Youness Lamzouri
email: lamzouri@mathstat.yorku.ca

AND Stephen Lester
email: sjlester@gmail.com

Maksym Radziwill
email: maksym.radziwill@gmail.com

AN EFFECTIVE UNIVERSALITY THEOREM FOR THE RIEMANN ZETA FUNCTION

Introduction

In 1914 Fekete constructed a formal power series ∞ n=1 a n x n with the following universal property: For any continuous function f on [-1, 1] (with f (0) = 0) and given any ε > 0 there exists an integer N > 0 such that sup -1≤x≤1 n≤N a n x n -f (x) < ε.

In the 1970's Voronin [START_REF] Voronin | A theorem on the distribution of values of the Riemann zeta-function[END_REF] discovered the remarkable fact that the Riemann zetafunction satisfies a similar universal property. He showed that for any r < 1 4 , any non-vanishing continuous function f in |z| ≤ r, which is analytic in the interior, and for arbitrary ε > 0, there exists a T > 0 such that (1.1) max

|z|≤r ζ( 3 4 + iT + z) -f (z) < ε.
Voronin obtained a more quantitative description of this phenomena, stated below.

Voronin's universality theorem. Let 0 < r < 1 4 be a real number. Let f be a nonvanishing continuous function in |z| ≤ r, that is analytic in the interior. Then, for any ε > 0, where meas is Lebesgue's measure on R.

There are several extensions of this theorem, for example to domains more general than compact discs (such as any compact set K contained in the strip 1/2 < Re(s) < 1 and with connected complement), or to more general L-functions. For a complete history of this subject, we refer the reader to [START_REF] Matsumoto | A survey on the theory of universality for zeta and L-functions[END_REF].

The assumption that f (z) = 0 is necessary: if f were allowed to vanish then an application of Rouche's theorem would produce at least T zeros ρ = β + iγ of ζ(s) with β > 1 2 + ε and T ≤ γ ≤ 2T , contradicting the simplest zero-density theorems. Subsequent work of Bagchi [START_REF] Bagchi | Statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series[END_REF] clarified Voronin's universality theorem by setting it in the context of probability theory (see [START_REF] Kowalski | Bagchi's theorem for families of automomorphic forms[END_REF] for a streamlined proof). Viewing ζ( 34 +it+z) with t ∈ [T, 2T ] as a random variable X T in the space of random analytic functions (i.e X T (z) = ζ( 34 + iU T + z) with U T uniformly distributed in [T, 2T ]), Bagchi showed that as T → ∞ this sequence of random variables converges in law (in the space of random analytic functions) to a random Euler product,

ζ(s, X) := p 1 - X(p) p s -1
with {X(p)} p a sequence of independent random variables uniformly distributed on the unit circle (and with p running over prime numbers). This product converges almost surely for Re(s) > 1 2 and defines almost surely a holomorphic function in the half-plane Re(s) > σ 0 for any σ 0 > 1 2 (see Section 2 below). The proof of Voronin's universality is then reduced to showing that the support of ζ(s + 3/4, X) in the space of random analytic functions contains all non-vanishing analytic f : {z : |z| < r} → C\{0}. Moreover it follows from Bagchi's work that the limit in Voronin's universality theorem exists for all but at most countably many ε > 0.

In this paper, we present an alternative approach to Bagchi's result using methods from hard analysis. As a result we obtain, for the first time, a rate of convergence in Voronin's universality theorem. We also give an explicit description for the limit in terms of the random model ζ(s, X). Theorem 1.1. Let 0 < r < 1 4 . Let f be a non-vanishing continuous function on |z| ≤ (r + 1/4)/2 that is holomorphic in |z| < (r + 1/4)/2. Let ω be a real-valued continuously differentiable function with compact support. Then, we have

1 T 2T T ω max |z|≤r |ζ( 3 4 + it + z) -f (z)| dt = E ω max |z|≤r |ζ( 3 4 + z, X) -f (z)| + O (log T ) -(3/4-r) 11
+o (1) , where the constant in the O depends on f, ω and r.

If the random variable Y r,f = max |z|≤r |ζ( 3 4 + z, X) -f (z)| is absolutely continuous, then it follows from the proof of Theorem 1.1 that for any fixed ε > 0 we have

1 T • meas T ≤ t ≤ 2T : max |z|≤r ζ( 3 4 + it + z) -f (z) < ε = P max |z|≤r ζ( 3 4 + z, X) -f (z) < ε + O (log T ) -(3/4-r) 11
+o (1) .

Unfortunately, we have not been able to even show that Y r,f has no jump discontinuities. We conjecture the latter to be true, and one might even hope that Y r,f is absolutely continuous.

A slight modification of the proof of Theorem 1.1 allows for more general domains than the disc |z| ≤ r. Furthermore, if ω ≥ 1 (0,ε) (where 1 S is the indicator function of the set S), then it follows from Voronin's universality theorem that the main term in Theorem 1.1 is positive. Explicit lower bounds for the limit in (1.2) (in terms of ε) are contained in the papers of Good [START_REF] Good | On the distribution of the values of Riemann's zeta function[END_REF] and Garunkstis [START_REF] Garunkstis | The effective universality theorem for the Riemann zeta-function[END_REF].

Our approach is flexible, and can be generalized to other L-functions in the t-aspect, as well as to "natural" families of L-functions in the conductor aspect. The only analytic ingredients that are needed are zero density estimates, and bounds on the coefficients of these L-functions (the so-called Ramanujan conjecture). In particular, the techniques of this paper can be used to obtain an effective version of a recent result of Kowalski [START_REF] Kowalski | Bagchi's theorem for families of automomorphic forms[END_REF], who proved an analogue of Voronin's universality theorem for families of L-functions attached to GL 2 automorphic forms. In fact, using the zero-density estimates near 1 that are known for a very large class of L-functions (including those in the Selberg class by Kaczorowski and Perelli [START_REF] Kaczorowski | The Selberg class: a survey[END_REF], and for families of L-functions attached to GL n automorphic forms by Kowalski and Michel [START_REF] Kowalski | Zeros of families of automorphic L-functions close to 1[END_REF]), one can prove an analogue of Theorem 1.1 for these L-functions, where we replace 3/4 by some σ < 1 (and r < 1 -σ).

The main idea in the proof of Theorem 1.1 is to cover the boundary of the disc |z| ≤ r with a union of a growing (with T ) number of discs, while maintaining a global control of the size of |ζ (s + z)| on |z| ≤ r. It is enough to focus on the boundary of the disc thanks to the maximum modulus principle. The behavior of ζ(s + z) with z localized to a shrinking disc is essentially governed by the behavior at a single point z = z i in the disc. This allows us to reduce the problem to understanding the joint distribution of a growing number of shifts log ζ(s+z i ) with the z i well-spaced, which can be understood by computing the moments of these shifts and using standard Fourier techniques.

It seems very difficult to obtain a rate of convergence which is better than logarithmic in Theorem 1.1. We have at present no understanding as to what the correct rate of convergence should be.

Key Ingredients and detailed results

We first begin with stating certain important properties of the random model ζ(s, X). Let {X(p)} p be a sequence of independent random variables uniformly distributed on the unit circle. Then we have We extend the X(p) multiplicatively to all positive integers by setting X(1) = 1 and X(n

-log 1 - X(p) p s = X(p) p s + h X (p,
) := X(p 1 ) a 1 • • • X(p k ) a k , if n = p a 1 1 . . . p a k k . Then we have (2.3) E X(n)X(m) = 1 if m = n, 0 otherwise.
Furthermore, for any complex number s with Re(s) > 1/2 we have almost surely that

ζ(s, X) = ∞ n=1 X(n) n s .
To compare the distribution of ζ(s + it) to that of ζ(s, X), we define a probability measure on [T, 2T ] in a standard way, by

P T (S) := 1 T meas(S), for any S ⊆ [T, 2T ].
The idea behind our proof of effective universality is to first reduce the problem to the discrete problem of controlling the distribution of many shifts log ζ(s j + it) with all of the s j contained in a compact set inside the strip 1 2 < Re(s) < 1. One of the main ingredients in this reduction is the following result which allows us to control the maximum of the derivative of the Riemann zeta-function. This is proven in Section 4.

Proposition 2.1. Let 0 < r < 1/4 be fixed. Then there exist positive constants b 1 , b 2 and b 3 (that depend only on r) such that

P T max |z|≤r |ζ ( 3 4 + it + z)| > e V exp -b 1 V 1 1-σ(r) (log V ) σ(r) 1-σ(r)
where σ(r) = 3 4 -r, uniformly for V in the range b 2 < V ≤ b 3 (log T ) 1-σ /(log log T ). We also prove an analogous result for the random model ζ(s, X), which holds for all sufficiently large V . Proposition 2.2. Let 0 < r < 1/4 be fixed and σ(r) = 3 4 -r. Then there exist positive constants b 1 and b 2 (that depend only on r) such that for all V > b 2 we have

P max |z|≤r |ζ ( 3 4 + z, X)| > e V exp -b 1 V 1 1-σ(r) (log V ) σ(r) 1-σ(r)
.

Once the reduction has been accomplished, it remains to understand the joint distribution of the shifts {log ζ(s 1 + it), log ζ(s 2 + it), . . . , log ζ(s J + it)} with J → ∞ as T → ∞ at a certain rate, and s 1 , . . . , s J are complex numbers with 1 2 < Re(s j ) < 1 for all j ≤ J. Heuristically, this should be well approximated by the the joint distribution of the random variables {log ζ(s 1 , X), log ζ(s 2 , X), . . . , log ζ(s J , X)}. In order to establish this fact (in a certain range of J), we first prove, in Section 5, that the moments of the joint shifts log ζ(s j + it) are very close to the corresponding ones of log ζ(s j , X), for j ≤ J.

Theorem 2.1. Fix 1/2 < σ 0 < 1. Let s 1 , s 2 , . . . , s k , r 1 , . . . , r be complex numbers in the rectangle σ 0 < Re(z) < 1 and |Im(z)| ≤ T (σ 0 -1/2)/4 . Then, there exist positive constants c 3 , c 4 , c 5 and a set

E(T ) ⊂ [T, 2T ] of measure T 1-c 3 , such that if k, ≤ c 4 log T / log log T then 1 T [T,2T ]\E(T ) k j=1 log ζ(s j + it) j=1 log ζ(r j -it) dt = E k j=1 log ζ(s j , X) j=1 log ζ(r j , X) + O T -c 5 .
Having obtained the moments we are in position to understand the characteristic function,

Φ T (u, v) := 1 T 2T T exp i J j=1 (u j Re log ζ(s j + it) + v j Im log ζ(s j + it)) dt,
where u = (u 1 , . . . , u J ) ∈ R J and v = (v 1 , . . . , v J ) ∈ R J . We relate the above characteristic function to the characteristic function of the probabilistic model,

Φ rand (u, v) := E exp i J j=1 (u j Re log ζ(s j , X) + v j Im log ζ(s j , X)) .
This is obtained in the following theorem, which we prove in Section 6.

Theorem 2.2. Fix 1/2 < σ < 1. Let T be large and J ≤ (log T ) σ be a positive integer. Let s 1 , s 2 , . . . , s J be complex numbers such that min(Re(s j )) = σ and max(|Im(s j )|) < T (σ-1/2)/4 . Then, there exist positive constants

c 1 (σ), c 2 (σ), such that for all u, v ∈ R J such that max(|u j |), max(|v j |) ≤ c 1 (σ)(log T ) σ /J we have Φ T (u, v) = Φ rand (u, v) + O exp -c 2 (σ) log T log log T .
Using this result, we can show that the joint distribution of the shifts log ζ(s j +it) is very close to the corresponding joint distribution of the random variables log ζ(s j , X). The proof depends on Beurling-Selberg functions. To measure how close are these distributions, we introduce the discrepancy D T (s 1 , . . . , s J ) defined as

sup (R 1 ,...,R J )⊂C J P T log ζ(s j + it) ∈ R j , ∀j ≤ J -P log ζ(s j , X) ∈ R j , ∀j ≤ J
where the supremum is taken over all (R 1 , . . . , R J ) ⊂ C J and for each j = 1, . . . , J the set R j is a rectangle with sides parallel to the coordinate axes. Our next theorem, proven in Section 7, states a bound for the above discrepancy. This generalizes Theorem 1.1 of [START_REF] Lamzouri | Discrepancy bounds for the distribution of the Riemann zeta-function and applications[END_REF], which corresponds to the special case J = 1.

Theorem 2.3. Let T be large, 1 2 < σ < 1 and J ≤ (log T ) σ/2 be a positive integer. Let s 1 , s 2 , . . . , s J be complex numbers such that

1 2 < σ := min j (Re(s j )) ≤ max j (Re(s j )) < 1 and max j (| Im(s j )|) < T (σ-1 2 )/4 .
Then, we have

D T (s 1 , . . . , s J ) J 2 (log T ) σ .
With all of the above tools in place we are ready to prove Theorem 1.1. This is accomplished in the next section.

Effective universality: Proof of Theorem 1.1

In this section, we will prove Theorem 1.1 using the results described in Section 2. First, by the maximum modulus principle, the maximum of |ζ( 34 + it + z) -f (z)| in the disc {z : |z| ≤ r} must occur on its boundary {z : |z| = r}. Our idea consists of first covering the circle |z| = r with J discs of radius ε and centres z j , where z j ∈ {z : |z| = r} for all 1 ≤ j ≤ J, and J 1/ε. We call each of the discs D j . Then, we observe that (3.1) max

j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ max |z|≤r |ζ( 3 4 + it + z) -f (z)| ≤ max j≤J max z∈D j |ζ( 3 4 + it + z) -f (z)|.
Using Proposition 2.1, we shall prove that for all j ≤ J (where J is a small power of log T ) we have max

z∈D j |ζ( 3 4 + it + z) -f (z)| ≈ |ζ( 3 4 + it + z j ) -f (z j )|
for all t ∈ [T, 2T ] except for a set of points t of very small measure. We will then deduce that the (weighted) distribution of max

|z|≤r |ζ( 3 4 + it + z) -f (z)| is very close to the corresponding distribution of max j≤J |ζ( 3 4 + it + z j ) -f (z j )|, for t ∈ [T, 2T
]. We will also establish an analogous result for the random model ζ(s, X) along the same lines, by using Proposition 2.2 instead of Proposition 2.1. Therefore, to complete the proof of Theorem 1.1 we need to compare the distributions of max j≤J |ζ( 34

+ it + z j ) -f (z j )| and max j≤J |ζ( 3 4 + z j , X) -f (z j )|.
Using Theorem 2.3 we prove Proposition 3.1. Let T be large, 0 < r < 1/4 and J ≤ (log T ) (3/4-r)/7 be a positive integer. Let z 1 , . . . , z J be complex numbers such that |z j | ≤ r. Then we have

P T max j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ u -P max j≤J |ζ( 3 4 + z j , X) -f (z j )| ≤ u u (J log log T ) 6/5 (log T ) (3/4-r)/5 .
Proof. Fix a positive real number u. Let A J (T ) be the set of those t for which | arg ζ( 3 4 + it + z j )| ≤ log log T for every j ≤ J. Since Re( 3 4 + it + z j ) ≥ 3 4 -r and Im( 3 4 + it + z j ) = t + O(1), then it follows from Theorem 1.1 and Remark 1 of [START_REF] Lamzouri | On the distribution of extreme values of zeta and L-functions in the strip 1/2 < σ < 1[END_REF] that for each j ≤ J we have

(3.2) P T | arg ζ( 3 4 + it + z j )| ≥ log log T exp -(log log T ) ( 1 1 (log T ) 4 .
Therefore, we obtain

P T ([T, 2T ] \ A J (T )) ≤ J j=1 P T | arg ζ( 3 4 + it + z j )| ≥ log log T J (log T ) 4 1 (log T ) 2 ,
and this implies that (3.3)

P T max j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ u = P T max j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ u , t ∈ A J (T ) + O 1 (log T ) 2 .
For each j ≤ J consider the region

U j = {z : |e z -f (z j )| ≤ u , |Im(z)| ≤ log log T } .
We cover U j with K area(U j )/ε 2 log log T /ε 2 squares R j,k with sides of length ε = ε(T ), where ε is a small positive parameter to be chosen later. Let K j denote the set of k ∈ {1, 2, . . . , K} such that the intersection of R j,k with the boundary of U j is empty and write K c j for the relative complement of K j with respect to {1, 2, . . . , K}.

Note that |K c j | log log T /ε. By construction,   k∈K j R j,k   ⊂ U j ⊂ k≤K R j,k .
Therefore (3.3) can be expressed as

P T ∀j ≤ J, ∀k ≤ K : log ζ( 3 4 + it + z j ) ∈ R j,k + E 1
where by Theorem 2.3

E 1 j≤J k∈K c j P T log ζ( 3 4 + it + z j ) ∈ R j,k j≤J k∈K c j P T log ζ( 3 4 + z j , X) ∈ R j,k + 1 (log T ) 3/4-r J • log log T ε ε 2 + 1 (log T ) 3/4-r , (3.4)
and in the last step we used the fact that log ζ(s, X) is an absolutely continuous random variable (see for example Jessen and Wintner [START_REF] Jessen | Distribution functions and the Riemann zeta function[END_REF]). We conclude that (3.5)

P T max j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ u = P T ∀j ≤ J, ∀k ≤ K : log ζ( 3 4 + it + z j ) ∈ R j,k + O εJ log log T + J log log T ε(log T ) 3/4-r .
Additionally, it follows from Theorem 2.3 that the main term of this last estimate equals

(3.6) P ∀j ≤ J, ∀k ≤ K : log ζ( 3 4 + z j , X) ∈ R j,k + O J 2 (log log T ) 2 ε 4 (log T ) 3/4-r .
We now repeat the exact same argument but for the random model ζ(s, X) instead of the zeta function. In particular, instead of (3.2) we shall use that

P | arg ζ( 3 4 + z j , X)| ≥ log log T exp -(log log T ) ( 1 4 +r) -1 1 (log T ) 4 ,
which follows from Theorem 1.9 of [START_REF] Lamzouri | On the distribution of extreme values of zeta and L-functions in the strip 1/2 < σ < 1[END_REF]. Thus, similarly to (3.5) we obtain

P ∀j ≤ J, ∀k ≤ K : log ζ( 3 4 + z j , X) ∈ R j,k = P max j≤J |ζ( 3 4 + z j , X) -f (z j )| ≤ u + O εJ log log T + J log log T ε(log T ) 3/4-r .
Combining the above estimate with (3.5) and (3.6) we conclude that

P T max j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ u = P max j≤J |ζ( 3 4 + z j , X) -f (z j )| ≤ u + O J 2 (log log T ) 2 ε 4 (log T ) 3/4-r + εJ log log T .
Finally, choosing

ε = J log log T (log T ) 3/4-r 1/5
completes the proof.

Proof of Theorem 1.1. We wish to estimate

(3.7) 1 T 2T T ω max |z|≤r |ζ( 3 4 + it + z) -f (z)| dt
with f an analytic non-vanishing function, and where ω is a continuously differentiable function with compact support.

Recall that the maximum of |ζ( 3 4 +it+z)-f (z)| on the disc {z : |z| ≤ r} must occur on its boundary {z : |z| = r}, by the maximum modulus principle. Let ε ≤ (1/4 -r)/4 be a small positive parameter to be chosen later, and cover the circle |z| = r with J 1/ε discs D j of radius ε and centres z j , where z j ∈ {z : |z| = r} for all j ≤ J.

Let S V (T ) denote the set of those t ∈ [T, 2T ] such that max

|z|≤(r+1/4)/2 |ζ ( 3 4 + it + z)| ≤ e V
where V ≤ log log T is a large parameter to be chosen later, and let L := max |z|≤(r+1/4)/2 |f (z)|. Then for t ∈ S V (T ), and for all z ∈ D j we have (3.8)

ζ( 3 4 + it + z) -f (z) -ζ( 3 4 + it + z j ) -f (z j ) = z z j ζ ( 3 4 + it + s) -f (s)ds ≤ |z -z j | • max |z|≤(r+1/4)/2 |ζ ( 3 4 + it + z)| + L ≤ ε(e V + L) ≤ Cεe V ,
for some large absolute constant C, depending at most on L. Define

θ(t) := max |z|≤r |ζ( 3 4 + it + z) -f (z)| -max j≤J |ζ( 3 4 + it + z j ) -f (z j )|.
Then, it follows from (3.1) and (3.8) that for all t ∈ S V (T ) we have

(3.9) 0 ≤ θ(t) ≤ Cεe V .
Therefore, using this estimate together with Proposition 2.1 and the fact that ω is bounded, we deduce that (3.7) equals

(3.10) 1 T t∈S V (T ) ω max j≤J |ζ( 3 4 + it + z j ) -f (z j )| + θ(t) dt + O e -V 2 = 1 T t∈S V (T ) ω max j≤J |ζ( 3 4 + it + z j ) -f (z j )| dt + O |E 2 | + e -V 2 = 1 T 2T T ω max j≤J |ζ( 3 4 + it + z j ) -f (z j )| dt + O |E 2 | + e -V 2 ,
where

E 2 = 1 T t∈S V (T ) θ(t) 0 ω max j≤J |ζ( 3 4 + it + z j ) -f (z j )| + x dx • dt εe V ,
using the fact that ω is bounded on R together with (3.9). Furthermore, observe that

(3.11) 1 T 2T T ω max j≤J |ζ( 3 4 + it + z j ) -f (z j )| dt = - 1 T 2T T ∞ max j≤J |ζ( 3 4 +it+z j )-f (z j )| ω (u)du • dt = - ∞ 0 ω (u) • P T max j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ u du.
Since ω has a compact support, then ω (u) = 0 if u > A for some positive constant A. Furthermore, it follows from Proposition 3.1 that for all 0 ≤ u ≤ A we have

P T max j≤J |ζ( 3 4 + it + z j ) -f (z j )| ≤ u = P max j≤J |ζ( 3 4 + z j , X) -f (z j )| ≤ u + O (J log log T ) 6/5 (log T ) (3/4-r)/5 .
Inserting this estimate in (3.11) gives that (3.12) 1

T 2T T ω max j≤J |ζ( 3 4 + it + z j ) -f (z j )| dt = - ∞ 0 ω (u) • P max j≤J |ζ( 3 4 + z j , X) -f (z j )| ≤ u du + O (J log log T ) 6/5 (log T ) (3/4-r)/5 = E ω max j≤J |ζ( 3 4 + z j , X) -f (z j )| + O (J log log T ) 6/5 (log T ) (3/4-r)/5 .
To finish the proof, we shall appeal to the same argument used to establish (3.10), in order to compare the (weighted) distributions of max j≤J |ζ( 3 4 + z j , X) -f (z j )| and max |z|≤r |ζ( 3 4 + z, X) -f (z)|. Let S V (X) denote the event corresponding to max

|z|≤(r+1/4)/2 |ζ ( 3 4 + z, X)| ≤ e V ,
and let S c V (X) be its complement. Then, it follows from Proposition 2.2 that P S c V (X) exp(-V 2 ). Moreover, similarly to (3.8) one can see that for all outcomes in S V (X) we have, for all z ∈ D j

ζ( 3 4 + z, X) -f (z) -ζ( 3 4 + z j , X) -f (z j ) = z z j ζ ( 3 4 + s, X) -f (s)ds εe V .
Thus, since the maximum of |ζ( 3 4 + z, X) -f (z)| for |z| ≤ r occurs (almost surely) on the boundary |z| = r, then following the argument leading to (3.10), we conclude that

E ω max |z|≤r |ζ( 3 4 + z, X) -f (z)| = E 1 S V (X) ω max |z|≤r |ζ( 3 4 + z, X) -f (z)| + O e -V 2 = E 1 S V (X) ω max j≤J |ζ( 3 4 + z j , X) -f (z)| + O εe V + e -V 2 = E ω max j≤J |ζ( 3 4 + z j , X) -f (z)| + O εe V + e -V 2 .
Finally, combining this estimate with (3.10) and (3.12), and noting that J 1/ε we deduce that 5 .

1 T 2T T ω max |z|≤r |ζ( 3 4 + it + z) -f (z)| dt = E ω max |z|≤r |ζ( 3 4 + z, X) -f (z)| + O εe V + e -V 2 + O (log log T ) 6/5 ε 6/5 (log T ) (3/4-r)/
Choosing ε = (log T ) -(3/4-
Λ(n) n 3/4+it+s log n 2k dt 1 δ |z|=r+2δ 1 T 2T T n≤y Λ(n) n 3/4+it+z log n 2k dt|dz| e V /2 c 8 (r) k 1-σ (r) (log k) σ (r) 2k (4.4)
where σ (r) = 3 4 -r -2δ, and k ≤ c 9 log T / log log T , for some sufficiently small constant c 9 > 0. We now choose k

= c 6 (r)V 1 (1-σ(r)) (log V ) σ(r) 1-σ(r) (so that k σ (r)
k σ(r) ) where c 6 (r) is a sufficiently small absolute constant. Using (4.2) and (4.3) along with Chebyshev's inequality and the above estimate we conclude that there exists c 7 (r) > 0 such that

P T max |z|≤r |ζ ( 3 4 + it + z)| > e V P T max |s|≤r+δ n≤y Λ(n) n 3/4+it+s log n > V 4 + T 1-(1/4-r)/5 e V /2 4 V • c 8 (r) k 1-σ (r) (log k) σ (r) 2k + T 1-(1/4-r)/5 exp -c 6 V 1 1-σ(r) (log V ) σ(r) 1-σ(r) for V ≤ c 7 (log T ) 1-σ(r) / log log T .
We now prove Proposition 2.2 along the same lines. The proof is in fact easier than in the zeta function case, since we can compute the moments of log ζ(s, X), for any s with Re(s) > 1/2. r) , where c 6 is sufficiently small (note that k σ (r) k σ(r) ), then apply (4.6) to complete the proof.

P max |s|≤r+δ | log ζ( 3 4 + s, X)| > V /2 ≤ 2 V 2k • E max |s|≤r+δ |ζ( 3 4 + s, X)| 2k 2 V 2k e V /2 |z|=r+2δ E | log ζ( 3 4 + z, X)| 2k • |dz| e V /2 2c 8 (r)k 1-σ (r) V (log k) σ (r) 2k , where σ (r) = 3 4 -r -2δ. Let σ(r) = 3 4 -r and take k = c 6 V 1 1-σ(r) (log V ) σ(r) 1-σ(

Moments of joint shifts of log ζ(s): Proof of Theorem 2.1

The proof of Theorem 2.1 splits into two parts. In the first part we derive an approximation to k j=1 log ζ(s j + it) by a short Dirichlet polynomial. In the second part we compute the resulting meanvalues and obtain Theorem 2.1.

Approximating k

j=1 log ζ(s j +it) by short Dirichlet polynomials. Fix 1/2 < σ 0 < 1, and let δ := σ 0 -1/2. Let k ≤ log T be a positive integer and s 1 , s 2 , . . . , s k be complex numbers (not necessarily distinct) in the rectangle {z : σ 0 ≤ Re(z) < 1, and |Im(z)| ≤ T δ/4 }. We let s = (s 1 , . . . , s k ), and define

F s (n) = n 1 ,n 2 ,...,n k ≥2 n 1 n 2 •••n k =n k =1 Λ(n ) n s log(n )
.

Then for all complex numbers z with Re(z) > 1 -σ 0 we have

k =1 log ζ(s + z) = ∞ n=1 F s (n) n z .
The main result of this subsection is the following proposition.

Proposition 5.1. Let T be large, s 1 , . . . , s k be as above, and E(T ) be as in Lemma 5.3 below. Then, there exist positive constants

a(σ 0 ), b(σ 0 ) such that if k ≤ a(σ 0 )(log T )/ log log T and t ∈ [T, 2T ] \ E(T ) then k j=1 log ζ(s j + it) = n≤T δ/8 F s (n) n it + O T -b(σ 0 ) .
This depends on a sequence of fairly standard lemmas which we now describe.

Lemma 5.1. With the same notation as above, we have

|F s (n)| ≤ (2 log n) k n σ 0 . Proof. We have |F s (n)| ≤ 1 n σ 0 (log 2) k n 1 ,n 2 ,...,n k ≥2 n 1 n 2 •••n k =n k =1 Λ(n ) ≤ 2 k n σ 0   m|n Λ(m)   k ≤ (2 log n) k n σ 0 .
Lemma 5.2. Let y ≥ 2 and |t| ≥ y + 3 be real numbers. Suppose that the rectangle {z :

σ 0 -δ/2 < Re(z) ≤ 1, |Im(z) -t| ≤ y + 2} is free of zeros of ζ(z).
Then, for all complex numbers s such that Re(s) ≥ σ 0 -δ/4 and |Im(s)| ≤ y we have

log ζ(s + it) σ 0 log |t|.
Proof. This follows from Theorem 9.6 B of Titchmarsh.

Lemma 5.3. Let s 1 , . . . , s k be as above. Then, there exists a set E(T ) ⊂ [T, 2T ] with measure meas(E(T )) T 1-δ/8 , and such that for all t ∈ [T, 2T ] \ E(T ) we have ζ(s j + it + z) = 0 for every 1 ≤ j ≤ k and every z in the rectangle {z :

-δ/2 < Re(z) ≤ 1, |Im(z)| ≤ 3T δ/4 }.
Proof. For every 1 ≤ j ≤ k, let E j (T ) be the set of t ∈ [T, 2T ] such that the rectangle {z : -δ/2 < Re(z) ≤ 1, |Im(z)| ≤ 3T δ/4 } has a zero of ζ(s j + it + z). Then, by the classical zero density estimate N (σ, T ) T 3/2-σ (log T ) 5 , we deduce that meas(E j (T )) T δ/4 T 3/2-σ 0 +δ/2 (log T ) 5 < T 1-δ/4 (log T ) 5 .

We take E(T ) = ∪ k j=1 E j (T ). Then E(T ) satisfies the assumptions of the lemma, since meas(E(T ))

T 1-δ/4 (log T ) 6 T 1-δ/8 .

We are now ready to prove Proposition 5.1.

Proof of Proposition 5.1.

Let x = T δ/8 +1/2. Let c = 1-σ 0 +1/ log T , and Y = T δ/4 .
Then by Perron's formula, we have for t

∈ [T, 2T ] \ E(T ) 1 2πi c+iY c-iY k j=1 log ζ(s j + it + z) x z z dz = n≤x F s (n) n it + O x c Y ∞ n=1 |F s (n)| n c | log(x/n)| .
To bound the error term of this last estimate, we split the sum into three parts: n ≤ x/2, x/2 < n < 2x and n ≥ 2x. The terms in the first and third parts satisfy | log(x/n)| ≥ log 2, and hence their contribution is

x 1-σ 0 Y ∞ n=1 |F s (n)| n c ≤ x 1-σ 0 Y ∞ n=1 Λ(n) n σ 0 +c log n k ≤ x 1-σ 0 (2 log T ) k Y T -b(σ 0 ) ,
or some positive constant b(σ 0 ), if a(σ 0 ) is sufficiently small. To handle the contribution of the terms x/2 < n < 2x, we put r = x -n, and use that | log(x/n)| |r|/x. Then by Lemma 5.1 we deduce that the contribution of these terms is

x 1-σ 0 (3 log x) k Y r≤x 1 r x 1-σ 0 (3 log x) k+1 Y T -b(σ 0 ) .
We now move the contour to the line Re(s) = -δ/4. By Lemma 5.3, we do not encounter any zeros of ζ(s j + it + z) since t ∈ [T, 2T ] \ E(T ). We pick up a simple pole at z = 0 which leaves a residue k j=1 log ζ(s j + it). Also Lemma 5.2 implies that for any z on our contour we have

| log ζ(s j + it + z)| ≤ c(σ 0 ) log T,
for all j where c(σ 0 ) is a positive constant. Therefore, we deduce that

1 2πi c+iY c-iY k j=1 log ζ(s j + it + z) x z z dz = k j=1 log ζ(s j + it) + E 1 ,
where

E 1 = 1 2πi -δ/4-iY c-iY + -δ/4+iY -δ/4-iY + c+iY -δ/4+iY k j=1 log ζ(s j + it + z) x z z dz x 1-σ 0 (c(σ 0 ) log T ) k Y + x -δ/4 (c(σ 0 ) log T ) k log Y T -b(σ 0 ) ,
as desired.

An Asymptotic formula for the moment of products of shifts of log ζ(s).

Proof of Theorem 2.1. Let E 1 (T ) and E 2 (T ) be the corresponding exceptional sets for s and r respectively as in Lemma 5.3, and let E(T ) = E 1 (T ) ∪ E 2 (T ). First, note that if t ∈ [T, 2T ] \ E(T ) then by Proposition 5.1 and Lemma 5.2 we have n≤x

F s (n) n it (c(σ 0 ) log T ) k , and m≤x F r (m) m -it (c(σ 0 )) log T ) ,
for some positive constant c(σ 0 ). Let x = T (σ 0 -1/2)/8 . Then, it follows from Proposition 5.1 that (5.1) 1

T [T,2T ]\E(T ) k j=1 log ζ(s j + it) j=1 log ζ(r j -it) dt = 1 T [T,2T ]\E(T ) n≤x F s (n) n it m≤x F r (m)m it dt dt + O T -b(σ 0 ) (log T ) max(k, ) = 1 T 2T T n≤x F s (n) n it m≤x F r (m)m it dt + O T -b(σ 0 )/2 .
Furthermore, we have

(5.2) 1 T 2T T n≤x F s (n) n it m≤x F r (n)m it dt = m,n≤x F s (n)F r (m) 1 T 2T T m n it dt.
The contribution of the diagonal terms n = m equals n≤x F s (n)F r (n). On the other hand, by Lemma 5.1 the contribution of the off-diagonal terms n = m is

(5.3) 1 T m,n≤x m =n (2 log n) k (2 log m) (mn) σ 0 1 | log(m/n)| x 3-2σ 0 (2 log x) k+ T T -1/2 , since | log(m/n)| 1/
x. Furthermore, it follows from (2.3) that (5.4)

E k j=1 log ζ(s j , X) j=1 log ζ(r j , X) = ∞ n=1 F s (n)F r (n) = n≤x F s (n)F r (n) + E 2 ,
where

E 2 ≤ n>x (2 log n) k+ n 2σ 0 .
Since the function (log t) β /t α is decreasing for t > exp(β/α), then with the choice α = (2σ 0 -1)/2 we obtain

E 2 ≤ (2 log x) k+ x α n>x 1 n 1+α (2 log x) k+ x 2α x -α .
Combining this with (5.2), (5.3), and (5.4) completes the proof. where C is a suitably large constant. Then, Φ T (u, v) equals

1 T [T,2T ]\E(T ) exp i J j=1 (u j Re log ζ(s j + it) + v j Im log ζ(s j + it)) dt + O T -c 3 = 2N -1 n=0 i n n! • 1 T [T,2T ]\E(T ) J j=1 (u j Re log ζ(s j + it) + v j Im log ζ(s j + it)) n dt + E 3 , (6.1)
where

E 3 T -c 3 + 1 (2N )! 2c 1 (log T ) σ J 2N 1 T [T,2T ]\E(T ) J j=1 | log ζ(s j + it)| 2N dt.
Now, by Theorem 2.1 along with Lemma 4.3, we obtain that for all 1 ≤ j ≤ J

(6.2) 1 T [T,2T ]\E(T ) | log ζ(s j + it)| 2N dt E | log ζ(s j , X)| 2N ≤ c 8 (σ)N 1-σ (log N ) σ 2N ,
for some positive constant c 8 = c 8 (σ). Furthermore, by Minkowski's inequality we have

1 T [T,2T ]\E(T ) J j=1 | log ζ(s j + it)| 2N dt ≤ c 8 J N 1-σ (log N ) σ 2N .
Therefore, we deduce that for some positive constant c 9 = c 9 (σ), we have

E 3 T -c 3 + c 9 (log T ) σ (N log N ) σ 2N e -N .
Next, we handle the main term of (6.1). Let ũj = (u j + iv j )/2 and ṽj = (u j -iv j )/2. Then by Theorem 2.1 we obtain 1

T [T,2T ]\E(T ) J j=1 (u j Re log ζ(s j + it) + v j Im log ζ(s j + it)) n dt 1 T [T,2T ]\E(T ) J j=1 ( ũj log ζ(s j + it) + ṽj log ζ(s j -it)) n dt = k 1 ,...,k 2J ≥0 k 1 +•••+k 2J =n n k 1 , k 2 , . . . , k 2J J j=1 ũj k j J =1 ṽ k J+ × 1 T [T,2T ]\E(T ) J j=1 (log ζ(s j + it)) k j J =1 (log ζ(s -it)) k J+ dt = k 1 ,...,k 2J ≥0 k 1 +•••+k 2J =n n k 1 , k 2 , . . . , k 2J J j=1 ũj k j J =1 ṽ k J+ × E J j=1 (log ζ(s j , X)) k j J =1 (log ζ(s , X)) k J+ + O T -c 5 2c 1 (log T ) σ n , = E J j=1 (u j Re log ζ(s j , X) + v j Im log ζ(s j , X)) n + O T -c 5 2c 1 (log T ) σ n .
Inserting this estimate in (6.1), we derive

Φ T (u, v) = 2N -1 n=0 i n n! E J j=1 (u j Re log ζ(s j , X) + v j Im log ζ(s j , X)) n + O e -N = Φ rand (u, v) + E 4 .
where

E 4 e -N + 1 (2N )! 2c 1 (log T ) σ J 2N E   J j=1 | log ζ(s j , X)| 2N   e -N
by (6.2) and Minkowski's inequality. This completes the proof.

Discrepancy estimates for the distribution of shifts

The deduction of Theorem 2.3 from Theorem 2.2 uses Beurling-Selberg functions. For z ∈ C let

H(z) = sin πz π 2 ∞ n=-∞ sgn(n) (z -n) 2 + 2 z and K(z) = sin πz πz 2 .
Beurling proved that the function B + (x) = H(x) + K(x) majorizes sgn(x) and its Fourier transform has restricted support in (-1, 1). Similarly, the function B -(x) = H(x) -K(x) minorizes sgn(x) and its Fourier transform has the same property (see Vaaler [START_REF] Vaaler | Some extremal functions in Fourier analysis[END_REF] Lemma 5). Let ∆ > 0 and a, b be real numbers with a < b. Take I = [a, b] and define

F I (z) = 1 2 B -(∆(z -a)) + B -(∆(b -z)) .
The function F I has the following remarkable properties. First, it follows from the inequality

B -(x) ≤ sgn(x) ≤ B + (x) that (7.1) 0 ≤ 1 I (x) -F I (x) ≤ K(∆(x -a)) + K(∆(b -x)).
Additionally, one has

(7.2) F I (ξ) = 1 I (ξ) + O 1 ∆ if |ξ| < ∆, 0 if |ξ| ≥ ∆.
The first estimate above follows from (7.1) and the second follows from the fact that the Fourier transform of B -is supported in (-1, 1). Before proving Theorem 2.3 we first require the following lemmas. Proof. It suffices to prove the lemma for ∆ = 1. Also, note that we only need to show that F I (x) ≥ -1. From the identity

∞ n=-∞ 1 (n -z) 2 = π sin πz 2 it follows that for y ≥ 0 (7.3) H(y) = 1 -K(y)G(y)
where

G(y) = 2y 2 ∞ m=0 1 (y + m) 2 -2y -1.
In Lemma 5 of [START_REF] Vaaler | Some extremal functions in Fourier analysis[END_REF], Vaaler shows for y ≥ 0 that (7.4) 0 ≤ G(y) ≤ 1.

Also, note that for each m ≥ 1, and 0 < y ≤ 1 one has m

(y+m) 3 ≤ m m-1 t (y+t) 3 dt so that for 0 < y ≤ 1 (7.5) G (y) = 4y m≥1 m (y + m) 3 -2 ≤ 4y ∞ 0 t (y + t) 3 dt -2 = 0.
First consider the case a ≤ x ≤ b. By (7.3) we get that in this range

F I (x) = 1 2 (2 -K(x -a)(G(x -a) + 1) -K(b -x)(G(b -x) + 1)) ,
which along with (7.4) implies F I (x) ≥ -1 for a ≤ x ≤ b. Now consider the case x < a. Since H is an odd function (7.3) and (7.4) imply

F I (x) = 1 2 (K(x -a)(G(a -x) -1) -K(b -x)(G(b -x) + 1)) ≥ 1 2 (-K(x -a) -2K(x -b)) , which is ≥ -1 if K(x -b) ≤ 1/2. If K(x -b) ≥ 1/2 we also have K(x -a) > K(x -b)
and 0 < b -x < 1. By this and (7.5) we have in this range as well that

F I (x) ≥ 1 2 (K(x -b)(G(a -x) -G(b -x) -2)) ≥ -1.
Hence, F I (x) ≥ -1 for x < a. The remaining case when x > b follows from a similar argument.

Lemma 7.2. Fix 1/2 < σ < 1, and let s be a complex number such that Re(s) = σ and | Im(s)| ≤ T Using the first bound and applying Theorem 2.2 with J = 1 establishes the first claim.

The second claim follows similarly by using the second bound and Theorem 2.2.

Proof of Theorem 2.3. First, we claim that it suffices to estimate the discrepancy over (R 1 , . . . , R J ) such that for each j we have where the last bound follows from Theorem 1.1 and Remark 1 of of [START_REF] Lamzouri | On the distribution of extreme values of zeta and L-functions in the strip 1/2 < σ < 1[END_REF]. Repeating this argument gives P T log ζ(s j +it) ∈ R j , ∀j ≤ J)-P T log ζ(s j +it) ∈ R j , ∀j ≤ J J exp -log T .

R j ⊂ [- √ log T , √ log T ]×[- √ log T , √ log
Similarly, P log ζ(s j , X) ∈ R j , ∀j ≤ J)-P log ζ(s j , X) ∈ R j , ∀j ≤ J J exp -log T .

Hence, the error from restricting to ( R 1 , . . . , R J ) is negligible and establishes the claim. Using (7.8) and (7.9) in (7.6) completes the proof.

1 T( 3 4 +

 14 • meas T ≤ t ≤ 2T : max |z|≤r ζit + z) -f (z) < ε > 0, 2010 Mathematics Subject Classification. Primary 11M06. The first and third authors are partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.

6 .

 6 The characteristic function of joint shifts of log ζ(s) Proof of Theorem 2.2. Let E(T ) be as in Theorem 2.1. Let N = [log T /(C(log log T ))]

Lemma 7 . 1 .

 71 For x ∈ R we have |F I (x)| ≤ 1.

1 4 •

 4 (σ-12 ) . Then there exists a positive constant c 1 (σ) such that for |u| ≤ c 1 (σ)(log T ) σ we haveΦ T (u, 0) exp -u 5 log u and Φ T (0, u) exp -u 5 log u .Proof. By a straightforward modification of Lemma 6.3 of[START_REF] Lamzouri | Discrepancy bounds for the distribution of the Riemann zeta-function and applications[END_REF] one has that E exp iu Re log ζ(s, X) exp -u 5 log u , and E exp iu Im log ζ(s, X) exp -u 5 log u .

  T ]. To see this consider ( R 1 , . . . , R J ) where R j = R j ∩[-√ log T , √ log T ]×[-√ log T , √ log T ]. It follows that P T log ζ(s j + it) ∈ R j , ∀j ≤ J -P T log ζ(s 1 + it) ∈ R 1 , log ζ(s j + it) ∈ R j , 2 ≤ j ≤ J P T | log ζ(s 1 + it)| ≥ log T exp -log T ,

Let ∆ = c 1 FFFF( 7 . 6 ) 1 4F I 1 2T T 1 I 1 F 1 TT 1 I 1 FFF

 176112T11111 (σ)(log T ) σ /J and R j = [a j , b j ] × [c j , d j ] for j = 1, . . . , J, with |b ja j |, |d j -c j | ≤ 2 √ log T . Also, write I j = [a j , b j ] and J j = [c j , d j ]. By Fourier inversion, (7.2), and Theorem 2.2I j Re log ζ(s j + it) F J j Im log ζ(s j + it) dt = R 2J J j=1 I j (u j ) F J j (v j ) Φ T (u, v) du dv = |u j |,|v j |≤∆ j=1,2,...,J J j=1 I j (u j ) F J j (v j ) Φ rand (u, v) du dv + O 2∆ log T I j Re log ζ(s j , X) F J j Im log ζ(s j , X) + O exp -c 2 log T 2 log log T .Next note that K(ξ) = max(0, 1 -|ξ|). Applying Fourier inversion, Theorem 2.2 with J = 1, and Lemma 7.2 we have that1 T 2T T K ∆ • Re log ζ(s + it) -α dt = 1 Φ T (ξ, 0) dξ 1 ∆ ,where α is an arbitrary real number and s ∈ C satisfies σ ≤ Re(s) < 1 and| Im(s)| < T Re log ζ(s 1 + it) dt = 1 T Re log ζ(s 1 + it) dt + O(1/∆).Lemma 7.1 implies that |F I j (x)|, |F J j (x)| ≤ 1 for j = 1, . . . , J. Hence, by this and (7I j Re log ζ(s j + it) F J j Im log ζ(s j + it) dt = 2T Re log ζ(s j + it) F J 1 Im log ζ(s j + it) × J j=2 I j Re log ζ(s j + it) F J j Im log ζ(s j + it) dt + O(1/∆).Iterating this argument and using an analog of (7.7) for Im log ζ(s+it), which is proved in the same way, I j Re log ζ(s j + it) F J j Im log ζ(s j + it) dt = P T log ζ(s j + it) ∈ R j , ∀j ≤ J + O J ∆ . I j Re log ζ(s j , X) F J j Im log ζ(s j , X) = P log ζ(s j , X) ∈ R j , ∀j ≤ J + O J ∆ .

  Proof of Proposition 2.2. Let δ = e -V /2 . Since ζ( 3 4 + s, X) is almost surely analytic in |s| ≤ r + 2δ, then by Cauchy's estimate we have almost surely that

				max |z|≤r	|ζ ( 3 4 + z, X)| ≤	1 δ	max |s|≤r+δ	|ζ( 3 4 + s, X)|.
	Therefore, we obtain
	(4.5)	P max |z|≤r	|ζ ( 3 4 + z, X)| > e V ≤P max |s|≤r+δ ≤P max |s|≤r+δ	|ζ( 3 4 + s, X)| > δe V | log ζ( 3 4 + s, X)| >	2 V	.
	Let k be a positive integer. By (2.2) log ζ( 3 4 + s, X) converges almost surely to a
	holomorphic function in |s| ≤ r + 2δ. Using Cauchy's integral formula as in (4.4), we
	obtain almost surely that
		max |s|≤r+δ	| log ζ( 3 4 + s, X)|	2k	1 δ |z|=r+2δ	log ζ( 3 4 + z, X)	2k • |dz|.
	Hence, applying Lemma 4.3 we get
	(4.6)

+r) -1

To bound the RHS we estimate large moments of log ζ( 34 + it + s). This is accomplished by approximating log ζ( 34 +it+s) by a short Dirichlet polynomial, uniformly for all s in the disc {|s| ≤ r + δ}. Using zero density estimates and large sieve inequalities, we can show that such an approximation holds for all t ∈ [T, 2T ], except for an exceptional set of t's with very small measure. We prove Lemma 4.1. Let 0 < r < 1/4 be fixed, and δ = (1/4 -r)/4. Let y ≤ log T be a real number. There exists a set I(T ) ⊂ [T, 2T ] with meas(I(T ))

T 1-δ y(log T ) 5 , such that for all t ∈ [T, 2T ] \ I(T ) and all |s| ≤ r + δ we have

To prove this result, we need the following lemma from Granville and Soundararajan [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF]. Lemma 4.2 (Lemma 1 of [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF]). Let y ≥ 2 and |t| ≥ y + 3 be real numbers. Let 1/2 ≤ σ 0 < 1 and suppose that the rectangle {z :

Then for any σ with σ 0 + 2/ log y < σ ≤ 1 we have

Proof of Lemma 4.1. Let σ 0 = 1/2 + δ. For j = 1, 2 let T j be the set of those t ∈ [T, 2T ] for which the rectangle {z : σ 0 < Re(z) ≤ 1, |Im(z) -t| < y + 1 + j} is free of zeros of ζ(z). Then, note that T 2 ⊆ T 1 , and for all t ∈ T 2 , we have t+Im(s) ∈ T 1 for all |s| ≤ r + δ. Hence, by Lemma 4.2 we have

for all t ∈ T 2 and all |s| ≤ r + δ. Let N (σ, T ) be the number of zeros of ζ(s) in the rectangle σ < Re(s) ≤ 1 and |Im(s)| ≤ T . By the classical zero density estimate N (σ, T ) T 3/2-σ (log T ) 5 (see for example Theorem 9.19 A ot Titchmarsh [START_REF] Titchmarsh | The theory of the Riemann zeta-function[END_REF]) we deduce that the measure of the complement of T 2 in [T, 2T ] is T 1-δ y(log T ) 5 .

We also require a minor variant of Lemma 3.3 of [START_REF] Lamzouri | Discrepancy bounds for the distribution of the Riemann zeta-function and applications[END_REF], whose proof we will omit.


Abdelmounaim Failali

Eléonor Acher

Valérie Vallet

Florent Réal

Dominique Guillaumont

Insights from quantum

come

Introduction

Amide derivatives represent an important class of molecules to recover selectively metal ions such as platinum-group metals, gold or 5f elements by solvent extraction. [1][2][3][4] Tertiary N,N-dialkyl amides are particularly well known for their strong extraction ability regarding uranium(VI) and plutonium(IV) and are regarded as a promising alternative extractant family to Tri-n-Butyl-Phosphate (TBP). [5][6][START_REF] Siddall | Application of amides as extractants, u Pont de Nemours[END_REF][START_REF] Condamines | [END_REF][9] A key property of amide derivatives is that it is possible to co-extract uranium and plutonium from nitric acid solutions and to further achieve U/Pu separation by decreasing the nitric acid concentration. [START_REF] Milanole | Dissymmetric N, N-dialkylamides used particularly for separating uranium (VI) from plutonium (IV), synthesis thereof and uses of same[END_REF] More recently, it was shown that carbamides are also capable of achieving such U(VI) and Pu(IV) a CEA, DES, ISEC, DMRC, Univ Montpellier, Marcoule, F-30207 Bagnols-sur-Ceze Cedex, France, E-mail: dominique.guillaumont@cea.fr a Université Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers, Atomes et Molécules, F-59000 Lille, France † Electronic Supplementary Information (ESI). Additional geometrical parameters. The Cartesian coordinates of all structures are available on the free platform Zenodo at the following link with DOI 10.5281/zenodo.4068222.

extraction and separation. [START_REF] Berger | [END_REF] Because of their keen industrial interest, a large number of N,N-dialkylamides have been synthesized and an extensive database of U(VI) and Pu(IV) distribution ratio between organic and aqueous phases and U(VI)/Pu(IV) selectivity factors is available. 6,[12][13][14][15] The strong dependence of the extraction properties on the chemical conditions, namely nitric acid and extractant concentrations, has been established. [START_REF] Condamines | [END_REF]16 Moreover, one of their most interesting features is that it is possible to adjust their physicochemical and extraction properties by altering the alkyl chains on either nitrogen or carbonyl sides. Extraction properties are particularly influenced by the size and bulkiness of their alkyl groups adjacent to the carbonyl group.

Variations of extraction properties are particularly strong for plutonium. 6,[START_REF] Condamines | [END_REF]17 This was first discovered in the 1960s by Siddall et al.. 6 This opens up the possibility to tune their extracting strength and selectivity toward metal ions by altering the length and branching of amide alkyl groups. Siddall et al. 6 have demonstrated that a simple modification of the acyl group can suppress Pu(IV) extraction. [18][19][20] In the series R = CH 3 , C 2 H 5 , CH(CH 3 ) 2 , C(CH 3 ) 3 , Pu(IV) distribution ratio D Pu continuously drops from 21 (R = CH 3 ) to 0.001 (R = C(CH 3 ) 3 ), while U(VI) distribution coefficient decreases much less significantly from 9.9 to 0.6 (upon extraction from a 3 M nitric acid aqueous solution to a toluene organic phase with 0.5 M of amide extractant). 6 Accordingly, U/Pu separation factor increases from 0.5 to ~600. Since this early work, monoamides have been investigated extensively for the separation of hexa-and tetravalent actinide ions. Monoamides with long alkyl chains that prevent third phase formation have been developed. DEHiBA (N,N-di(2-ethylhexyl)isobutyramide) and DEHBA (N,N-di(2-ethylhexyl)butyramide) are probably the most investigated derivatives so far. [START_REF] Condamines | [END_REF]9,12,15,[START_REF] Mahanty | Mohapatra[END_REF]22 As reported for the earlier monoamides with shorter alkyl chains, the branching on the acyl group for DEHiBA suppress plutonium(IV) extraction compared to DEHBA.

Notwithstanding their industrial interest, the strong influence of N,N-dialkylamide structure on extracting properties is yet to be rationalized. To the best of our knowledge, there is no clear explanation for what makes the monoamide structure have such a great impact on the extraction ability. For decades, it was assumed that amide derivatives operate through a solvation mechanism as in the PUREX process with TBP,where Pu(IV) and U(VI) are predominantly extracted as the Pu(NO 3 ) 4 L 2 and UO 2 (NO 3 ) 2 L 2 complexes in which two amide ligands bind Pu(IV) and U(VI) in the inner coordination sphere. [START_REF] Condamines | [END_REF] The strong variations of the Pu distribution coefficient as a function of the bulkiness of the acyl group was attributed to steric hindrance in the Pu coordination sphere. However, recent studies have revealed that plutonium extraction with amide derivatives is more complex than with TBP. 17,19 By coupling UV-Vis spectroscopy, EXAFS and quantum chemical calculations, it has been shown that the plutonium hexanitrato complex Pu(NO 3 ) 6

2can also be extracted in the organic phase with no amide ligand present in the first coordination sphere. It was deduced that two protonated amide ligands bind in the outer coordination sphere to ensure the charge neutrality and the stability of the complex in the organic phase. However the detailed structure of the outer-sphere species is not known and could not be characterized from experimental data. It was shown that the Pu(IV) coordination structure can switch from inner to outer complexation either by varying the experimental conditions (such as nitric acid concentration) or by slightly altering the amide alkyl group. 19,[23][24][25] Furthermore, the recent study of Pu(IV) with linear alkyl chains carbamides demonstrated that carbamides are strong extractants at high nitric acidity while promoting outersphere coordination. As a result, the strong variation of extraction properties cannot be solely rationalized from steric hindrance in the inner coordination sphere. As found for the recovery of gold with amides, outer-sphere interactions appear to be of primary importance. 3 Thus, the understanding the relationship between the structure of the ligands and their extraction ability is crucial to design new extraction processes. The aim of this work is to explore and rationalize Pu(IV) inner and outer-sphere interactions with amide and carbamide derivatives from scalar relativistic Density Functional Theory (DFT) calculations, that have proven to be reliable at predicting the geometries of Pu(IV) complexes. 24,[26][27][28] While the structures of the inner-sphere complexes with amide and carbamide ligands have been well characterized in the solid state from XRD and in solution from EXAFS, the structures of outer-sphere complexes are largely unknown, though carbamides are believed to promote the formation of outer-sphere complexes. [START_REF] Berger | [END_REF] DFT geometry optimizations and complexation energetics shall help shedding light on their 3D structures and relative stabilities.

Herein, we consider that the ligands extract Pu(IV) from nitric acid according to the following reactions:

Pu 4+ (aq) + 4 (NO 3 ) 2-(aq) + 2 L(org) ----[Pu(NO 3 ) 4 L 2 ](org) (1)
Pu 4+ (aq) + 6 (NO 3 ) 2-(aq) + 2 H + (aq) + 2 L(org) ----[Pu(NO 3 ) 6 ](HL) 2 (org) (2)
in which the complexing ligand is noted L; (aq) and (org) stand for the aqueous and the organic phases, respectively. Reaction 1 corresponds to the formation of an inner-sphere complex [Pu(NO 3 ) 4 L 2 ](org) with four bidentate nitrate anions and two ligands. Reaction 2 corresponds to the formation a Pu(VI)hexanitrato core Pu(NO 3 ) 6 2which is charge balanced by protonated HL + attached by hydrogen bonds (outer-sphere complexation). Note that we disregard the more hypothetical possibility of forming a complex with a hybrid coordination Pu(NO 3 ) 5 L(HL), corresponding to Pu(NO 3 ) 5 L -with one HL + protonated ligand hydrogen-bonded to one nitrate ion.

To rationalize the influence of the alkyl groups on Pu(IV) complexation, the nature of the alkyl groups in the monoamide (R 1 -R 3 ) and the carbamide (R 1 -R 4 ) ligands drawn in Figure 1 have been systematically varied following the list and labelling given in Table 1. PEE and IEE ligands can be considered as model ligands for the two isomeric DEHBA and DEHiBA monoamides.

Computational Methods

All molecular geometries have been fully optimized using the Gaussian quantum chemistry package at the DFT level, [START_REF] Frisch | Gaussian~16 Revision C.01[END_REF] employing the PBE0 functional, [START_REF] Adamo | [END_REF] that has been shown to yield accurate structural parameters of the [Pu(NO 3 ) 6 ] 2 -complex. 28,31 For each ligand, 4 to 10 initial conformations for [Pu(NO 3 ) 4 L 2 ] and [Pu(NO 3 ) 6 ](HL 2 ) complexes were constructed and optimized in the gas phase in order to determine the lowest energy structure. Def2-TZVP basis set has been used for light atoms (H, N, O, and C). 32 For plutonium, the 82 inner-shell core electrons were replaced by the relativistic 5f-in-core large-core pseudopotential (LPP) noted ECP82MWB, 33 together with the corresponding valence basis set ECP82MWB-AVTZ. 33 The use of an LPP for actinides eliminates the difficulties arising from the open 5f-shells (5 f 4 valence configuration for Pu(IV)) that may yield a multireference character for the ground state with spin contamination issues. It is also trustworthy as it offers satisfactory results for actinide-ligand bond distances and binding energies, with deviations to small-core results not larger than 0.025 Å (1.2%), and 0.92 eV (0.6%), for actinocene complexes. 34 To confirm the applicability of LPP to our systems, test calculations were performed using a plutonium small-core pseudopotential noted ECP60MWB and accompanying basis set. 35 Energy variations for reaction ( 1) and ( 2) are compared in Table S1 in S.I. section. The values differ by less than 5 kJ mol -1 and confirm the judicious choice of a 5f-in-core pseudopotential for the systems of interest.

Core polarization effects described through a core-polarization potential (CPP) were shown to be small 33 and can be safely disregarded. We have also neglected spin-orbit as Šulka et al. 28 found a small effect of 0.014 Å on the Pu -O NO3 bond lengths in the Pu(IV)-hexanitrato complex. Furthermore, as we are interested in relative complexation energies, we can forecast that the absolute errors induced by neglecting core-polarization effects and spin-orbit coupling will compensate when taking energy differences.

When calculating the relative complexation free energies for various monoamide or carbamide ligands, it is relevant to account for long-range dispersion contributions, that might be sizable in the outer-sphere complexes ([Pu(NO 3 ) 6 ](HL) 2 (org)). We have thus performed single-point energy calculations with two approaches (i) by adding to the PBE0 energies the Grimme's DFT-D3 dispersion correction 36 ; (ii) by using the MP2 wave-function based method as implemented in the Turbomole quantum chemistry package. 37 The later calculations used aug-cc-pVTZ 38 basis sets for the light atoms, keeping the 1s core electrons frozen, and the ECP82MWB-AVTZ 33 basis set for Pu, together with the resolution of identity 39 with the corresponding auxiliary basis sets. 32 .

To mimic experimental conditions, in which the Pu metal ions are extracted into an organic phase that contains the extractant molecules, nitric acid and aliphatic solvent, the impact of solvation effects on the complex geometries and relative complexation energies was explored using the Polarizable Continuum solvation Model (PCM) as implemented in Gaussian package (See Ref . 40 for a review on solvation models), for two solvents n-dodecane (ε r = 2.02) and Dimethylacetamide DMA (ε r = 37.78).

Vibrational frequencies calculation for the ligands and complexes were performed to verify that the geometries were minima, and to compute the zero-point energies and thermal corrections at 298.15 K using the rigid rotor-harmonic oscillator approximation, as implemented in Gaussian, either in the gas phase or in solution.

Finally, the affinities of the model extractants are compared by calculating the free energies of the following ligand-exchange reactions:

[Pu(NO 3 ) 4 (L1) 2 ] + 2 L2 ----[Pu(NO 3 ) 4 (L2) 2 ] + 2 L1 (3) [Pu(NO 3 ) 6 ](HL1) 2 + 2 L2 ----[Pu(NO 3 ) 6 ](HL2) 2 + 2 L1 (4)
For instance the relative complexation free energy ∆∆G reads:

∆∆G = G(Pu NO 3 L2) -G(Pu NO 3 L1) -2(G(L2) -G(L1)) (5)

Results and Discussion

All Pu(IV) complexes with amide ligands reported in Table 1 were considered. The calculations were performed in the gas phase for all the ligands and in solution for some selected ligands. In the industrial process, the organic phase contains extractant molecules in an aliphatic solvent. After contact with the acidic aqueous phase, some amount of extracted nitric acid is also present in the organic phase. n-dodecane is the best model for the aliphatic solvent but the presence of amides and nitric acid is expected to increase the polarity of the solution. Therefore, the DMA solvent model was also used to investigate the effect of increasing the polarity of the solution.

Inner-sphere complexation with monoamides

Fig. 2 Geometry of the inner-sphere Pu(NO 3 ) 4 (L) 2 complex with the PEE ligand optimized in the gas phase.

The optimized structure of [Pu(NO 3 ) 4 L 2 ] is represented in Figure 2 for PEE ligand and structural parameters are listed in Table 2 for all the ligands. Additional structural parameters are given in the ESI. In the most stable geometry of these systems, the coordination number of plutonium is 10, with the nitrate groups acting as bidentate ligands. Amide ligands act as monodentate ligands, which interact with plutonium through the carbonyl oxygen. Changing the ligand alkyl chains does not signifi-cantly alter the structure. In the gas phase, the Pu -O L distance varies from 2.34 Å to 2.35 Å while the mean value of Pu -O nit varies from 2.47 Å to 2.49 Å. Ligand interatomic distances, such as C -O or C -N distances, also remain similar for all the ligands. In the polar DMA solvent Pu -O L distances are shortened by 0.03 -0.04 Å compared to the gas phase while Pu -O nit distances are lengthened by 0.02 -0.03 Å. These calculated distances are all longer than the solid-phase distances reported from XRD of 2.26 Å and 2.44 Å respectively for Pu -O L and Pu -O nit distances in the Pu(IV)-tetranitrate complex with N,N-dibutyl-butanamide (DBBA). 19 The variation of the complexation free energies between the ligands and the tetra-nitrate plutonium core complex are reported in Table 3. As detailed in the computational methods section, the free energies are given as ligand-exchange complexation free energies defined by Eq. 5. The latter were also calculated adding the D3 empirical correction to account for dispersion interactions. On the carbonyl side, substituting the methyl by an ethyl group in R 1 position (from MMM to EMM or from MEE to EEE) has a small destabilizing effect (+7 kJ mol -1 ) which nearly disappears with dispersion corrections at the D3 level. Lengthening the alkyl chain further from ethyl to n-propyl stabilizes the complex at both levels of calculation (EEE → PEE). Conversely, increasing the branching from n-propyl to iso-propyl destabilizes strongly the complexation and has the most significant effect. On the nitrogen side, in R 2 and R 3 positions, substituting only one methyl into one ethyl has a small destabilizing effect while further changing the second methyl group into ethyl has the reverse stabilizing effect. The inclusion of D3 dispersion correction systematically increases the stability of the complexation with longer alkyl chains by a few kJ mol -1 as given by the comparison between PBE0 and PBE0-D3 results. This is indicative of the dispersion interactions induced by alkyl groups.

If we focus on PEE and IEE, which can be considered as model

Table 3 Free energy variations corresponding to the inner-sphere ligand exchange reaction (3), in kJ mol -1 , calculated in the gas phase at the PBE0 and PBE0-D3 levels of theory.

L1 → L2 PBE0 PBE0-D3 R 1 MMM → EMM 7 2 MEE → EEE 7 -2 EEE → PEE -5 -9 PEE → IEE 22 14 R 2 , R 3 MMM → MME -3 -4 MME → MEE -4 -5
ligands for DEHBA (N,N-di(2-ethylhexyl)butyramide) and DE-HiBA (N,N-di(2-ethylhexyl)isobutyramide) that have been largely studied for U(VI) and Pu(IV) extraction, [START_REF] Condamines | [END_REF]18 we can observe a +22 kJ mol -1 free energy difference at the PBE0 level and +14 kJ mol -1 at the PBE0-D3 level. For these two isomeric ligands, calculations at the MP2 level and with the n-dodecane and DMA solvent models were also performed. As the ligand-exchange complexes are of comparable shape and size, the errors associated with the solvent model are expected to largely cancel out. The relative complexation free energies between PEE and IEE are given in Table 4. The +22 kJ mol -1 free energy difference calculated in the gas phase is slightly decreased in solution, by 6 kJ mol -1 in ndodecane and by 10 kJ mol -1 in the more polar DMA solvent. The MP2 values give a slightly lower free energy difference than PBE0, but they are consistent with the PBE-D3 results, making us confident on the capability of the D3 correction in capturing dispersion effects. Thus, only the latter method will be used for outer-sphere complexes. According to the calculations, the free energy variation systematically indicates a less favorable complexation for the branched alkyl group than for the linear alkyl one in the case of [Pu(NO 3 ) 4 L 2 ] formation. These results are consistent with the sharp diminution of the distribution ratio moving from DEHBA to DEHiBA observed experimentally for Pu(IV) extraction at low nitric acidity where the inner-sphere complex is the predominant species. 12 They confirm the destabilizing steric hindrance effect, induced by the presence of voluminous branched alkyl group at the amide R 1 moiety, on plutonium tetranitrate complexation.

Outer-sphere complexation with monoamides In the outer-sphere Pu(NO 3 ) 6 (HL) 2 structures, the interaction of the amide ligands with plutonium nitrates complexes occurs via strong hydrogen bonds between protonated amides HL + and oxy- gen nitrate ions. The coordination number of plutonium is twelve with six bidentate nitrate ions in the first coordination sphere and the two protonated amides form hydrogen bonds with nitrate ions in the outer coordination sphere. Two possible hydrogen bonds interactions were considered resulting in two different geometries described in Figure 3 for PEE ligand. In the first geometry (I), hydrogen bonds are formed with oxygen nitrate of the outer sphere. This structure was described in our previous work. 19 In geometry II, hydrogen atoms are bound to nitrate oxygen atoms that are also coordinated to plutonium. This coordination mode can be deduced from the electrostatic potential which was determined for Pu(NO 3 ) 6 2anion. 41 It was shown that first-shell oxygen atoms may be considered as possible acceptor sites for non-covalent interactions even though such interactions were not observed in the crystal structure with pyridinium cations. 41 In addition, such a coordination mode has been shown to exist for uranyl sulfate complexes with tertiary amines. Hydrogen bonds between the protonated amines and coordinated oxygen sulfate atoms were found in the solid-state crystal structure 42 and in n-dodecane from molecular dynamics simulations. 43 According to our DFT calculations, geometry II is significantly more stable than geometry I both in the gas phase and in solution (See Table 5) for all the ligands except MMM. For the latter, geometries I and II are found energetically degenerate in the gas phase (∆G = 1 kJ mol -1 ) while geometry II is relatively stabilized by 14 kJ mol -1 and 22 kJ mol -1 in n-dodecane and DMA solvents, respectively. In both outer-sphere geometries I and II, the proton is located on the carbonyl group rather than on a nitrate anion. In the gas phase, the protonated ligands wrap the hexanitrato plutonium complex. Calculated interatomic distances are listed in the ESI for geometry I and reported in Table 6 for the most stable geometry II. For the latter the inner Pu(NO 3 ) 6 2coordination shell is marginally altered by the nature of the outer-sphere ligand. In all complexes, the average Pu-O distance to the nitrate anions which are not involved in hydrogen bonding (Pu -O nit1 ) is 2.52 Å. This distance is close to the average value reported in the Pu(NO 3 ) 6 2crystal structures (2.49 Å). 44 The average distance between Pu and the nitrate oxygens involved in the hydrogen bonds (Pu -O nit2 ) is 0.04-0.05 Å longer than the distance to the other nitrate oxygens (Pu -O nit1 ). The longest Pu -O nit2 distance is found with IEE ligand. The lengths of the hydrogen bonds between the nitrate anions and the protonated ligand increase with the lengthening of the alkyl group in any of the R 1 , R 2 or R 3 positions, as indicated from O nit -O L and O L -H distances. On the contrary, the hydrogen bond distance decreases from n-propyl to iso-propyl. This could indicate a stronger H-bond but the OHO angle also diminishes from 164°to 160°. This may indicate that the hydrogen bond is more constrained with IEE ligands. Another geometric difference arises from the position of the two amide groups in the complexes. In the gas-phase structures, each planar amide group is nearly parallel to one of the six trigonal planar nitrate anions (Figure 3) except with the bulky isopropyl group. As depicted in Figure 4, with IEE the planar orientation of the amide group deviates from the parallel orientation with the planar nitrate ion. As a consequence the distance between Pu and the nitrogen atom of the amide function (Pu -N L ) increases by about 0.4 Å for IEE compared to other ligands in the gas phase.

As shown in the optimized structures, a number of short interatomic close-contacts are found between C -H and NO 3 -oxygen atoms (< 2.80 Å), involving the C -H groups from alkyl chains of both nitrogen and carbonyl sides. The inclusion of solvent effects does not alter the plutoniumoxygen distances in the inner sphere but modifies the outersphere structure. Indeed, the two outer-shell ligands remain hydrogen bonded by the protonated amide group but move away from Pu(NO 3 ) 6 2 -. This effect amplifies with the solvent dielectric constant. As a consequence, the Pu -N L distances increase by 0.2 Å in n-dodecane and by 0.9 Å in DMA compared to the gas phase except for IEE. The solvent effect is weaker for IEE ligands because the deviation of the amide group from the parallel planar orientation is already strong in the gas phase. The hydrogen bond distance increases slightly in dodecane (by 0.04 Å to 0.07 Å) and strongly in DMA (by 0.13 Å to 0.22 Å). The largest increase corresponds to IEE ligands.

The complexation energies were calculated for the ligand exchange reaction (4) corresponding to the formation of O nit2 denotes oxygen from nitrate ions which are involved in hydrogen bond.

Table 7 Free energies variations corresponding to the outer-sphere ligand exchange reaction (4). The values are in kJ mol -1 and were calculated in the gas phase.

L1 → L2 PBE0 PBE0-D3 R 1 MMM → EMM 3 -1 MEE → EEE 3 -3 EEE → PEE -6 -11 PEE → IEE 34 39 R 2 , R 3 MMM → MME -6 -10 MME → MEE -3 0
[Pu(NO 3 ) 6 ](HL2) 2 relative to [Pu(NO 3 ) 6 ](HL1) 2 . Calculated values are reported in Table 7 for the most stable geometry II. On the carbonyl side, in R 1 position, substituting the methyl by an ethyl group or an ethyl by a n-propyl has very similar effects than those determined for inner-sphere complexation. Changing from MMM to EMM and MEE to EEE has a small destabilizing effect that is reversed when D3 dispersion corrections are included.

From EEE to PEE, a stabilizing effect is calculated at both levels of calculations. Increasing the branching from n-propyl to isopropyl has a strong destabilizing effect (∆∆G Gas = 39 kJ mol -1 at the PBE0-D3 level). Such a destabilizing effect was also found for inner complexation but is unexpectedly much larger for this coordination mode while steric hindrance is expected to be smaller in the outer coordination sphere. On the nitrogen side, the effects are different than those found for inner complexation. Substituting one methyl by in an ethyl group has a strong stabilizing effect up to 19 kJ mol -1 from MMM to MME at the PBE0-D3 level. Such effect was not found for innersphere complexation. The further addition of one ethyl group has no significant effect.

As for inner-sphere complexes, the complexation energy in the solvent phase was calculated for isomeric PEE and IEE derivatives. Relative energies in the gas phase and in solution are given in Table 8. The free energy difference between the two ligands is similar in the gas phase and in n-dodecane but decreases from 34 kJ mol -1 in the gas phase to 14 kJ mol -1 in the more polar DMA solution. These energy variations follow the structural changes from the gas phase to a polar solution with the diminution of close-contact interactions between C-H groups and the nitrate anions. The energy difference between PEE and IEE is significant and is consistent with the sharp diminution that has been measured for Pu(IV) distribution ratio from DEHBA to DEHiBA in high nitric acid concentration. The solvent effect is more significant for outer than inner-sphere complexation. This is consistent with solvent effects which are stronger on the outer-sphere structure. In the gas phase, the preferred conformation of the outersphere complex with nearly parallel nitrate and amide groups is disfavored with the bulky IEE. In polar solution, the ligands move away and steric constraints become smaller.

Complexation with carbamides

On the one hand, a carbamide is considered as being more basic than an amide function and is expected to bind a metal cation more strongly. The calculated proton affinities confirm the stronger electron donor capability of the carbonyl oxygen in carbamides compared to amides (See Table S1 in the ESI). On the other hand, tetra-alkyl carbamides are bulkier than tri-alkyl monoamides for similarly sized-alkyl groups and are expected to induce larger steric hindrance effect in the plutonium coordination sphere. In the present work, inner and outer-sphere complexations were investigated for two carbamide derivatives with methyl and ethyl groups (C4M and C4E).

In the inner-sphere carbamide complexes, the plutoniumoxygen bond distances are close to those calculated with monoamides within 0.02 Å (Table 9). The Pu -O L distance is slightly shorter with C4M than with any other monoamides while Pu -O L distance with C4E is the longest one. This is an indication of the competition between electronic donor and steric hindrance effects.

For the outer-sphere complexation, the energy differences between geometries I and II are not as large as for monoamides (see Table 11). At the PBE0 level, the preferred coordination mode is geometry I in the gas phase and II in n-dodecane. The inclusion of dispersion corrections are more significant for the tetra-ethyl carbamide than for monoamides derivatives. Dispersion attrac-tive interactions stabilize geometry I over geometry II in the gas phase and in n-dodecane. This is attributed to the weak interactions that take place between the alkyl chains and nitrate ions. Such interactions are becoming more important as the number of alkyl groups increases and are also more important in the gas phase and in n-dodecane than in the more polar DMA solution which counterbalances their strengths. Finally, considering the small energy difference in solution it is not possible to discriminate between both structures which may coexist in solution. The structural parameters are reported in Table 10 for geometry II and in the ESI for geometry I. The hydrogen bond distances are slightly longer with the carbamides than with the monoamides (in the gas phase, the lengthening of O nit -H distance is 0.03 Å from C4E to EEE and 0.05 Å for the O nit -O L distance). In geometry I, this is the reverse, the hydrogen bond distance is shorter with carbamide, O-H distance decreases by 0.08 Å from C4E to EEE. Complexation energies are given in Table 12. For outer-sphere complexes, the lowest energy geometry was taken at each level of calculation. Energies were calculated in the gas phase and in solution. Results are compared with the PEE monoamide. For inner-sphere complexation, the relative complexation free energies are strongly dependent on dispersion corrections. Namely, at the PBE0 level, when the length of the carbamide alkyl groups increases (from C4M to C4E) or from monoamide to carbamide (PEE to C4E), inner-sphere complexation is destabilized by 9 kJ mol -1 to 17 kJ mol -1 , in the gas phase as well as in solution.

In the presence of dispersion corrections, this is the reverse, inner sphere is stabilized, attractive dispersion forces overbalance steric hindrance effects. Dispersion interactions are likely to be overestimated from such calculations since the competitive dispersion interactions with alkyl chains from the solvent are not properly taken into account. However, these results emphasize the key role of weak interactions in these systems. For outer-sphere complexation, the lengthening of the alkyl group destabilizes the complexation in the gas phase. Solvent effects counterbalance steric hindrance effects in the outer sphere and favor C4E complexation compared to C4M. There is no strong influence of dispersion corrections. The replacement of PEE by C4E slightly favors outersphere complexation in the gas phase and in n-dodecane.

Conclusion

From this work it is possible to get a better understanding of the strong influence of amide structure on plutonium extraction. Experimentally, a simple modification of the alkyl chain attached to the carbonyl group can suppress plutonium extraction at low and high acidity even though amide operates through different coordination modes. Complexation energies were calculated for inner and outer-sphere complexations, which take place in weakly acidic and strong acidic conditions, respectively. For both inner and outer-sphere complexes, it is found that the introduction of a bulky alkyl group on the carbonyl side diminishes strongly the complexation energy. This is fully consistent with the measured monamide extraction properties. The influence of the bulkiness of the alkyl group is as or even more important for outer than for inner-sphere interactions. This was unexpected when considering that there is less flexibility and stronger steric constraints in the inner sphere. This is attributed to specific electrostatic interactions between the two outer-sphere amide ligands and two nitrate ions of Pu(NO 3 ) 6 2 -. By increasing the polarity of the solution, the outer-sphere protonated ligands remain attached to Pu(NO 3 ) 6 2but alkyl groups move away and their interactions with nitrate ions diminish. As a consequence, solvent effects are very significant for outer-sphere complexation while there are small for inner-sphere complexation. It gives the key possibility to tune the substituent effect by changing the polarity of the solution. When changing the ligand, from monoamide to carbamide, the results show that weak interactions such as dispersion forces may have a remarkable impact on ligand binding affinities and on plutonium(IV) extraction. In order to predict ligand affinity of different families, it will be essential to go beyond the static DFT quantum chemical calculations with implicit treatments of solvent effects and to perform explicit solvent simulations.

Fig. 1

 1 Fig.1Labelling of the R 1 , R 2 , R 3 and R 4 alkyl groups in the monoamide (left) and carbamide (right) ligands, as listed in Table1.

Fig. 3

 3 Fig. 3 Gas-phase optimized geometries of the outer-sphere [Pu(NO 3 ) 6 ](HPEE) 2 complex in the geometries I (left) and II (right).

Fig. 4

 4 Fig. 4 Perspective views of the outer-sphere complex [Pu(NO 3 ) 6 ](HIEE) 2 in geometry II optimized in the gas phase (left) and in DMA solvent (right).

Table

Table 1

 1 1. List of aliases for the substituted monoamide and carbamide ligands.

	Ligands	R 1	R 2	R 3	R 4
	MMM	CH 3	CH 3	CH 3	-
	EMM	C 2 H 5	CH 3	CH 3	-
	MME	CH 3	CH 3	C 2 H 5	-
	MEE	CH 3	C 2 H 5 C 2 H 5	-
	EEE	C 2 H 5	C 2 H 5 C 2 H 5	-
	PEE	C 3 H 7	C 2 H 5 C 2 H 5	-
	IEE	CH(CH 3 ) 2 C 2 H 5 C 2 H 5	-
	C4M	CH 3	CH 3	CH 3	CH 3
	C4E	C 2 H 5	C 2 H 5 C 2 H 5 C 2 H 5

Table 2

 2 Selected interatomic distances (average values in Å) in the [Pu(NO 3 ) 4 L 2 ] complexes optimized in the gas phase and within a solvent.

	Ligand	d Pu-O L	d Pu-O nit	d Pu-N nit	d O-C	d C-N
	gas phase
	MMM	2.352	2.474	2.932	1.250	1.327
	EMM	2.347	2.477	2.931	1.250	1.328
	MME	2.356	2.483	2.930	1.249	1.327
	MEE	2.354	2.480	2.931	1.250	1.327
	EEE	2.348	2.485	2.933	1.251	1.326
	PEE	2.347	2.483	2.934	1.252	1.327
	IEE	2.343	2.486	2.937	1.252	1.327
	n-dodecane solvent
	MMM	2.337	2.489	2.934	1.252	1.325
	EEE	2.332	2.495	2.940	1.256	1.323
	PEE	2.332	2.495	2.940	1.257	1.323
	IEE	2.327	2.498	2.943	1.256	1.325
	DMA solvent
	MMM	2.313	2.508	2.941	1.259	1.319
	EEE	2.315	2.506	2.942	1.261	1.320
	PEE	2.315	2.506	2.942	1.261	1.320
	IEE	2.300	2.513	2.962	1.263	1.320

Table 4

 4 Variation of the complexation free energies for the formation of [Pu(NO 3 ) 4 L 2 ] when going from the PEE to the IEE ligand. All values are in kJ mol -1 and were calculated in the gas phase and in solution.

		∆∆G(PEE → IEE)
	Gas PBE0	22
	Gas PBE0-D3	14
	Gas MP2	14
	Dodecane PBE0	16
	DMA PBE0

Table 5

 5 Free energy differences between the two outer-sphere geometries I and II for the [Pu(NO 3 ) 6 ](HL 2 ) monoamide complexes. The values are in kJ mol -1 and were calculated in the gas phase, in n-dodecane (Dod.) and in DMA solvents.

	Ligand	MMM EMM MME MEE EEE PEE IEE
	Gas PBE0	1	-17	-5 -11 -20 -22 -17
	Gas PBE0-D3	1		-4	-20 -23 -18
	Dod. PBE0	-14			-24 -23 -26
	DMA PBE0	-22			-13 -23 -26

Table 6

 6 Selected interatomic distances (average values d in Å) a and hydrogen bond angle (α O-H-O ) in °calculated in the [Pu(NO 3 ) 6 ](HL 2 ) complex in geometry II optimized the gas phase and in solution.d Pu-O nit1 d Pu-O nit2 d Pu-N L d O nit2 -O L d O nit -H α OHO

	gas
	MMM	2.524	2.565	5.973	2.504	1.506 163
	EMM	2.523	2.566	6.031	2.511	1.513 164
	MME	2.524	2.564	6.031	2.511	1.515 164
	MEE	2.524	2.564	6.006	2.517	1.521 164
	EEE	2.523	2.566	6.047	2.522	1.528 164
	PEE	2.523	2.565	6.085	2.524	1.530 164
	IEE	2.520	2.573	6.363	2.498	1.514 160
	solution n-dodecane
	MMM	2.524	2.562	6.178	2.546	1.555 165
	EEE	2.524	2.562	6.210	2.558	1.570 165
	PEE	2.524	2.562	6.232	2.559	1.571 165
	IEE	2.523	2.567	6.551	2.546	1.585 158
	solution DMA
	MMM	2.524	2.560	6.856	2.644	1.651 174
	EEE	2.524	2.560	6.811	2.650	1.660 173
	PEE	2.524	2.561	6.840	2.654	1.664 172
	IEE	2.525	2.559	6.993	2.666	1.732 156

a O nit1 denotes oxygen from nitrate ions which are not involve in hydrogen bond.

Table 8

 8 Variation of the complexation free energies for the formation of the [Pu(NO 3 ) 6 (HL) 2 ] complex when going from PEE to IEE ligands. The values are in kJ mol -1 and were calculated in the gas phase and in solvent.

		PBE0 PBE0-D3
	Gas phase 34	39
	Dodecane 29	35
	DMA	14	19

Table 9

 9 Selected interatomic distances (average values in Å) calculated in the [Pu(NO 3 ) 4 L 2 ] complexes with carbamides optimized in the gas phase.

	Ligand	d Pu-O L	d Pu-O nit	d Pu-N nit	d O-C	d C-N
	C4M	2.346	2.490	2.936	1.256	1.354
	C4E	2.358	2.493	2.939	1.252	1.349
	solution n-dodecane
	C4M	2.326	2.497	2.941	1.260	1.346
	C4E	2.345	2.499	2.943	1.267	1.352
	solution DMA
	C4M	2.298	2.510	2.952	1.268	1.342
	C4E	2.323	2.508	2.951	1.273	1.348

Table 10

 10 Pu-O nit1 d Pu-O nit2 d Pu-N L d Pu-N L d O nit2 -O L d O nit -H α OHO

	gas phase
	C4M 2.523	2.563	5.178 6.190	2.557	1.552 169
	CAE 2.516	2.580	5.608 7.016	2.573	1.561 174
	solution n-dodecane
	C4M 2.524	2.560	5.234 6.296	2.600	1.606 169
	C4E	2.529	2.571	5.695 7.071	2.613	1.613 174
	solution DMA
	C4M 2.525	2.561	5.513 6.667	2.693	1.712 169
	C4E	2.525	2.559	5.863 7.178	2.692	1.711 170
	a O nit1 denotes oxygen from nitrate ions which are not involved in hydrogen bond.
	O nit2 denotes oxygen from nitrate ions which are involved in hydrogen bond.

Selected interatomic distances (average values d in Å) a and hydrogen bond angle (α O-H-O ) in degrees calculated for the [Pu(NO 3 ) 6 ](HL 2 ) complex in geometry II in the gas phase and in solution. a d

Table 11

 11 Free energy differences between the two outer-sphere geometries I and II of the [Pu(NO 3 ) 6 ](HL 2 ) carbamide complexes. The values are in kJ mol -1 and were computed in the gas phase, in n-dodecane and in DMA solvents.

		C4M C4E
	Gas PBE0	4	7
	Dodecane PBE0	-4	-2
	DMA PBE0	1	-3
	Gas PBE0-D3	6	17
	Dodecane PBE0-D3 -3	7
	DMA PBE0-D3	-1	-3

Table 12

 12 Complexation free energies when going from C4M to C4E and from PEE to C4E ligands for reactions (3) and (4). The values in kJ mol -1 and were calculated in the gas phase and in solution at the PBE0 and PBE0-D3 levels of theory.

			inner-sphere		outer sphere
		C4M → C4E PEE → C4E C4M → C4E PEE → C4E
	Gas PBE0	9	17	11	5
	Dod. PBE0	11	10	4	6
	DMA PBE0	13	9	-2	-8
	Gas PBE0-D3	-13	-3	17	-1
	Dod. PBE0-D3 -11	-9	6	-3
	DMA PBE0-D3	-9	-9	-6	-10

1-9 J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

Acknowledgements

We acknowledge support by the French government through the Program "Investissement d'avenir" (Grants LABEX CaPPA/ANR-11-LABX-0005-01 and I-SITE ULNE/ANR16-IDEX-0004 ULNE), as well as by the Ministry of Higher Education and Research, Hauts de France council and European Regional Development Fund (ERDF) through the Contrat de Projets État Region (CPER CLIMIBIO). Furthermore, this work was granted access to the HPC resources of CINES/IDRIS/TGCC under the allocation 2019-2020 (A0070801859) made by GENCI. We also acknowledge the CEA for the Ph.D. grant given to A.F.

Notes and references

Conflicts of interest

There are no conflicts to declare.