a|dt.
(2.3)

In [START_REF] Borchsenius | Mean motions and values of the Riemann zeta function[END_REF], Borchsenius and Jessen proved the following asymptotic formula for this integral from which they deduced their result (1.3)

1 T 2T T log |ζ(σ + it) -a|dt ∼ E[log |ζ(σ, X) -a|], as T → ∞.
We improve on this result, obtaining the first effective error term for the integral (2.3).

Theorem 2.4. Let 1 2 < σ < 1 and a = 0 be a complex number. Then,

1 T 2T T log |ζ(σ + it) -a|dt = E[log |ζ(σ, X) -a|] + O (log log T ) 2 (log T ) σ .
We should note that apart from the factor (log log T ) 2 , the error term in Theorem 2.4 is the best we can get using our bound for the discrepancy D σ (T ) in Theorem 1.1.

There are two main ingredients in the proof of Theorem 2.4. First, we use our result on D σ (T ) to capture the main term. Secondly, to control the error term we need a completely uniform (but not necessarily very good) bound for the measure of those t for which ζ(σ + it) is very close to a. We achieve such an estimate by using the following

L 2k bound. Proposition 2.5. Let 1 2 < σ ≤ 2 be fixed. Let a ∈ C.
There exists an absolute constant C > 0 such that for every real number k > 0 we have

1 T 2T T | log |ζ(σ + it) -a|| 2k dt (Ck) 4k .
In order to study a-points to the left of the half-line, Selberg obtains a similar proposition when σ = 1 2 . His argument depends on the rapid rate of change of the phase of ζ(σ + it) when σ ≤ 1 2 (for σ < 1 2 this follows from the Riemann hypothesis) and does not generalize to any line with σ > 1 2 (see [START_REF] Tsang | The distribution of the values of the Riemann zeta-function[END_REF], Chapter 8, in particular the discussion on page 119). Our treatment depends on a careful use of Jensen's formula. Proposition 2.5 bears some resemblance to a result obtained by Guo to study zeros of ζ (s). Our result is more refined, in particular our treatment removes the loss of a power of log log T .

Preliminary Lemmas

In this section we collect together several preliminary results that will be useful in our subsequent work.

Lemma 3.1 (Lemma 2.2 of [5]). Let 1 2 < σ ≤ 1 be fixed and 3 ≤ Y ≤ T /2. For t ∈ [T, 2T ] except outside a set of measure T 5/4-σ/2 Y log 5 T we have log ζ(σ + it) = R Y (σ + it) + O Y -(σ-1/2)/2 log 3 T . (3.1) Lemma 3.2. Let 2 ≤ y ≤ z. For any positive integer k that is ≤ log T /(3 log z) we have 1 T 2T T y≤p≤z 1 p σ+it 2k dt k! y≤p≤z 1 p 2σ k + T -1/3 .
Additionally, for any positive integer k we have

E y≤p≤z X(p) p σ 2k k! y≤p≤z 1 p 2σ k .
Proof. The first assertion of the lemma is Lemma 4.2 of [START_REF]On the distribution of extreme values of zeta and L-functions in the strip 1 2 < σ < 1[END_REF]. Next, note that

E y≤p≤z X(p) p σ 2k = y≤p 1 ,...,p k ≤z y≤q 1 ,...,q k ≤z 1 (p 1 • • • p k q 1 • • • q k ) σ E X(p 1 ) • • • X(p k )X(q 1 ) • • • X(q k ) .
Since the X(p)'s are independent random variables uniformly distributed on the unit circle the only terms that contribute to the above sum are those where

p 1 • • • p k = q 1 • • • q k . The contribution from these terms is k! y≤p≤z 1 p 2σ k . Lemma 3.3. Let 1 2 < σ < 1 and A ≥ 1 be fixed. Also, let Y = (log T ) A and k be an integer that satisfies 2 ≤ k ≤ log T /(6A log log T ). Then there exists a constant a(σ) > 0 such that 1 T 2T T |R Y (σ + it)| 2k dt a(σ)k 1-σ (log k) σ 2k .
Additionally, for any integer k ≥ 2 we have

E |R Y (σ, X)| 2k a(σ)k 1-σ (log k) σ 2k .
Proof. We will only prove the first assertion; the second follows from a similar argument. Plainly,

2T T |R Y (σ + it)| 2k dt ≤9 k 2T T p≤k log k 1 p σ+it 2k dt + 2T T k log k≤p≤Y 1 p σ+it 2k dt + O(T log 2k ζ(2σ)) . (3.2)
By Lemma 3.2 and the prime number theorem we have

1 T 2T T k log k≤p≤Y 1 p σ+it 2k dt k! k log k≤p≤Y 1 p 2σ k + T -1/3 k k (k log 2k) 1-2σ (2σ -1) log k k . Next, note that for fixed 1/2 < σ < 1 1 T 2T T p≤k log k 1 p σ+it 2k dt ≤ p≤k log 2k 1 p σ 2k (k log k) 1-σ (1 -σ) log k 2k ,
by the prime number theorem. Inserting the two estimates above into (3.2) completes the proof. Lemma 3.4. Let 1 2 < σ < 1 and A ≥ 1 be fixed, and let Y = (log T ) A . Then there exists a constant B = B(σ, A) such that

P T |R Y (σ + it)| ≥ (log T ) 1-σ log log T exp -B log T log log T and P |R Y (σ, X)| ≥ (log T ) 1-σ log log T exp -B log T log log T .
Proof. We will only prove the first assertion; the second follows from a similar argument. Let 2 ≤ k ≤ log T /(6A log log T ) be an integer. Then, Lemma 3.3 implies that

P T |R Y (σ + it)| ≥ (log T ) 1-σ log log T ≤ (log T ) 1-σ log log T -2k 1 T 2T T |R Y (σ + it)| 2k dt Ck 1-σ log log T (log k) σ (log T ) 1-σ 2k . Choosing k = [log T /(C 1 log log T )], where C 1 = 6A(1 + C) 1/(1-σ)
, yields the desired bound.

Lemma 3.5. Let 1 2 ≤ σ ≤ 1 and A ≥ 1 be fixed, and let Y = (log T ) A . Then, for any positive integers k, such that k + ≤ (log T )/(6A log log T ), we have

1 T 2T T R Y (σ + it) k • R Y (σ + it) dt = E R Y (σ, X) k • R Y (σ, X) + O Y k+ T 5/6 .
Proof. Integrating term-by-term gives

1 T 2T T R Y (σ + it) k • R Y (σ + it) dt = p m 1 1 ,p m 2 2 ,...,p m k k ≤Y q n 1 1 ,q n 2 2 ,...,q n ≤Y 1 m 1 p m 1 σ r • • • m k p m k σ r n 1 q n 1 σ 1 • • • n q n σ 1 T 2T T q n 1 1 • • • q n p m 1 1 • • • p m k k it dt = Σ D + Σ O ,
where Σ D equals the sum over the terms where

p m 1 1 • • • p m k k = q n 1 1 • • • q n and
Σ O consists of the remaining terms. Observe that for non-negative integers a and b

E X(p) a X(p) b = 1 if a = b, 0 otherwise.
Using this and noting that {X(p)} p are independent random variables and that p 1 , . . . p k , q 1 , . . . , q are prime numbers it follows that

Σ D = E R Y (σ, X) k • R Y (σ, X) .
To complete the proof note that both products

p m 1 1 • • • p m k k and q n 1 1 • • • q n are ≤ T 1/6 so for each term in Σ O one has | log (p m 1 1 • • • p m k k )/(q n 1 1 • • • q n ) | T 1/6 . Thus, Σ O 1 T 5/6 p m 1 1 ,p m 2 2 ,...,p m k k ≤Y q n 1 1 ,q n 2 2 ,...,q n ≤Y 1 m 1 p m 1 σ r • • • m k p m k σ r n 1 q n 1 σ 1 • • • n q n σ < Y k+ T 5/6 .
T A(T ) exp z 1 R Y (σ + it) + z 2 R Y (σ + it) dt = j+ ≤N z j 1 z 2 j! ! 1 T A(T ) R Y (σ + it) j R Y (σ + it) dt + E 1 . (4.1)
To estimate E 1 we use the assumption that R Y (σ + it) ≤ (log T ) 1-σ / log log T for t ∈ A(T ) and also apply Stirling's formula to get that

E 1 j+ ≥N k j+ j! ! 1 T A(T ) R Y (σ + it) j+ dt ≤ n≥N k n n! (log T ) 1-σ log log T n j≤n n! j!(n -j)! ≤ n≥N 1 n! 2k(log T ) 1-σ log log T n ≤ n≥N 6k(log T ) 1-σ N log log T n e -N .
Let S(T ) = {T ≤ t ≤ 2T : t / ∈ A(T )}. If j + ≤ N then using Lemmas 3.3 and 3.4 along with the Cauchy-Schwarz inequality we get

1 T S(T ) R Y (σ + it) j R Y (σ + it) dt ≤ meas(S(T )) T 1/2 1 T 2T T R Y (σ + it) 2(j+ ) dt 1/2 exp -B log T 2 log log T C (j + ) 1-σ (log(j + + 2)) σ j+ ,
for some positive constants B = B(σ, A) and C = C(σ). Inserting this bound in equation (4.1) we deduce

1 T A(T ) exp z 1 R Y (σ + it) + z 2 R Y (σ + it) dt = j+ ≤N z j 1 z 2 j! ! 1 T 2T T R Y (σ + it) j R Y (σ + it) dt + E 2 , (4.2)
where

E 2 exp -B log T 2 log log T j+ ≤N k j+ j! ! C (j + ) 1-σ (log(j + + 2)) σ j+ + e -N exp -B log T 2 log log T n≤N 1 n! 2C kN 1-σ (log(N + 2)) σ n + e -N exp -B log T 2 log log T exp 2C kN 1-σ (log(N + 2)) σ + e -N exp -B log T 4 log log T + e -N , (4.3)
if D is suitably large and k ≤ c 0 (log T ) σ where c 0 is suitably small. Now, for all j + ≤ N , we have by Lemma 3.5 that

1 T 2T T R Y (σ + it) j R Y (σ + it) dt =E R Y (σ, X) j R Y (σ, X) + O Y j+ √ T .
Note that Y 2N T 1/4 if D is suitably large. By this, (4.2), and (4.3) we obtain

1 T A(T ) exp z 1 R Y (σ + it) + z 2 R Y (σ + it) dt = j+ ≤N z j 1 z 2 j! ! E R Y (σ, X) j R Y (σ, X) + O exp - B 4 log T log log T . (4.4)
Furthermore, by Lemma 3.3 we have

E |R Y (σ, X)| C 1-σ (log ) σ ,
for all ≥ 2. Therefore, the main term on the RHS of (4.4) equals

E exp z 1 R Y (σ, X) + z 2 R Y (σ, X) + E 3 ,
where

E 3 n≥N 1 n! 2Ckn 1-σ (log n) σ n ≤ n≥N 6Ck (n log n) σ n ≤ n≥N 6Ck (N log N ) σ n e -N .
This completes the proof.

Before proving Theorems 2.1 and 2.2, we need the following lemma which shows that the characteristic function of the random variable log ζ(σ, X) is well approximated by that of R Y (σ, X) in a certain range that depends on Y . Lemma 4.1. Let 1/2 < σ < 1 be fixed. Let Y be a large positive real number, and s be a complex number such that |s| ≤ Y σ-1/2 . Then we have

E (|ζ(σ, X)| s ) = E exp s Re R Y (σ, X) + O E |ζ(σ, X)| Re(s) |s| Y σ-1/2 . (4.5) Moreover, if u, v are real numbers such that |u| + |v| ≤ Y σ-1/2 , then Φ rand σ (u, v) = E exp iu Re R Y (σ, X) + iv Im R Y (σ, X) + O |u| + |v| Y σ-1/2 . (4.6)
Proof. Let z be a complex number with |z| ≤ Y σ-1/2 . Using that

n≥2 p n >Y 1 p σn 1 Y σ-1/2 ,
we obtain

E exp z Re log ζ(σ, X) = E exp z Re R Y (σ, X) + z Re p>Y X(p) p σ + O |z| Y σ-1/2 .
Furthermore, if p > Y then |z| < p σ and hence

E exp z Re X(p) p σ = E 1 + z Re X(p) p σ + O |z| 2 p 2σ = 1 + O |z| 2 p 2σ .
The independence of the X(p)'s together with the fact that p>Y p -2σ Y 1-2σ imply that

E exp z Re log ζ(σ, X) = E exp z Re R Y (σ, X) + O |z| Y σ-1/2 , (4.7)
from which (4.5) follows. To obtain (4.6) one also uses that

E exp z Im log ζ(σ, X) = E exp z Im R Y (σ, X) + O |z| Y σ-1/2 ,
which can be obtained along similar lines.

Proof of Theorem 2.1. Let Y = (log T ) B/(σ-1/2) where B = B(A) is a suitably large constant that will be chosen later. Then it follows from Lemma 3.1 that

log ζ(σ + it) = R Y (σ + it) + O 1 (log T ) B/2-3 , (4.8
Φ σ,T (u, v) = 1 T B(T ) exp iu Re log ζ(σ + it) + iv Im log ζ(σ + it) dt + O T -d(σ) = 1 T B(T ) exp iu Re R Y (σ + it) + iv Im R Y (σ + it) dt + O 1 (log T ) B/2-4 = 1 T 2T T exp iu Re R Y (σ + it) + iv Im R Y (σ + it) dt + O 1 (log T ) B/2-4 .
Let A(T ) be as in Proposition 2.3. Then, by Lemma 3.4 and Proposition 2.3, taking

z 1 = i 2 (u -iv) and z 2 = i 2 (u + iv) there, we get 1 T 2T T exp iu Re R Y (σ + it) + iv Im R Y (σ + it) dt = 1 T A(T ) exp iu Re R Y (σ + it) + iv Im R Y (σ + it) dt + O 1 (log T ) B = E exp iu Re R Y (σ, X) + iv Im R Y (σ, X) + O 1 (log T ) B .
Finally, using (4.6) we deduce

E exp iu Re R Y (σ, X) + iv Im R Y (σ, X) = Φ rand σ (u, v) + O 1 (log T ) B-1 .
Choosing B = 2A + 8, and collecting the above estimates completes the proof.

Proof of Theorem 2.2. As in the proof of Theorem 2.1, let Y = (log T ) B/(σ-1/2) where B = 2A + 8, and B(T ) be the set of t ∈ [T, 2T ] such that (4.8) holds. Then, by Lemma

meas [T, 2T ] \ B(T )

T 1-d(σ) for some constant d(σ) > 0. Moreover, let A(T ) be as in Proposition 2.3. We define

E(T ) := [T, 2T ] \ A(T ) ∩ B(T ) .
Then, it follows from Lemma 3.4 that

meas E(T ) T exp -b 0 log T log log T , (4.9)
for some positive constant b 0 = b 0 (σ, A).

Now, by (4.8) we get

1

T [T,2T ]\E(T ) |ζ(σ + it)| z dt = 1 T A(T )∩B(T ) exp z Re R Y (σ + it) + O 1 (log T ) A dt = 1 T A(T )∩B(T ) exp z Re R Y (σ + it) dt + E 4 , (4.10
) where On the other hand, since meas [T, 2T ] \ B(T ) T 1-d(σ) , and

E 4 1 T (log T ) A A(T ) exp Re(z) Re R Y (σ + it) dt 1 (log T ) A E exp Re(z) Re R Y (σ, X) 1 (log T ) A E |ζ(σ, X)| Re(z) ,
|R Y (σ + it)| ≤ (log T ) 1-σ / log log T for all t ∈ A(T ) we deduce that 1 T A(T ) - A(T )∩B(T ) exp z Re R Y (σ + it) dt T -d(σ) exp Re(z) (log T ) 1-σ log log T T -d(σ)/2 .
(4.12) Furthermore, combining Proposition 2.3 and Lemma 4.1 we obtain z) .

1 T A(T ) exp z Re R Y (σ + it) dt = E exp z Re R Y (σ, X) + O 1 (log T ) A = E |ζ(σ, X)| z + O 1 (log T ) A E |ζ(σ, X)| Re(
(4.13) The result follows upon inserting the estimates (4.11), (4.12) and (4.13) in (4.10).

L

2k norm of log ζ(σ + it) -a: Proof of Proposition 2.5
As a special case of Lemma 2.2.1 of Guo [START_REF] Rang | On the zeros of the derivative of the Riemann zeta function[END_REF], which itself is a generalization of a lemma of Landau (see [START_REF] Landau | Über die Wurzeln der Zetafunktion[END_REF] or Lemma α from Chapter III of [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]), we have

Lemma 5.1. Let 0 < r 1. Also, let s 0 = σ 0 + it and suppose f (z) is analytic in |z -s 0 | ≤ r. Define M r (s 0 ) = max |z-s 0 |≤r f (z) f (s 0 ) + 3 and N r (s 0 ) = | -s 0 |≤r 1,
where the last sum runs over the zeros, , of f (z) in the closed disk of radius r centered at s 0 . Then for 0 < δ < r/2 and |z -s 0 | ≤ r -2δ we have

f f (z) = |ρ-s 0 |≤r-δ 1 s -ρ + O 1 δ 2 log M r (s 0 ) + N r-δ (s 0 )(log 1/δ + 1) .
Here and throughout, we denote by ρ a = β a +iγ a the a-points of ζ(s). In the following we take

f (z) = f a (z) = (ζ(z) -a)/(1 -a) if a = 1, 2 z (ζ(z) -1) if a = 1. (5.1)
We also choose

δ = (σ -1/2)/5 and r = σ 0 -(σ + 1/2)/2, ( 5.2)
where σ 0 is taken to be large enough (depending on a) so that |f (σ 0 + it)| ≥ 1/10 and min ρa |s 0 -ρ a | ≥ 1/10 uniformly in t. For |z -s 0 | ≤ r -2δ Lemma 5.1 yields

ζ (z) ζ(z) -a = |ρa-s 0 |≤r-δ 1 z -ρ a + O 1 δ 2 (log M r (s 0 ) + N r-δ (s 0 ))(log 1/δ + 1) . (5.3)
Lemma 5.2. Let 1/2 < σ ≤ 2 be fixed. Also, let δ, r, and σ 0 be as in (5.2). For t sufficiently large we have

log |ζ(σ + it) -a| = |ρa-s 0 |≤r-δ log |σ + it -ρ a | + O(log M r (s 0 )).
Proof. Let f (z) be as in (5.1). First, note that Jensen's formula gives

r 0 N x (s 0 ) dx x = 1 2π 2π 0 log |f (re iθ + s 0 )| dθ -log |f (s 0 )|.
Observe that

r 0 N x (s 0 ) dx x ≥ r r-δ N x (s 0 ) dx x ≥ δ r N r-δ (s 0 ).
By this and the bound log |f (s 0 )| ≥ log 1/10 we have Applying (5.4) to the error term completes the proof.

N r-δ (s 0 ) ≤ r δ 1 2π
Lemma 5.3. Let 1 2 < σ ≤ 2 be fixed. Also, let r, δ, and σ 0 be as in (5.2). Then there exists an absolute constant C > 0 such that for any real number k ≥ 1 2 we have 1

T 2T T |ρa-s 0 |≤r-δ | log |σ + it -ρ a || 2k dt Γ(2k + 1)(C log M r+δ (s 0 )) 2k .
Proof. Define D R (z) to be the closed disc of radius R centered at z. For n = T , . . . , 2T + 1 let

D n = 1/ √ δ +1 =0 D r (σ 0 + i(n + • √ δ)).
Observe that

D r-δ (σ 0 +in) z : n ≤ Im z ≤ n+ √ δ, σ 0 -(r -δ) ≤ Re z ≤ σ 0 +r -δ ⊂ D r (σ 0 +in).
Next, note that

{z : n + √ δ ≤ Im z ≤ n + 2 √ δ, σ 0 -(r -δ) ≤ Re z ≤ σ 0 + r -δ} ⊂ D r (σ 0 + i(n + √ δ)),
and so on. Hence, by construction

n≤t≤n+1 D r-δ (σ 0 + it) ⊂ D n .
This implies that

2T T |ρa-s 0 |≤r-δ | log |σ + it -ρ a || 2k dt ≤ 2T +1 n= T n+1 n |ρa-s 0 |≤r-δ | log |σ + it -ρ a || 2k dt ≤ 2T +1 n= T n+1 n ρa∈Dn | log |σ + it -ρ a || 2k dt.
Applying Minkowski's inequality to the right-hand side we get that

2T T |ρa-s 0 |≤r-δ | log |σ + it -ρ a || 2k dt ≤ 2T +1 n= T ρa∈Dn n+1 n | log |σ + it -ρ a || 2k dt 1/(2k) 2k . (5.5) Let R = σ 0 -(σ + 1/2)/2. To bound |ζ(s n )| 2 we note that |ζ(s n )| 2 ≤ 4 π(σ -1 2 ) 2 D R (σ 0 +itn)
|ζ(x + iy)| 2 dx dy.

(5.9) (For a proof of this inequality see the lemma preceding Theorem 11.9 of Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]). Let S j = {s n : n ≡ j (mod

(4 R + 2))}. If s m , s n ∈ S j and m = n then |m -n| ≥ 4 R +2; so that |t m -t n | ≥ 2R +1. This implies that D R (σ 0 +it n )∩D R (σ 0 +it m ) = ∅.
Thus, since the disks are disjoint we see that by (5.9) we have

sn∈S j |ζ(s n )| 2 2σ 0 -1 2 •(σ+ 1 2 ) 1 2 •(σ+ 1 2 ) 2T +R +R+1 T -R -R-1 |ζ(u + it)| 2 dt du.
Applying, the well-known mean value estimate for ζ(s) to the inner integral (see Theorem 7.2(A) of [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]) we have (uniformly in j)

sn∈S j |ζ(s n )| 2 T.
Also, {s n } = j S j . Thus,

2T +1 n= T |ζ(s n )| 2 = 4 R +1 j=0 sn∈S j |ζ(s n )| 2 T.
Inserting this into (5.8) completes the proof.

Lemma 5.5. Let 1 2 < σ ≤ 2 be fixed. Also, let r, δ, and σ 0 be as in (5.2). Then, there exists an absolute constant C > 0 such that for any real number k ≥ 1

2T T (log M r+δ (s 0 )) 2k dt T (Ck) 2k .
Proof. Apply Jensen's inequality to the function (log(x + e 2k-1 )) 2k and then Lemma 5.4 to see that

1 T 2T T (log M r+δ (s 0 )) 2k dt < 1 4 k T 2T T (log((M r+δ (s 0 )) 2 + e 2k-1 )) 2k dt ≤ 1 4 k log 1 T 2T T (M r+δ (s 0 )) 2 dt + e 2k-1 2k (Ck) 2k ,
for some absolute constant C. Lemma 6.2. Let R = {z = x + iy ∈ C : a 1 < x < a 2 and b 1 < y < b 2 }, and L > 0 be a real number. For any z = x + iy ∈ C we have

1 R (z) = W L,R (z)+O sin 2 (πL(x -a 1 )) (πL(x -a 1 )) 2 + sin 2 (πL(x -a 2 )) (πL(x -a 2 )) 2 + sin 2 (πL(y -b 1 )) (πL(y -b 1 )) 2 + sin 2 (πL(y -b 2 )) (πL(y -b 2 )) 2
where W L,R (z) equals

1 2 Re L 0 L 0 G u L G v L e 2πi(ux-vy) f a 1 ,a 2 (u)f b 1 ,b 2 (v) -e 2πi(ux+vy) f a 1 ,a 2 (u)f b 1 ,b 2 (v) du u dv v .
Proof. Here and throughout we shall denote by 1 α,β the indicator function of the interval (α, β). Observe that

1 α,β (x) = sgn(x -α) -sgn(x -β) 2 + O δ(x -α) + δ(x -β) ,
where δ(x) is the Dirac delta function (it equals 1 when x = 0, and zero otherwise). Furthermore, it follows from Lemma 6.1 that

1 α,β (x) = Im L 0 G u L e 2πiux f α,β (u) du u + O sin 2 (πL(x -α)) (πL(x -α)) 2 + sin 2 (πL(x -β)) (πL(x -β)) 2 .
(6.1) The result follows from the fact that 1 R (z) = 1 a 1 ,a 2 (x)1 b 1 ,b 2 (y) together with (6.1) and the identity Im(w 1 )Im(w 2 ) = 1 2 Re(w 1 w 2 -w 1 w 2 ). (6.

2)

The last ingredient we need in order to establish Theorem 1.1 is the following lemma.

Lemma 6.3. Let 1 2 < σ ≤ 1. Let u be a large positive real number, then

E exp iu Re log ζ(σ, X) exp - u 5 log u , and
E exp iu Im log ζ(σ, X) exp - u 5 log u .
Proof. First, note that E(e is Re X(p) ) = E(e is Im X(p) ) = J 0 (s) for all s ∈ R and all primes p, where J 0 (s) is the Bessel function of order 0. We shall prove only the first inequality since the second can be derived similarly. We have

E exp iu Re log ζ(σ, X) = p E exp -iu Re log 1 - X(p) p σ .
Since |E (exp (-iu Re log (1 -X(p)p -σ ))) | ≤ 1 for all primes p, then

E exp iu Re log ζ(σ, X) ≤ √ u≤p≤u E exp iu p σ Re X(p) + O u p 2σ = exp O u 3/2-σ √ u≤p≤u/2 J 0 u p σ .
Now, using that |J 0 (x)| ≤ e -1/2 for all x ≥ 2, along with the prime number theorem we obtain

E exp iu Re log ζ(σ, X) exp - 1 2 π(u/2) + O u 3/2-σ exp - u 5 log u ,
as desired.

Proof of Theorem 1.1. To shorten our notation we let

Ψ T (R) = P T log ζ(σ + it) ∈ R , and Ψ(R) = P log ζ(σ, X) ∈ R .
Let R be a rectangle with sides parallel to the coordinate axes, and

R = R ∩ [-log 2 T, log 2 T ] × [-log 2 T, log 2 T ]. If 1/2 < σ < 1 we use the large deviation re- sult (1.
2) (and its analogue for arg ζ(σ + it)) to obtain that

Ψ T R = Ψ T R + O 1 (log T ) 2 , and Ψ(R) = Ψ R + O 1 (log T ) 2 . ( 6.3)
When σ = 1, we obtain (6.3) by using the large deviation result of Granville and Soundararajan [START_REF] Granville | Extreme values of |ζ(1 + it)|, The Riemann zeta function and related themes[END_REF], which states that in the range τ ≤ log 3 T + γ + o(1) we have

log P T log |ζ(1 + it)| > τ = (1 + o(1)) log P log |ζ(1, X)| > τ = -exp (c 1 e τ -τ -c 2 ) (1 + o(1)),
for some positive constants c 1 , c 2 . The analogous result for arg ζ(1 + it) follows from the work of Lamzouri [START_REF]Extreme values of arg L(1, χ)[END_REF]. Let S be the set of rectangles R ⊂ [-log 2 T, log 2 T ] × [-log 2 T, log 2 T ] with sides parallel to the coordinate axes. Then, we deduce that

D σ (T ) = sup R∈S |Ψ T (R) -Ψ(R)| + O 1 (log T ) 2 .
Let R be a rectangle in S and L a positive real number to be chosen later. Then it follows from Lemma 6.2 that

Ψ T R = 1 T 2T T W L,R log ζ(σ +it) dt+O I T (L, a 1 )+I T (L, a 2 )+J T (L, b 1 )+J T (t, b 2 ) (6.4
) where

I T (L, s) = 1 T 2T T sin 2 πL(Re log ζ(σ + it) -s) (πL(Re log ζ(σ + it) -s)) 2 dt, and
J T (L, s) = 1 T 2T T sin 2 πL(Im log ζ(σ + it) -s) (πL(Im log ζ(σ + it) -s)) 2 dt.
We choose L = c(log T ) σ for a suitably small constant c = c(σ) > 0 if 1/2 < σ < 1, and L = log T /(50(log 2 T ) 2 ) when σ = 1. Then it follows from Theorem 2.1 and equation (2.2) that for all |u|, |v| ≤ L we have

Φ σ,T (u, v) = Φ rand σ (u, v) + O 1 (log T ) 5 . (6.5)
First, we handle the main term of (6.4)

1 T 2T T W L,R log ζ(σ + it) dt = 1 2 Re L 0 L 0 G u L G v L Φ σ,T (2πu, -2πv)f a 1 ,a 2 (u)f b 1 ,b 2 (v) -Φ σ,T (2πu, 2πv)f a 1 ,a 2 (u)f b 1 ,b 2 (v) du u dv v .
(6.6) Inserting the estimate (6.5) in equation (6.6) and using that

|f α,β (u)| = 1 2 2πβu 2παu e -it dt ≤ πu|β -α|, (6.7)
we obtain 1

T 2T T W L,R log ζ(σ + it) dt =E W L,R log ζ(σ, X) + O meas 2 (R) L 2 (log T ) 5 =E W L,R log ζ(σ, X) + O 1 (log T ) 2 ,
(6.8) where meas 2 denotes the two-dimensional Lebesgue measure. Furthermore we infer from Lemma 6.2

E W L,R log ζ(σ, X) =E 1 R log ζ(σ, X) + O I rand (L, a 1 ) + I rand (L, a 2 ) + J rand (L, b 1 ) + J rand (L, b 2 ) , ( 6
.9) where

I rand (L, s) = E sin 2 πL(Re log ζ(σ, X) -s) (πL(Re log ζ(σ, X) -s)) 2 ,
and

J rand (L, s) = E sin 2 πL(Im log ζ(σ, X) -s) (πL(Im log ζ(σ, X) -s)) 2 . Note that E 1 R log ζ(σ, X) = P (log ζ(σ, X) ∈ R).
Moreover, in order to bound I rand (L, s) and J rand (L, s) we use the following identity

sin 2 (πLx) (πLx) 2 = 2(1 -cos(2πLx)) L 2 (2πx) 2 = 2 L 2 L 0 (L -v) cos(2πxv)dv. (6.10)
Indeed, using (6.10) along with Lemma 6.3 we obtain

I rand (L, s) = E Re L 0 2(L -v) L 2 exp 2πiv Re log ζ(σ, X) -s dv = Re L 0 2(L -v) L 2 e -2πivs Φ rand σ (2πv, 0)dv 1 L 1 + L 2 exp - v log v dv 1 L . (6.11)
uniformly for all s ∈ R. Similarly, one obtains that J rand (L, s) 1/L. Therefore, inserting these estimates in (6.9) and using (6.8) we deduce

1 T 2T T W L,R log ζ(σ + it) dt = P (log ζ(σ, X) ∈ R) + O 1 L . (6.12)
Now it remains to bound the error term on the right hand side of (6.4). Using the identity (6.10) along with equations (6.5) and (6.11) we obtain

I T (L, s) = Re 1 T 2T T L 0 2(L -v) L 2 exp 2πiv Re log ζ(σ + it) -s dvdt = Re L 0 2(L -v) L 2 e -2πivs Φ σ,T (2πv, 0)dv = Re L 0 2(L -v) L 2 e -2πivs Φ rand σ (2πv, 0)dv + O 1 (log T ) 5 1 L ,
uniformly for all s ∈ R. Moreover, the bound J T (L, s) 1/L can be obtained along the same lines. Combining these estimates with (6.4) and (6.12) completes the proof.

Large deviations: Proof of Theorem 1.3

For z ∈ C we define

M (z) = log E(|ζ(σ, X)| z ).
Further, let κ be the unique positive solution to the equation M (k) = τ. One of the main ingredients in the proof of Theorem 1.3 is the following proposition which is established using the saddle-point method.

Proposition 7.1. Let 1 2 < σ < 1. Uniformly for τ ≥ 1 we have

P(log |ζ(σ, X)| > τ ) = E (|ζ(σ, X)| κ ) e -τ κ k 2πM (κ) 1 + O κ 1-1 σ log κ .
7.1. Preliminaries. Let χ(y) = 1 if y > 1 and be equal to 0 otherwise. Then we have the following smooth analogue of Perron's formula, which is a slight variation of a formula of Granville and Soundararajan (see [START_REF] Granville | Extreme values of |ζ(1 + it)|, The Riemann zeta function and related themes[END_REF]).

Lemma 7.2. Let λ > 0 be a real number and N be a positive integer. For any c > 0 we have for y > 0

0 ≤ 1 2πi c+i∞ c-i∞
y s e λs -1 λs

N ds s -χ(y) ≤ 1 2πi c+i∞ c-i∞
y s e λs -1 λs N 1 -e -λN s s ds.

Proof. For any y > 0 we have

1 2πi c+i∞ c-i∞
y s e λs -1 λs

N ds s = 1 λ N λ 0 • • • λ 0 1 2πi c+i∞ c-i∞ ye t 1 +•••+t N s ds s dt 1 • • • dt N
so that by Perron's formula we obtain

1 2πi c+i∞ c-i∞
y s e λs -1 λs

N ds s =      = 1 if y ≥ 1, ∈ [0, 1] if e -λN ≤ y < 1, = 0 if 0 < y < e -λN .
Therefore we deduce that

1 2πi c+i∞ c-i∞
y s e -λN s e λs -1 λs

N ds s ≤ χ(y) ≤ 1 2πi c+i∞ c-i∞
y s e λs -1 λs

N ds s (7.1)
which implies the result.

Lemma 7.3. Let s = k + it
where k is a large positive number. Then, in the range |t| ≥ k we have

E (|ζ(σ, X)| s ) exp -|t| 1/σ-1 E |ζ(σ, X)| k .
Proof. For simplicity we suppose that t > 0. First, note that

E (|ζ(σ, X)| s ) = p E 1 - X(p) p σ -s
.

Therefore, for any y ≥ 2 we have

|E (|ζ(σ, X)| s )| E (|ζ(σ, X)| k ) ≤ p>y E 1 -X(p) p σ -k-it E 1 -X(p) p σ -k (7.2)
Moreover, since for p > |s| 1/(2σ)

1 -2 ReX(p) p σ + 1 p 2σ -s/2 = exp s p σ • Re X(p) 1 + |s| p 2σ we have E 1 - X(p) p σ -s = E 1 -2 ReX(p) p σ + 1 p 2σ -s/2 = I 0 s p σ 1 + O |s| p 2σ , (7.3)
where I 0 (z) := 1 2π 2π 0 e z cos(θ) dθ is the modified Bessel function of order 0. Let y = t 2/σ . since I 0 (z) = 1 + z 2 /4 + O(|z| 4 ) for |z| ≤ 1, we deduce that for all primes p > y

E 1 -X(p) p σ -s E 1 -X(p) p σ -k = exp s 2 -k 2 4p 2σ + O t p 2σ + t 4 p 4σ .
Since Re(s 2 -k 2 ) = -t 2 , it follows from the prime number theorem and equation (7.2) that

|E (|ζ(σ, X)| s )| E (|ζ(σ, X)| k ) ≤ exp - t 2 4 p>y 1 p 2σ + O t p>y 1 p 2σ + t 4 p>y 1 p 4σ ≤ exp -c(σ) t 2/σ-2 log t + O t 2/σ-3 ,
for some constant c(σ) > 0. This implies the result.

Let f (u) := log I 0 (u). Then, a classical estimate (see for example Lemma 3.1 of [START_REF]Extreme values of arg L(1, χ)[END_REF]) asserts that f (u) u 2 if 0 ≤ u ≤ 1 and f (u) u if u ≥ 1 (where we write g(u) h(u) if we have both g(u) h(u) and h(u) g(u)). Similarly, we have the following standard estimates.

Lemma 7.4. We have

f (u) u if 0 ≤ u ≤ 1 1 if u ≥ 1. f (u) 1 if 0 ≤ u ≤ 1 u -1 if u ≥ 1. f (u) u if 0 ≤ u ≤ 1 u -2 if u ≥ 1.
Next, we have the following proposition from which we deduce an asymptotic formula for the saddle-point κ in terms of τ . Proposition 7.5. For large positive real numbers k, we have

M (k) = g 0 (σ) k 1/σ log k 1 + O 1 log k , (7.4)
where

g 0 (σ) := ∞ 0 f (u) u 1/σ+1 du, and
M (k) = g 1 (σ) k 1/σ-1 log k 1 + O 1 log k . (7.5)
where

g 1 (σ) := ∞ 0 f (u) u 1/σ du. Similarly we have M (k) σ k 1/σ-2 log k , and
M (k + it) σ k 1/σ-3 log k , (7.6)
for all |t| ≤ k.

Proof. The first estimate (7.4) follows from Proposition 3.2 of [START_REF]On the distribution of extreme values of zeta and L-functions in the strip 1 2 < σ < 1[END_REF]. The other estimates can be proved along the same lines.

Corollary 7.6. Let τ be a large real number and let κ be the solution to M (k) = τ . Then

κ = g 2 (σ)τ σ/(1-σ) (log τ ) σ/(1-σ) 1 + O log 2 τ log τ ,
where

g 2 (σ) = σ (1 -σ)g 1 (σ) σ/(1-σ)
. Combining Proposition 7.1, Proposition 7.5 and Corollary 7.6 we recover the following result, which was obtained by the first author in [START_REF]On the distribution of extreme values of zeta and L-functions in the strip 1 2 < σ < 1[END_REF].

Corollary 7.7. Let 1 2 < σ < 1. There exists a constant A(σ) > 0 such that uniformly for τ ≥ 2 we have Therefore, combining this estimate with equations (7.7), (7.8) and (7.9) we deduce that

P(log |ζ(σ, X)| > τ ) = exp -A(σ)τ 1 (1-σ) (log τ ) σ (1-σ) (1 + o(1)) .
P(log |ζ(σ, X)| > τ ) - 1 2πi κ+iκ κ-iκ E (|ζ(σ, X)| s ) e -τ s e λs -1 λs 2 ds λκ + e -κ 1/σ-1 λκ E (|ζ(σ, X)| κ ) e -τ κ .
(7.10)

On the other hand, it follows from (7.6) that for |t| ≤ κ we have

log E |ζ(σ, X)| κ+it = log E (|ζ(σ, X)| κ ) + itM (κ) - t 2 2 M (κ) + O κ 1/σ-3 log κ |t| 3 .
Also, note that e λs -1

λs 2 = 1 κ 1 -i t κ + O λκ + t 2 κ 2 .
Hence, using that M (κ) = τ we obtain

E (|ζ(σ, X)| s ) e -τ s e λs -1 λs 2 = 1 κ E (|ζ(σ, X)| κ ) e -τ κ exp - t 2 2 M (κ) 1 -i t κ + O λκ + t 2 κ 2 + κ 1/σ-3 log κ |t| 3 .
Therefore, we obtain

1 2πi κ+iκ κ-iκ E (|ζ(σ, X)| s ) e -τ s e λs -1 λs 2 ds = 1 κ E (|ζ(σ, X)| κ ) e -τ κ 1 2π κ -κ exp - t 2 2 M (κ) 1 + O λκ + t 2 κ 2 + κ 1/σ-3 log κ |t| 3 dt
since the integral involving it/κ vanishes. Further, we have

1 2π κ -κ exp - t 2 2 M (κ) dt = 1 2πM (κ) 1 + O exp - 1 2 κ 2 M (κ) , and κ -κ |t| n exp - t 2 2 M (κ) dt 1 M (κ) (n+1)/2 .
Thus, using (7.6) we deduce that

1 2πi κ+iκ κ-iκ E (|ζ(σ, X)| s ) e -τ s e λs -1 λs 2 ds = E (|ζ(σ, X)| κ ) e -τ κ κ 2πM (κ) 1 + O λκ + κ 1-1 σ log κ . (7.11)
Finally, it follows from (7.6) that κ M (κ) σ κ 1/(2σ) (log κ) -1/2 . Thus, combining the estimates (7.10) and (7.11) and choosing λ = κ -3 completes the proof.

Proof of Theorem 1. Then we have the following lemma.

Lemma 8.1. Let 1 2 < σ < 1 be fixed. We have,

1 T 2T T log |ζ(σ + it) -a|dt = 1 T t∈S(T ) log |ζ(σ + it) -a|dt + O (log 2 T ) 2 (log T ) σ .
and and hence the claim. The proof of the second statement is similar.

E(log |ζ(σ, X) -a|) = E(1 F • log |ζ(σ, X) -a|) + O (log 2 T ) 2 (log T ) σ . Proof. Note 1 T t / ∈S(T ) log |ζ(σ + it) -a|dt ≤ ≤ 1 T • meas{T ≤ t ≤ 2T : t / ∈ S(T )} 1-1/2k • 1 T 2T T | log |ζ(σ + it) -a|| 2k dt 1/2k
We let S 1 (T ) be the set of points t ∈ S(T ) such that log |ζ(σ + it)| > log |a| + δ, and S 2 (T ) = S(T )\S 1 (T ). Similarly, F 1 is the sub-event of F where log |ζ(σ, X)| > log |a|+δ and F 2 = F \ F 1 . Moreover, we define Also, let

Φ 1 (u, v) = 1 T meas{t ∈ S 1 (
Ψ(u) = 1 T meas{t ∈ S 1 (T ) : log |ζ(σ + it)| ≤ u} Ψ(u) = P F 1 and log |ζ(σ, X)| ≤ u .
Let g(u, v) := log(e u+iv -a) and h(u, v) := Re(g(u, v)). Note that h is twice differentiable in the region of R 2 where u -log |a| > δ.

We are now going to show that

t∈S(T ) log |ζ(σ + it) -a|dt and E[1 F • log |ζ(σ, X) -a|]
match up to a small error term. For this we will need to integrate by parts. We establish the three necessary lemmas below.

Lemma 8.2. We have

1 T t∈S 1 (T ) log |ζ(σ + it) -a|dt = log 2 T -log 2 T log 2 T log |a|+δ Φ 1 (u, v) ∂ 2 h(u, v) ∂u∂v dudv - meas(S 1 (T )) T h(log 2 T, log 2 T ) + 1 T t∈S 1 (T ) h log 2 T, arg ζ(σ + it) + h log |ζ(σ + it)|, log 2 T dt, and
E 1 F 1 • log |ζ(σ, X) -a| = log 2 T -log 2 T log 2 T log |a|+δ Φ1 (u, v) ∂ 2 h(u, v) ∂u∂v dudv -P(F 1 )h(log 2 T, log 2 T ) + E 1 F 1 • h log 2 T, arg ζ(σ, X) + E 1 F 1 • h log |ζ(σ, X)|, log 2 T .
Proof. We only prove the first identity since the second can be obtained along similar lines. We have

log 2 T -log 2 T log 2 T log |a|+δ Φ 1 (u, v) ∂ 2 h(u, v) ∂u∂v dudv = 1 T log 2 T -log 2 T log 2 T log |a|+δ t∈S 1 (T ) log |ζ(σ+it)|≤u arg ζ(σ+it)≤v ∂ 2 h(u, v) ∂u∂v dtdudv = 1 T t∈S 1 (T ) log 2 T arg ζ(σ+it) log 2 T log |ζ(σ+it)| ∂ 2 h(u, v) ∂u∂v dudvdt = 1 T t∈S 1 (T ) log 2 T arg ζ(σ+it) ∂ ∂v h log 2 T, v - ∂ ∂v h log |ζ(σ + it)|, v dvdt = 1 T t∈S 1 (T ) log |ζ(σ + it) -a|dt + 1 T t∈S 1 (T ) h log 2 T, log 2 T dt - 1 T t∈S 1 (T ) h log 2 T, arg ζ(σ + it) + h log |ζ(σ + it)|, log 2 T dt. Lemma 8.3. Let 1 2 < σ < 1 be fixed. We have 1 T t∈S 1 (T ) h log |ζ(σ+it)|, log 2 T dt = E 1 F 1 •h log |ζ(σ, X)|, log 2 T +O log 2 T (log T ) σ , and 1 T t∈S 1 (T ) h log 2 T, arg ζ(σ + it) dt = E 1 F 1 • h log 2 T, arg ζ(σ, X) + O log 2 T (log T ) σ .
Proof. We only prove the first estimate since the second is similar. We have

1 T t∈S 1 (T ) h log |ζ(σ + it)|, log 2 T dt = log 2 T log |a|+δ h(u, log 2 T )dΨ(u).
Integrating by parts, the right-hand side equals

Ψ(u)h(u, log 2 T ) log 2 T log |a|+δ - log 2 T log |a|+δ h (u, log 2 T )Ψ(u)du = Ψ(u)h(u, log 2 T ) log 2 T log |a|+δ - log 2 T log |a|+δ h (u, log 2 T ) Ψ(u)du + E 5 = E 1 F 1 • h log |ζ(σ, X)|, log 2 T + E 5 where E 5 1 (log T ) σ log 2 T + log 2 T log |a|+δ |h (u, log 2 T )|du , (8.1)
which follows from the discrepancy estimate Ψ(u) -Ψ(u) (log T ) -σ , along with the bounds h(log and |e u+iv -a| 2 = e 2u + |a| 2 -2 Re(ae u-iv ) = (e u -|a|) 2 + 2|a|e u 1 -cos(v -arg a) . We split the range of integration over v into intervals [-π + 2πk + arg a, π + 2πk + arg a] with |k| ≤ (log 2 T )/π. Since the integrand is non-negative, we deduce that Combining these estimates we deduce that is twice differentiable in σ for 1 2 < σ < 1. Proof. See Theorem 14 of [START_REF] Borchsenius | Mean motions and values of the Riemann zeta function[END_REF].

Proof of Theorem 1.4. Let 1 2 < σ < 1 and ρ a = β a + iγ a denote an a-point of ζ(s). We know that there is σ 0 = σ 0 (a) such that β a < σ 0 for all a-points ρ a . By Littlewood's lemma (see equation (9.9.1) of Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]), we have (8.2) Furthermore, a standard application of the Jensen's formula shows that (see for example section 9.4 of Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF] or equation (8.4) of Tsang's thesis [START_REF] Tsang | The distribution of the values of the Riemann zeta-function[END_REF]) Recall that f a (σ) = E[log |ζ(σ, X)-a|] is twice differentiable in σ by Lemma 8.5. Hence, we derive 1 h σ+h σ βa>u T ≤γa≤2T

1 du = T 2π • f (σ) -f (σ + h) h + O T (log 2 T ) 2 (log T ) σ • 1 h = - T 2π • f (σ) + O hT + T (log 2 T ) 2 (log T ) σ • 1 h .
Therefore,

βa≥σ+h T ≤γa≤2T 1 ≤ - T 2π • f (σ) + O hT + T (log 2 T ) 2 (log T ) σ • 1 h ≤ βa≥σ T ≤γa≤2T
1.

2010

  Mathematics Subject Classification. Primary 11M26; Secondary 11N13, 60F10. The first author is supported in part by an NSERC Discovery grant. The third author was partially supported by NSF grant DMS-1128155. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n o 320755. log ζ(σ + it), i.e. D σ (T ) := sup R P T log ζ(σ + it) ∈ R -P log ζ(σ, X) ∈ R ,

  Re log ζ(σ + it) and Im log ζ(σ + it) can be very well approximated by the corresponding characteristic function of the random model log ζ(σ, X). For u, v ∈ R we define Φ σ,T (u, v) := 1 T 2T T exp iu Re log ζ(σ + it) + iv Im log ζ(σ + it) dt, and Φ rand σ (u, v) := E exp iu Re log ζ(σ, X) + iv Im log ζ(σ, X) .

Proposition 2 . 3 .

 23 Let 1 2 < σ < 1 and A ≥ 1 be fixed. Let Y = (log T ) A . There exist positive constants b 5 = b 5 (σ, A) and b 6 = b 6 (σ, A) such that for all complex numbers z 1 , z 2 with |z 1 |, |z 2 | ≤ b 5 (log T ) σ we have

4 .

 4 Complex moments of ζ(σ + it): Proofs of Theorems 2.1 and 2.2 We begin by proving Proposition 2.3. Proof of Proposition 2.3. Let k = max{|z 1 |, |z 2 |}, and N = [log T /(D(log log T ))] where D is a suitably large constant. Write 1

  ) for all t ∈ [T, 2T ] except a set of measure T1-d(σ) for some constant d(σ) > 0. Let B(T ) be the set of t ∈ [T, 2T ] such that (4.8) holds. Note that |e ib -e ia | = | b a e ix dx| ≤ |b -a|. Therefore, we obtain

( 4 .

 4 11) by Proposition 2.3 and Lemma 4.1.

2π 0 log

 0 |f (re iθ +s 0 )| dθ -log |f (s 0 )| ≤ r δ (log M r (s 0 )+log 10). (5.4) Applying this estimate in (5.3) and noting that δ 1 and r 1 we have ζ (z) ζ(z) -a = |ρa-s 0 |≤r-δ 1 z -ρ a + O(log M r (s 0 )) for |z -s 0 | ≤ r -2δ. In particular, this formula is valid along the line segment that connects s to s 0 . Hence, integrating the above equation from s to s 0 and taking real parts gives log |ζ(s) -a| -log |ζ(s 0 ) -a| = |ρa-s 0 |≤r-δ (log |s -ρ a | -log |s 0 -ρ a |) + O(log M r (s 0 )). By the choice of σ 0 we have log |ζ(s 0 ) -a| = O(1) and log |s 0 -ρ a | = O(1). Thus, log |ζ(s) -a| = |ρa-s 0 |≤r-δ log |s -ρ a | + O(N r-δ (s 0 ) + log M r (s 0 )).

7. 2 .EE( 7 . 7 )E

 277 Proof of Proposition 7.1 and Theorem 1.3. Proof of Proposition 7.1. Let 0 < λ < 1/(2κ) be a real number to be chosen later. Using Lemma 7.2 with N = 1 (|ζ(σ, X)| s ) e -τ s e λs -1 λs ds s -P(log |ζ(σ, X)| > τ ) (|ζ(σ, X)| s ) e -τ s e λs -1 λs 1 -e -λs s ds. Since λκ < 1/2 we have |e λs -1| ≤ 3 and |e -λs -1| ≤ 2. Therefore, using Lemma 7.3 we obtain κ-iκ κ-i∞ + κ+i∞ κ+iκ E (|ζ(σ, X)| s ) e -τ s e λs -1 λs ds s e -κ 1/σ-1 λκ E (|ζ(σ, X)| κ ) e -τ κ , (|ζ(σ, X)| s ) e -τ s e λs -1 λs 1 -e -λs s ds e -κ 1/σ-1 λκ E (|ζ(σ, X)| κ ) e -τ κ . (7.9) Furthermore, if |t| ≤ κ then (1 -e -λs )(e λs -1) λ 2 |s| 2 . Hence we derive κ+iκ κ-iκ E (|ζ(σ, X)| s ) e -τ s e λs -1 λs 1 -e -λs s ds λκE (|ζ(σ, X)| κ ) e -τ κ .

3 .E 1 T 8 . 3 8. 1 .

 31831 As before, κ denotes the unique solution to M (k) = τ . Let N be a positive integer and 0 < λ < min{1/(2κ), 1/N } be a real number to be chosen later.Let A = 10, E(T ), and b 4 = b 4 (σ, 10) be as in Theorem 2.2. Let Y = (b 4 (log T ) σ )/2. Note that, if T is large enough then by Corollary 7.6 we have κ ≤ Y . Let s be a complex number with Re(s) = κ and | Im(s)| ≤ Y . Then, it follows from Theorem 2.2 that1 T [T,2T ]\E(T ) |ζ(σ + it)| s dt = E (|ζ(σ, X)| s ) + O E (|ζ(σ, X)| κ ) (log T ) 10 . (|ζ(σ, X)| s ) e -τ s e λs -[T,2T ]\E(T ) |ζ(σ + it)| s dt e -τ s e λs -1 λs N ds s .Then, using equation (7.1) we obtainP(log |ζ(σ, X)| > τ ) ≤ I(σ, τ ) ≤ P(log |ζ(σ, X)| > τ -λN ),(7.13)andP T log |ζ(σ+it)| > τ +O δ(T ) ≤ J T (σ, τ ) ≤ P T log |ζ(σ+it)| > τ -λN +O δ(T ) , (7.14) where δ(T ) = exp -c 0 (σ) log T log log T , for some positive constant c 0 (σ), by equation (4.9). Further, using that |e λs -1| ≤ 3 we obtain κ-iY κ-i∞ + κ+i∞ κ+iY E (|ζ(σ, X)| s ) e -τ s e λs -1 λs N ds s 3 λY N E (|ζ(σ, X)| κ ) e -τ κ . (7.15) The result follows from these estimates together with the fact that P(log |ζ(σ, X)| > τ ) (δ(T )) 1/2 in our range of τ , by Corollary 7.3. Distribution of a-points: Proof of Theorem 1.Preliminaries. Let S(T ) be the set of points T ≤ t ≤ 2T such that max log |ζ(σ + it)| , arg ζ(σ + it) < log 2 T and log |ζ(σ + it)| -log |a| > δ, where δ = 1/(log T ) σ . Similarly let F be the event, max log |ζ(σ, X)| , arg ζ(σ, X) < log 2 T and log |ζ(σ, X)| -log |a| > δ.

  T ) : log |ζ(σ + it)| ≤ u and arg ζ(σ + it) ≤ v} Φ1 (u, v) = P F 1 and log |ζ(σ, X)| ≤ u and arg ζ(σ, X) ≤ v .

1 (

 1 -|a|) 2 + 2|a|e u 1 -cos(v -arg a) dv e u log 2 T π 0 e u -|a|) 2 + 2|a|e u (1 -cos v) dv,by a simple change of variable and since the integrand is an even function of v. Furthermore, using that 1 -cos v ≥ v 2 /10 for 0 ≤ v ≤ π we obtain thatπ 0 1 (e u -|a|) 2 + 2|a|e u (1 -cos v) dv ≤ π 0 1 (e u -|a|) 2 + |a|e u v 2 /5 dv.Now, by making the change of variable -|a|) 2 + |a|e u v 2

1 1 e 8 . 2 . 4 .Finally, using a similar approach one obtains 1 T 2 T-log 2 T-log 2 TΦ 2 1 TLemma 8 . 5 .

 1182412222185 dx by making the change of variable x = u -log |a| and since the integrand is positive. The lemma follows upon noting that -x/2 dx log 2 T. Proofs of Theorems 2.4 and 1.Proof of Theorem 2.4. In view of Lemma 8.1 we only need to prove that1 T t∈S(T ) log |ζ(σ + it) -a|dt = E 1 F • log |ζ(σ, X) -a| + O (log 2 T ) 2 (log T ) σ .Recall that S(T ) = S 1 (T )∪S 2 (T ) and F = F 1 ∪F 2 . Combining the discrepancy estimate meas(S 1 (T ))T -P(F 1 ) (log T ) -σwith Lemmas 8.2, 8.3 and 8.4, we obtain1 T t∈S 1 (T ) log |ζ(σ + it) -a|dt = E 1 F 1 • log |ζ(σ, X) -a| + O (log 2 T ) 2 (log T ) σ . t∈S 2 (T ) log |ζ(σ + it) -a|dt = E 1 F 2 • log |ζ(σ, X) -a| + O (log 2 T ) 2 (log T ) σ ,where instead of Lemma 8.2 we use 1T t∈S 2 (T ) log |ζ(σ + it) -a|dt = log log |a|-δ (u, v) ∂ 2 h(u, v) ∂u∂v dudv -meas(S 2 (T )) T h(-log 2 T, -log 2 T ) + t∈S 2 (T ) h -log 2 T, arg ζ(σ + it) + h log |ζ(σ + it)|, -log 2 T dt, with Φ 2 (u, v) = 1 T meas{t ∈ S 2(T ) : log |ζ(σ + it)| ≥ u and arg ζ(σ + it) ≥ v}. For the proof of Theorem 1.4 we need an auxiliary lemma. Let a = 0. The function f a (σ) := E[log |ζ(σ, X) -a|]

T 1 2π σ 0 σ

 10 log |ζ(σ + it) -a|dt -1 2π 2T T log |ζ(σ 0 + it) -a|dt + arg ζ(α + 2iT ) -a -arg ζ(α + iT ) -a dα.

σ 0 σ

 0 arg ζ(α + 2iT ) -a -arg ζ(α + iT ) -a dα a log T. Let 0 < h < min(σ -1 2 , 1 -σ).Inserting this last estimate in equation (8.2) and using Theorem 2.4 we obtain σ+h σ βa>u T ≤γa≤2T1 du = T 2π • E[log |ζ(σ, X)-a|]-E[log |ζ(σ+h, X)-a|] +O T (log 2 T ) 2 (log T ) σ .

  Moreover, the same asymptotic formula holds when |ζ(σ + it)| z and |ζ(σ, X)| z are replaced by exp z(arg ζ(σ + it)) and exp z(arg ζ(σ, X)) respectively. When computing complex moments of ζ(σ + it) the first step is to use the classical zero density estimates to approximate log ζ(σ + it) by a short Dirichlet polynomial for all t ∈ [T, 2T ] except for a set of small measure (see Lemma 3.1 below). Let

  2 T, log 2 T ) log 2 T and h(log |a| + δ, log 2 T ) log(1/δ) log 2 T. Now, we have|h (u, log 2 T )| = | Re(g (u, log 2 T ))| ≤ |g (u, log 2 T )| ≤ e u |e u -|a|| .Further, by making the change of variable x = u -log |a|, we get Inserting this estimate in (8.1) completes the proof.Lemma 8.4. Let 1 2 < σ < 1 be fixed. We have

	log 2 T log |a|+δ	|h (u, log 2 T )|du	δ	2 log 2 T	e x e x -1	dx	δ	1	dx x	+ log 2 T	log 2 T.
	log 2 T -log 2 T	log 2 T log |a|+δ	Φ 1 (u, v)	∂ 2 h(u, v) ∂u∂v	dudv =	log 2 T -log 2 T	log 2 T log |a|+δ	Φ1 (u, v)	∂ 2 h(u, v) ∂u∂v	dudv
	+ O (log 2 T ) log 2 T -log 2 T log 2 T log |a|+δ Φ 1 (u, v) ∂ 2 h(u, v) ∂u∂v dudv = log 2 T -log 2 T log 2 T log |a|+δ Φ1 (u, v)	∂ 2 h(u, v) ∂u∂v	dudv
									+ O	1 (log T ) σ	log 2 T log |a|+δ	log 2 T -log 2 T	∂ 2 h(u, v) ∂u∂v	dvdu .
	Note that
			∂ 2 h(u, v) ∂u∂v	= Re	∂ 2 g(u, v) ∂u∂v	≤	∂ 2 g(u, v) ∂u∂v	e u |e u+iv -a| 2 ,

2

(log T ) σ .

Proof. By the discrepancy estimate Φ 1 (u, v) -Φ1 (u, v) (log T ) -σ , we obtain that

We now estimate the inner integral on the right-hand side. We have for n ≤ t ≤ n (5.7)

Next, note that the set D n consists of 1/ √ δ = O(1) disks, each of radius r. Arguing as in (5.4), we see that each one contains δ -1 log M r+δ (s 0 ) log M r+δ (s 0 ) zeros. Hence, by this, (5.5), (5.6), and (5.7) we see that

for some absolute constant C > 0.

Lemma 5.4. Let 1 2 < σ ≤ 2 be fixed. For any fixed σ 0 > 1 and R = σ 0 -σ we have

Next, let D R (z) be the disk of radius R centered at z. Also, let s n = σ n + it n be a point at which |ζ(s)| achieves its maximum value on the set ∪ n≤t≤n+1 D R (s 0 ). Thus,

Hence, we have

Proof of Proposition 2.5. First we consider the case k ≥ 1. Note that by Lemma 5.2 we have

Hence, for this case, we see that Proposition 2.5 follows from the above inequality, Lemma 5.3, and Lemma 5.5. For 0 < k < 1 the proposition follows from an application of Hölder's inequality.

6. Bounding the discrepancy: Proof of Theorem 1.1

In order to prove Theorem 1.1 we shall appeal to the following Lemma of Selberg (Lemma 4.1 of [START_REF] Tsang | The distribution of the values of the Riemann zeta-function[END_REF]), which provides a smooth approximation for the signum function. Selberg used this lemma in his proof that log ζ( 12 + it) has a limiting two-dimensional Gaussian distribution (see [START_REF] Tsang | The distribution of the values of the Riemann zeta-function[END_REF] and [START_REF] Selberg | Old and new conjectures and results about a class of Dirichlet series[END_REF]). Recall that the signum function is defined by

Lemma 6.1 (Selberg, Lemma 4.1 of [START_REF] Tsang | The distribution of the values of the Riemann zeta-function[END_REF]). Let L > 0. Define

Then for all x ∈ R we have

For any rectangle R in the complex plane, let 1 R denote its indicator function. Using Lemma 6.1 we derive a smooth approximation for 1 R which will be used to prove Theorem 1.1. For any α, β ∈ R, we define

2 .

Then, we prove Similarly, using (7.12) we get Further, note that |(e λs -1)/λs| ≤ 3, which is easily seen by looking at the cases |λs| ≤ 1 and |λs| > 1. Therefore, combining equations (7.12), (7.15) and (7.16) we obtain

(7.17) Furthermore, it follows from Proposition 7.1 and Proposition 7.5 that (7.20) Combining this last estimate with (7.13), (7.14), and (7.19) we obtain

We substitute σ -h for σ and use

We pick h = (log 2 T ) • (log T ) -σ/2 to conclude that

From this the claim follows.

Appendix: Lower bounds for the discrepancy

According to the work of Ingham [START_REF] Ingham | Mean-Value Theorems in the Theory of the Riemann Zeta-Function[END_REF],

We notice that

) for some δ > 0, then by integration by parts

which contradicts the previous equation. Therefore D σ (T ) = Ω(T 1-2σ-ε ). We notice that the term T 2-2σ arises from the χ factors in the approximate functional equation. Therefore the observed discrepancy D σ (T ) = Ω(T 1-2σ-ε ) ultimately arises because the probabilistic model ζ(σ, X) does not take into account the χ factors in the approximate functional equation (or equivalently because independence is ruined for the harmonics n it and m it with n, m close to T ).

As to the second assertion, if we have that D σ (T ) = O(T 1-2σ+ε ), then again an integration by parts shows that From this it follows that the number of zeros of ζ(s) in the region β > σ+ε is T 2-2σ+ε . This completes the proof of Proposition 1.2.