Is the Riemann zeta function in a short interval a 1-RSB spin glass ?
IIS THE RIEMANN ZETA FUNCTION IN A SHORT INTERVALA 1-RSB SPIN GLASS ?
LOUIS-PIERRE ARGUIN AND WARREN TAI
Abstract.
Fyodorov, Hiary & Keating established an intriguing connection between the maximaof log-correlated processes and the ones of the Riemann zeta function on a short interval of thecritical line. In particular, they suggest that the analogue of the free energy of the Riemann zetafunction is identical to the one of the Random Energy Model in spin glasses. In this paper, theconnection between spin glasses and the Riemann zeta function is explored further. We study arandom model of the Riemann zeta function and show that its two-overlap distribution correspondsto the one of a one-step replica symmetry breaking (1-RSB) spin glass. This provides evidence thatthe local maxima of the zeta function are strongly clustered. Introduction and Main Result
Background.
The Riemann zeta function is defined on C by(1) ζ ( s ) = (cid:88) n ≥ n s = (cid:89) p primes (cid:0) − p − s (cid:1) − if Re s > ζ ( s ) = χ ( s ) ζ (1 − s ) , χ ( s ) = 2 s π s − sin (cid:16) π s (cid:17) Γ(1 − s ) . Trivial zeros are located at negative even integers where χ ( s ) = 0. The non-trivial zeros arerestricted to the critical strip 0 ≤ Re s ≤
1. The Riemann hypothesis states that they all lie onthe critical line Re s = 1 /
2. A weaker statement, yet with deep implications on the distribution ofthe primes, is the Lindel¨of hypothesis which stipulates that the maximum of ζ on a large interval[0 , T ] of the critical line grows slower than any power of T , i.e. ζ (1 / T ) is O( T ε ) for any ε > ?tao ] for a proof that the constant is non-negative. Third, Fyodorov, Hiary& Keating [22] and Fyodorov & Keating [23] recently unraveled a striking connection betweenthe local statistics of the large values of the Riemann zeta function on the critical line and theextremes of a class of disordered systems, the log-correlated processes , that includes among others Date : September 13, 2018.
Key words and phrases.
Riemann zeta function, Disordered systems, Spin glasses. a r X i v : . [ m a t h . P R ] O c t L.-P. ARGUIN AND W. TAI (cid:45) (cid:45) Figure 1.
The value of − log | ζ (1 / i ( T + h )) | for T = 10000 and h ∈ [0 , τ is sampled uniformly on a largeinterval [ T, T ], then the maximum on a short interval, say [0 , τ is(2) max h ∈ [0 , log | ζ (1 / τ + h ) | = log log T −
34 log log log T + M T , where ( M T ) is a sequence of random variables converging in distribution. The deterministic order ofthe maximum corresponds exactly to the one of a log-correlated process, such as a branching randomwalk and the two-dimensional Gaussian free field, see for example [1, 26] for more background onthis class of processes. The precise value of the leading order can be predicted heuristically sincethe process for log ζ has effectively log T distinct values on [0 ,
1] (because there are on average log T zeros on [0 , | ζ (1 / i ( τ + h )) | shouldbe close to Gaussian with variance log log T as predicted by Selberg’s Central Limit Theorem [35].The log-correlations already appear at the level of the typical values from the multivariate CLTproved in [13]. The first order of the conjecture (2) was proved recently in parallel: conditionally onthe Riemann hypothesis in [28], and unconditionally in [3]. The evidence in favor of the conjecturelaid out by Fyodorov & Keating [23] suggests that the large values of the Riemann zeta functionlocally behaves like a disordered system of the spin-glass type characterized by an energy landscapewith multiple minima, see Figure 1.1. In particular, by considering − log | ζ (1 / τ + h )) | as theenergy of a disordered system on the state space [0 , T →∞ T log (cid:18) log T · (cid:90) | ζ ( 12 + i( τ + h)) | β dh (cid:19) = (cid:40) β if β < β if β ≥ T independent Gaussian variables of variance log log T .In this paper, we explore the connection with spin glasses further by providing evidence thatlog | ζ | behaves locally like a spin glass with one-step replica symmetry breaking (1-RSB), cf. The-orem 1. More precisely, we study a simple random model introduced by Harper [24] for the largevalues of log | ζ | . We show that two points sampled from the Gibbs measure at low temperaturehave correlation coefficients (or overlap ) 0 or 1 in the limit, similarly to a 1-RSB spin glass. Weexpect that part of our approach could be extended to prove a similar result for the Riemann zetafunction itself as stated in Conjecture 2 below. S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 3
The model and main result.
Let ( U p , p primes) be IID uniform random variables on theunit circle in C . We write E for the expectation over the U p ’s. We study the stochastic process(4) X h = (cid:88) p ≤ T Re( U p p − ih ) p / , h ∈ [0 , . We drop the dependence on T in the notation for simplicity. The process ( X h , h ∈ [0 , | ζ (1 / τ + h )) | , h ∈ [0 , h ∈ [0 , X h corresponds to the one in (2), asproved in [2]. Roughly speaking, the process X h corresponds to the leading order of the logarithmof the Euler product (1) with the identification (cid:0) p − i τ , p primes (cid:1) ←→ ( U p , p primes) . It is easily checked by computing the joint moments that the above identification is exact as T → ∞ in the sense of finite-dimensional distribution.The covariance can be calculated using the explicit distribution of the U p ’s:(5) E [ X h X h (cid:48) ] = (cid:88) p ≤ T (cid:90) π (cid:16) e i( θ − h log p ) + e − i( θ − h log p ) (cid:17) · (cid:16) e i( θ − h (cid:48) log p ) + e − i( θ − h (cid:48) log p ) (cid:17) d θ π = 12 (cid:88) p ≤ T cos( | h − h (cid:48) | log p ) p . We are interested in the correlation coefficient or overlap (in the spin glass terminology):(6) ρ ( h, h (cid:48) ) = E [ X h X h (cid:48) ] (cid:113) E [ X h ] E [ X h (cid:48) ] , for a given pair ( h, h (cid:48) ).Any sum over primes can be estimated using the Prime Number Theorem [27], which gives thedensity of the primes up to very good errors,(7) { p ≤ x : p prime } = (cid:90) x y d y + O( xe − c √ log x ) . (The error term, which is already more than sufficient for our purpose, is improved under theRiemann hypothesis.) In particular, this can be used to rewrite the covariances as (see Lemma 5below for details),(8) E [ X h ] = 12 (cid:88) p ≤ T p − = 12 log log T + O(1) E [ X h X h (cid:48) ] = 12 log | h − h (cid:48) | − + O(1) . The process ( X h ) is said to be log-correlated , since the covariance decays approximately like thelogarithm of the distance. The correlation coefficients as a function of the distance become(9) ρ ( h, h (cid:48) ) = log | h − h (cid:48) | − log log T + o(1) , for | h − h (cid:48) | ≥ (log T ) − .Throughout the paper, we will use the notation f ( T ) = o( g ( T )) if f ( T ) /g ( T ) → f ( T ) =O( g ( T )) if f ( T ) /g ( T ) is bounded. We will sometimes use f ( T ) (cid:28) g ( T ) for short if f ( T ) = O( g ( T ))(the Vinogradov notation).The main result of this paper is the limiting distribution of the correlation coefficient when h and h (cid:48) are sampled from the Gibbs measure. This is referred to as the two-overlap distribution in L.-P. ARGUIN AND W. TAI the spin-glass terminology. We denote the the Gibbs measure by(10) G β,T ( A ) = (cid:90) A e βX h Z β,T d h Z β,T = (cid:90) e βX h d h . Theorem 1.
For every β > and for any interval I ⊆ [0 , , lim T →∞ E (cid:104) G × β,T { ( h, h (cid:48) ) : ρ ( h, h (cid:48) ) ∈ I } (cid:105) = 2 β I (0) + (1 − β ) I (1) . where I is the indicator function of the set I . In other words, when h, h (cid:48) are sampled independentlyfrom the Gibbs measure G β,T , the random variable ρ ( h, h (cid:48) ) is Bernoulli-distributed with parameter /β in the limit T → ∞ . The limit is exactly the two-overlap distribution of a 1-RSB spin glass. In view of the relation (9)between the correlation coefficient and the distance | h − h (cid:48) | , the result means that the large valuesof X h must lie at a distance O(1) or O((log T ) − ). The mesoscopic distances (log T ) − α , 0 < α < | ζ | exhibits1-RSB for β large enough. Conjecture 2.
Consider G β ( t ) = | ζ (1 / t ) | β Z β ( t ) = (cid:90) G β ( t + h )d h . For β > , and any interval I ⊆ [0 , , if τ is sampled uniformly on [ T, T ] : lim T →∞ E (cid:34)(cid:90) { ( h,h (cid:48) ): ρ ( h,h (cid:48) ) ∈ I } G β ( τ + h ) · G β ( τ + h (cid:48) ) Z β ( τ ) d h d h (cid:48) (cid:35) = 2 β I (0) + (1 − β ) I (1) . In other words, points h, h (cid:48) whose ζ -value is of the order of log log T are at a distance of O(1) or O((log T ) − ) . The above conjecture implies a strong clustering of the high values of ζ at a scale (log T ) − akin to the one observed in log-correlated process [5]. In turns, this phenomenon has importantconsequences for the joint statistics of high values which should be Poissonian at a suitable scaleas for log-correlated processes [6, 11]. In particular, it is expected that the statistics of the Gibbsweights is Poisson-Dirichlet [8, 9], and that the Gibbs measure converges to an atomic measure on[0 , Acknowledgements . L.-P. A. is supported by NSF CAREER 1653602, NSF grant DMS-1513441,and a Eugene M. Lang Junior Faculty Research Fellowship. W. T. is partially supported by NSFgrant DMS-1513441. Both authors would like to thank Fr´ed´eric Ouimet for useful comments on afirst version of the paper. L.-P. A. is indebted to Chuck Newman for his constant support and hisscientific insights throughout the years.1.3.
Main Propositions and Proof of the Theorem 1.
The proof of Theorem 1 is based on amethod developed for log-correlated Gaussian processes by Arguin & Zindy [8, 9]. It was adaptedfrom a method of Bovier & Kurkova [15, 16] for Generalized Random Energy Models (GREM’s).The main idea is to relate the distribution of the overlaps with the free energy of a perturbedprocess. In the present case, the process is not Gaussian and the method has to be modified. To
S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 5 this aim, consider the process at scale α , for 0 < α <
1, where the sum over primes is truncated atexp((log T ) α ),(11) X h ( α ) = (cid:88) log p ≤ (log T ) α Re( U p p − ih ) p / , h ∈ [0 , . Note that X h (1) = X h . For a small parameter | u | <
1, we consider the free energy of the perturbedprocess X h + uX h ( α ) at scale α :(12) F T ( β ; α, u ) = E (cid:20) log (cid:90) exp (cid:0) β ( X h + uX h ( α ) (cid:1) d h (cid:21) . The connection between the free energy (12) and the distribution of the correlation coefficients isthrough Gaussian integration by parts. Of course, for the process X h , this step is only approximate.It follows closely the work of Carmona & Hu [17] and Auffinger & Chen [10] on the universality ofthe free energy and overlap distributions in the Sherrington-Kirkpatrick model. Proposition 3.
For any < α < , (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α E (cid:104) G × β,T (cid:8) ( h, h (cid:48) ) : ρ ( h, h (cid:48) ) ≤ y (cid:9)(cid:105) d y − β log log T ∂F T ∂u ( β ; α, (cid:12)(cid:12)(cid:12)(cid:12) = o(1) . The free energy of the perturbed process is calculated using Kistler’s multiscale second momentmethod [26]. The treatment is similar to the one of Arguin & Ouimet [7] for the perturbed Gaussianfree field. The same result can be obtained by adapting the method of Bolthausen, Deuschel &Giacomin [12] and Daviaud [18] to the model as was done in [8, 9]. Kistler’s method is simpler andmore flexible. The result is better stated by first defining(13) f ( β, σ ) = (cid:40) β σ / β ≤ /σ , βσ − β ≥ /σ . Proposition 4.
For every β > and | u | < , the following limit holds lim T →∞ T F T ( β ; α, u ) = (cid:40) f (cid:0) β, (1 + u ) α + (1 − α ) (cid:1) if u < , αf ( β, (1 + u ) ) + (1 − α ) f ( β, if u ≥ . The theorem follows from the above two propositions. They are proved in Sections 3 and 4respectively. Estimates on the model needed for the proofs are given in Section 2.
Proof of Theorem 1.
We need to show that the distribution of ρ ( h, h (cid:48) ) converges weakly to β δ +(1 − β ) δ where δ a stands for the Dirac measure at a . Write x β,T ( s ) for E [ G × β,T (cid:8) ( h, h (cid:48) ) : ρ ( h, h (cid:48) ) ≤ s (cid:9) ].By compactness of the space of probability measures on [0 , x β,T )that converges weakly to x β as T → ∞ . We show that the limit x β is unique and equals x β ( s ) = 2 /β for 0 ≤ s <
1, thereby proving the claimed convergence.By definition of weak convergence, x β,T ( s ) converges to x β ( s ) at all points of continuity of s .Since x β is non-decreasing, this implies convergence almost everywhere. Thus, the dominatedconvergence theorem implies(14) lim T →∞ (cid:90) α x β,T ( s ) d s = (cid:90) α x β ( s ) d s , for 0 < α < T →∞ (cid:90) α x β,T ( s ) d s = lim T →∞ β log log T ∂F T ∂u ( β ; α, . L.-P. ARGUIN AND W. TAI
Since (cid:0) ((log log T ) − F T ( β ; α, u ) (cid:1) T is a sequence of convex functions of u , the limit of the derivativesis the derivative of the limit at any point of differentiability. Here the limit of the expectation ofthe free energy is given by Proposition 4, for u small enough so that β > /σ whenever β > T →∞ T F T ( β ; α, u ) = β (cid:16) (1 + u ) α + (1 − α ) (cid:17) / − u < αβ (1 + u ) + (1 − α ) β − u ≥ u = 0. Therefore, equations (14), (15)and (16) altogether imply (cid:90) α x β ( s ) d s = α β , for 0 < α < < α < α (cid:48) < α (cid:48) − α (cid:90) α (cid:48) α x β ( s ) d s = 2 β . By taking α (cid:48) − α →
0, we conclude from the Lebesgue differentiation theorem that x β ( s ) = 2 /β almost everywhere. Since x β is non-decreasing and right-continuous, this implies that x β ( s ) = 2 /β for every 0 ≤ s < (cid:3) Estimates on the model of zeta
In this section, we gather the estimates on the model of zeta needed for the proof of Propositions3 and 4. Most of these results are contained in [2]. We include them for completeness since we willneed to deal with a perturbed version of the process ( X h ). It is is important to point out that most(but not all!) of these estimates can be obtained for zeta itself with some more work, see [3].The essential input from number theory for the model is the Prime Number Theorem (7). It showsthat the density of the primes is approximately 1 / log p . This implies, for example, that (cid:80) p p − a < ∞ for a >
1. The equation (8) expressing the log-correlations for h (cid:54) = h (cid:48) is straightforward from thefollowing lemma by taking ∆ = | h − h (cid:48) | and by splitting the sum (5) into the ranges log p ≤ | h − h (cid:48) | − and | h − h (cid:48) | − < log p ≤ log T . Lemma 5.
Let ≤ P < Q < ∞ . Then for ∆ > , we have (17) (cid:88) P ≤ p primes ≤ Q cos(∆ · log p ) p = (cid:90) QP cos(∆ · log v ) v log v d v + O( e − c √ log P )= (cid:40) log log Q − log log P + O(1) for ∆ · log Q ≤ , O( · log P ) + O( e − c √ log P ) for ∆ · log P ≥ .Proof. Denote by Li( x ) = (cid:82) x y d y the logarithmic integal. Write E ( x ) for the function of boundedvariation π ( x ) − Li( x ) giving the error, and f ( x ) for cos(∆ · log x ) x . Clearly, we have (cid:88) P ≤ p ≤ Q f ( p ) = (cid:90) QP f ( x ) π ( dx ) = (cid:90) QP f ( x )log x dx + (cid:90) QP f ( x ) E ( dx ) . It remains to estimate the error term. By integration by parts, (cid:90) QP f ( x ) E ( dx ) = E ( Q ) f ( Q ) − E ( P ) f ( P ) − (cid:90) QP E ( x ) f (cid:48) ( x ) dx . S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 7
Note that f ( x ) is of the order of 1 /x and f (cid:48) ( x ) is of the order of 1 /x . Since E ( x ) = O ( xe − c √ log x ),the first claimed equality follows. For the dichotomy in the second equality, in the case ∆ · log Q ≤ y = log x (cid:90) QP f ( x )log x d x = (cid:90) log Q log P cos(∆ · y ) y d y = (cid:90) log Q log P (cid:18) y + O(∆ · y ) (cid:19) d y . The result follows by integration. In the case ∆ · log P ≥
1, we integrate by parts to get (cid:90) QP f ( x )log x dx = sin(∆ · y )∆ · y (cid:12)(cid:12)(cid:12) log Q log P + (cid:90) log Q log P sin(∆ · y )∆ · y d y . Both terms are O( · log P ) as claimed. (cid:3) Proposition 4 gives an expression for the free energy (12) of the perturbed process at scale α .For simplicity, we denote this process by(18) (cid:101) X h = (1 + u ) X h ( α ) + X h ( α,
1) for X h ( α,
1) = X h − X h ( α ), h ∈ [0 , X h at u = 0. The finite-dimensional distributions of ( (cid:101) X h ) can be explicitlycomputed. In fact, it is not hard to compute explicitly the moment generating function for anyincrement of ( X h ). We will only need the two-dimensional case. Proposition 6.
Let ≤ α < α ≤ . Consider X h ( α , α ) = X h ( α ) − X h ( α ) . We have for λ, λ (cid:48) ∈ R and h, h (cid:48) ∈ [0 , , E (cid:2) exp (cid:0) λX h ( α , α ) + λ (cid:48) X h (cid:48) ( α , α (cid:1)(cid:3) = C ( λ, λ (cid:48) ) · exp (cid:88) log p ≤ (log T ) α log p> (log T ) α p (cid:16) λ + λ (cid:48) + 2 λλ (cid:48) cos( | h − h (cid:48) | log p ) (cid:17) , where C = C ( λ, λ (cid:48) ) is bounded if λ and λ (cid:48) are bounded uniformly in T .Proof. The expression can be evaluated explicitly as follows. Since the U p ’s are independent, wecan first restrict the computation to a single p . Straightforward manipulations yield E (cid:104) exp (cid:16) p − / λ · Re( U p p − i h ) + p − / λ (cid:48) · Re( U p p − i h (cid:48) ) (cid:17)(cid:105) = E (cid:2) exp( aU p + ¯ aU p ) (cid:3) for a = (2 p / ) − ( λp − i h + λ (cid:48) p − i h (cid:48) ). By expanding the exponentials and using the fact that U p isuniform on the unit circle, we get(19) E (cid:2) exp( aU p + ¯ aU p ) (cid:3) = ∞ (cid:88) n =0 n (cid:88) k =0 a k ¯ a n − k n ! (cid:18) nk (cid:19) E [ U kp U n − kp ]= ∞ (cid:88) m =0 m !) (cid:32) λ + λ (cid:48) + 2 λλ (cid:48) cos( | h − h (cid:48) | log p )4 p (cid:33) m = 1 + (cid:32) λ + λ (cid:48) + 2 λλ (cid:48) cos( | h − h (cid:48) | log p )4 p (cid:33) + O( p − ) , where the O-term depends on λ, λ (cid:48) . The second equality follows from the fact that the expectationis non-zero only if k = n/
2. It remains to take the product over the range of p . The claim thenfollows from the fact that the sum of p − is finite by (7). (cid:3) L.-P. ARGUIN AND W. TAI
Proposition 6 yields Gaussian bounds in the large deviation regime we are interested in. Indeed,by Chernoff’s bound (optimizing over λ ), it implies that, for γ > P ( X h ( α , α ) > γ log log T ) (cid:28) exp (cid:18) − γ log log T ( α − α ) (cid:19) = (log T ) − γ α − α , where we used Lemma 5 to estimate the sum over primes. This supports the heuristic that X h ( α , α ) is approximately Gaussian of variance α − α log log T . This implies for (cid:101) X h in (18)(21) P (cid:16) (cid:101) X h > γ log log T (cid:17) (cid:28) exp (cid:18) − γ log log T (1 + u ) α + (1 − α ) (cid:19) = (log T ) − γ u )2 α +(1 − α ) . The same can be done for two points h, h (cid:48) . Using Lemma 5 again, we get(22) P ( X h ( α , α ) > γ log log T, X h (cid:48) ( α , α ) > γ log log T ) (cid:28) exp (cid:16) − γ log log T ( α − α ) (cid:17) if | h − h (cid:48) | ≤ (log T ) − α ,exp (cid:16) − γ log log T ( α − α ) (cid:17) if | h − h (cid:48) | ≥ (log T ) − α .This can be interpreted as follows. The increments are (almost) independent if the distance betweenthe points is larger than the relevant scales of the increments, and are (almost) perfectly correlatedif the distance is smaller than the scales.It is important to note that if α >
0, then a stronger estimate than the one of Proposition6 holds. This is because the sum over primes in (19) is then negligible since it is the tail of asummable series. This means that the constant C ( λ, λ (cid:48) ) is then 1 + O(1). This gives a preciseGaussian estimate by inverting the moment generating function (or the Fourier transform if wepick λ, λ (cid:48) ∈ C ). We omit the proof for conciseness and we refer to [2] where this is done using ageneral version of the Berry-Esseen theorem. Proposition 7 (see Propositions 2.9, 2.10, 2.11 in [2]) . For < α < α ≤ and < γ < , wehave for h ∈ [0 , , P ( X h ( α , α ) > γ log log T ) (cid:29) √ log log T exp (cid:18) − γ log log T ( α − α ) (cid:19) = (log T ) − γ α − α +o(1) . Moreover, if | h − h (cid:48) | > (log T ) − α , then P ( X h ( α , α ) > γ log log T, X h (cid:48) ( α , α ) > γ log log T ) = (1 + o(1)) P ( X h ( α , α ) > γ log log T ) . Since the process ( X h , h ∈ [0 , X h ( α , α )over an interval of width smaller than (log T ) − α behaves like a single value X h ( α , α ). This isdone in [2] by a chaining argument and we omit the proof for conciseness. Lemma 8 (Corollary 2.6 in [2]) . Let ≤ α < α ≤ . For every h ∈ [0 , and γ > , we have P (cid:18) max | h − h (cid:48) |≤ (log T ) − α X h (cid:48) ( α , α ) > γ log log T (cid:19) (cid:28) (log T ) − γ α − α . In particular, we have P (cid:18) max | h − h (cid:48) |≤ (log T ) − (cid:101) X h (cid:48) > γ log log T (cid:19) (cid:28) (log T ) − γ α (1+ u )+(1 − α ) . S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 9 Proof of Proposition 3
As mentioned in Section 1.3, the proof of Proposition 3 is based on an approximate Gaussianintegration by parts as in [17] and [10]. The following lemma is an adaptation for complex randomvariables of Lemma 4 in [17] .
Lemma 9.
Let ξ be a complex random variable such that E [ | ξ | ] < ∞ , and E [ ξ ] = E [ ξ ] = 0 . Let F : C → C be a twice continuously differentiable function such that for some M > , (cid:13)(cid:13) ∂ z F (cid:13)(cid:13) ∞ , (cid:13)(cid:13) ∂ z F (cid:13)(cid:13) ∞ < M , where (cid:107) f (cid:107) ∞ = sup z ∈ C | f ( z, z ) | . Then (cid:12)(cid:12)(cid:12) E (cid:2) ξF ( ξ, ξ ) (cid:3) − E [ | ξ | ] E (cid:2) ∂ z F ( ξ, ξ ) (cid:3)(cid:12)(cid:12)(cid:12) (cid:28) M E [ | ξ | ] . Proof.
Since E [ ξ ] = E [ ξ ] = 0, the left-hand side can be written as(23) E (cid:2) ξ (cid:0) F ( ξ, ξ ) − F (0 , − ξ∂ z F (0 , − ξ∂ z F (0 , (cid:1)(cid:3) − E [ | ξ | ] E (cid:104)(cid:16) ∂ z F ( ξ, ξ ) − ∂ z F (0 , (cid:17)(cid:105) . By Taylor’s theorem in several variables and the assumptions, the following estimates hold (cid:12)(cid:12) F ( ξ, ξ ) − F (0 , − ξ∂ z F (0 , − ξ∂ z F (0 , (cid:12)(cid:12) (cid:28) M | ξ | (cid:12)(cid:12) ∂ z F ( ξ, ξ ) − ∂ z F (0 , (cid:12)(cid:12) (cid:28) M | ξ | . Therefore the norm of (23) gives (cid:12)(cid:12)(cid:12) E (cid:2) ξF ( ξ, ξ ) (cid:3) − E [ | ξ | ] E (cid:2) ∂ z F ( ξ, ξ ) (cid:3)(cid:12)(cid:12)(cid:12) (cid:28) M ( E [ | ξ | ] + E [ | ξ | ] E [ | ξ | ]) . The claim then follows by H¨older’s inequality. (cid:3)
As in [17], the lemma can be applied to relate the derivative of the free energy to the two-pointcorrelations of the process.
Proposition 10.
For any p ≤ T , we have ∂∂u E (cid:20) log (cid:90) exp (cid:0) β ( X h ( T ) + u Re( U p p − ih − / ) (cid:1) d h (cid:21) (cid:12)(cid:12)(cid:12) u =0 = β E (cid:34)(cid:90) [0 , − cos( | h − h (cid:48) | log p ) p d G × β,T ( h, h (cid:48) ) (cid:35) + O( p − / ) . Proof.
Write for short ω p ( h ) = (2 p / ) − p − i h . Direct differentiation yields at u = 0(24) (cid:90) U p ω p ( h ) d G β,T ( h ) + (cid:90) U p ω p ( h ) d G β,T ( h ) . We make the dependence on U p in the measure G β,T explicit. For this, define Y p ( h ) = β (cid:88) q ≤ Tp (cid:54) = q Re (cid:18) U q q − ih q / (cid:19) . Clearly, Y p ( h ) is independent of U p by definition. Consider F p ( z, z ) = (cid:82) ω p ( h ) exp( βω p ( h ) z + βω p ( h ) z + Y p ( h )) d h (cid:82) exp( βω p ( h (cid:48) ) z + βω p ( h (cid:48) ) z + Y p ( h (cid:48) )) d h (cid:48) . Note that with this definition, the first integral in (24) is U p F p ( U p , U p ) and the second is its complexconjugate. This shows that the derivative of the expectation at u = 0 is E (cid:2) U p · F p ( U p , U p ) + U p · F p ( U p , U p ) (cid:3) . It remains to apply Lemma 9 with the function F p ( z, z ) and ξ = U p . Write for short for a function H on [0 , (cid:104) H (cid:105) ( z,z ) = (cid:82) H ( h ) e β ( z ω p ( h )+ z ω p ( h ))+ Y p ( h ) d h (cid:82) e β ( z ω p ( h )+ z ω p ( h ))+ Y p ( h ) d h . Direct differentiation of the above yields(25) ∂ z (cid:104) H (cid:105) ( z,z ) = β (cid:16) (cid:104) Hω p (cid:105) ( z,z ) − (cid:104) H (cid:105) ( z,z ) (cid:104) ω p (cid:105) ( z,z ) (cid:17) . In particular, for H = ω p , we get(26) ∂ z F p ( z, z ) = β (cid:16)(cid:10) | ω p | (cid:11) ( z,z ) − | (cid:104) ω p (cid:105) ( z,z ) | (cid:17) . When evaluated at z = U p , this is by definition of ω p (27) ∂ z F p ( U p , U p ) = β (cid:90) [0 , ( p − − p − cos( | h − h (cid:48) | log p )) d G × β,T ( h, h (cid:48) ) . Clearly, | ω p | ≤ p − / . Therefore the second derivatives are easily checked to be bounded by O( p − / )by applying the formula (25) to each term of (26). The statement of the lemma then follows fromLemma 9 and (27), after noticing that the second term of (24) is the conjugate of the first. (cid:3) The proof of Proposition 3 is an application of Proposition 10 to a range of primes.
Proof of Proposition 3.
Recall the definition of ρ ( h, h (cid:48) ) in equations (6) and (9). On one hand,Fubini’s theorem directly implies that(28) (cid:90) α G × β,T (cid:8) ( h, h (cid:48) ) : ρ ( h, h (cid:48) ) ≤ y (cid:9) d y = (cid:90) [0 , (cid:18)(cid:90) α { ρ ( h,h (cid:48) ) ≤ r } d r (cid:19) d G β,T ( h, h (cid:48) )= (cid:90) [0 , (cid:0) α − ρ ( h, h (cid:48) ) (cid:1) { ρ ( h,h (cid:48) ) ≤ α } d G β,T ( h, h (cid:48) ) . It remains to check on the other hand that the derivative in the proposition is close to the expec-tation of the above. Direct differentiation of (12) at u = 0 yields by Proposition 10(29) ∂F T ∂u ( β ; α,
0) = β (cid:90) [0 , (cid:88) log p ≤ (log T ) α E (cid:2) p − (1 − cos( | h − h (cid:48) | log p )) d G × β,T ( h, h (cid:48) ) (cid:3) +O( (cid:88) p ≤ e (log T ) α p − / ) . The error term is of order one by (7). Similarly, if | h − h (cid:48) | ≤ (log T ) − α , the sum in the integral isby (17) (cid:88) log p ≤ (log T ) α p − (1 − cos( | h − h (cid:48) | log p )) = α log log T − α log log T + O(1) = O(1) . On the other hand, if | h − h (cid:48) | > (log T ) − α , the sum can be divided into three parts (cid:88) log p ≤ (log T ) α p − − (cid:88) log p ≤| h − h (cid:48) | − p − cos( | h − h (cid:48) | log p ) − (cid:88) | h − h (cid:48) | − < log p ≤ (log T ) α p − cos( | h − h (cid:48) | log p ) . When equation (17) is applied to each of the parts, this equals α log log T − log | h − h (cid:48) | − + O(1) . S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 11
Furthermore, recall from (9) that ρ ( h, h (cid:48) ) log log T differs from log | h − h (cid:48) | − by o(log log T ). Thisimplies that the conditions on log | h − h (cid:48) | − can be replaced by ρ ( h, h (cid:48) ) log log T at a cost of a termo(log log T ) (since the sum would differ by a range of log p of at most o(log T ) primes). All theseobservations together imply1log log T (cid:88) log p ≤ (log T ) α p − (1 − cos( | h − h (cid:48) | log p )) = (cid:18) α − log | h − h (cid:48) | − log log T (cid:19) { ρ ( h,h (cid:48) ) ≤ α } + o(1) . We finally conclude by putting the right side back in the integral of (29) and by using (9) that2 β log log T ∂F T ∂u ( β ; α,
0) = (cid:90) [0 , (cid:0) α − ρ ( h, h (cid:48) ) (cid:1) E (cid:2) { ρ ( h,h (cid:48) ) ≤ α } d G × β,T ( h, h (cid:48) ) (cid:3) + o(1) . This matches the first claim (28) by an error o(1) thereby proving the proposition. (cid:3) Proof of Proposition 4
We write (cid:101) X h = (1 + u ) X h ( α ) + X h ( α,
1) as in equation (18). The limit of the free energy of thisprocess is obtained by Laplace’s method once the measure of high points is known, cf. Lemma 12.The proof of Lemma 12 is based on a similar computation of [7] for the two-dimensional Gaussianfree field based on Kistler’s multiscale second moment method [26]. But first, we need an a priori restriction on the maximum of the process ( (cid:101) X h ). The maximum depends on the value of theparameter u as expected from GREM models. With this in mind, we define(30) γ (cid:63) = (cid:16) (1 + u ) α + (1 − α ) (cid:17) / if u ≤ u ) α + (1 − α ) if u > u = 0 and that γ (cid:63) > u >
0, and γ (cid:63) < u < (cid:101) X h . Lemma 11.
For any ε > , lim T →∞ P (cid:18) max h ∈ [0 , (cid:101) X h > (1 + ε ) γ (cid:63) log log T (cid:19) = 0 . Proof.
This is a consequence of Lemma 8 which shows that the large values of X h ( α ) are wellapproximated by points at a distance (log T ) − α . In the case u ≤
0, we use the lemma with α = 1.Without loss of generality, suppose that log T is an integer and consider I k , k ≤ log T , a collectionof intervals of length (log T ) − that partitions [0 , P (cid:18) max h ∈ [0 , (cid:101) X h > (1 + ε ) γ (cid:63) log log T (cid:19) ≤ log T (cid:88) k =1 P (cid:18) max h ∈ I k (cid:101) X h > (1 + ε ) γ (cid:63) log log T (cid:19) . Lemma 8 applied to (cid:101) X h then implies P (cid:18) max h ∈ [0 , (cid:101) X h > (1 + ε ) γ (cid:63) log log T (cid:19) (cid:28) (log T ) exp − (1 + ε ) (cid:16) (1 + u ) α + (1 − α ) (cid:17) log log T )(1 + u ) α + (1 − α ) ≤ (log T ) − (1+ ε ) , which goes to 0 as claimed.In the case u >
0, an extra restriction is needed since the large values of X h ( α ) are themselveslimited. Proceeding as above, without loss of generality, assume that (log T ) α , (log T ) − α andlog T are integers. Consider the collection of intervals J j , j ≤ (log T ) α , that partitions [0 ,
1] into intervals of length (log T ) − α . Each J j is again partitioned into intervals I jk , k ≤ (log T ) − α , oflength (log T ) − (1 − α ) . Then Lemma 8 implies(31) P (cid:18) ∃ j : max h ∈ J j X h ( α ) > (1 + ε ) · α log log T (cid:19) → . Therefore, the probability of the maximum of (cid:101) X h can be restricted as follows: P (cid:18) max h ∈ [0 , (cid:101) X h > (1 + ε ) γ (cid:63) log log T (cid:19) = P (cid:16) ∃ h ∈ [0 ,
1] : (cid:101) X h > (1 + ε ) γ (cid:63) log log T, X h ( α ) ≤ (1 + ε ) · α log log T (cid:17) + o(1) (cid:28) (cid:88) j,k (1+ ε ) · α log log T (cid:88) q =0 P (cid:18) max h ∈ J j X h ( α ) ∈ [ q, q + 1] , max h ∈ I jk X h ( α, > (1 + ε ) γ (cid:63) log log T − (1 + u )( q + 1) (cid:19) . The last inequality is obtained by a union bound on the partition ( I jk ) and by splitting the values ofthe maximum of X h ( α ) on the range [0 , (1 + ε ) α log log T ]. (Note that X h ( α ) is symmetric thus themaximum is greater than 0 with large probability.) By independence between ( X h ( α ) , h ∈ [0 , X h ( α, , h ∈ [0 , (cid:28) exp (cid:18) − q α log log T (cid:19) · exp − (cid:16) (1 + ε ) γ (cid:63) log log T − (1 + u )( q + 1) (cid:17) (1 − α ) log log T . On the interval [0 , (1 + ε ) α log log T ], this is maximized at the endpoint q = (1 + ε ) α log log T . (Thisis where the case u < q there is within the interval. See Remark 13 formore on this.) Putting this back in (32) and summing over j, k , and q finally give the estimate: P (cid:18) max h ∈ [0 , (cid:101) X h > (1 + ε ) γ (cid:63) log log T (cid:19) (cid:28) (log log T ) · (log T ) α · e − (cid:16) (1+ ε ) α log log T (cid:17) α log log T · (log T ) − α · e − (cid:16) (1+ ε )(1 − α ) log log T (cid:17) − α ) log log T (cid:28) (log log T ) · log T − (1+ ε ) = o(1) . This concludes the proof of the lemma. (cid:3)
Consider for 0 < α ≤ | u | < γ -high points(33) E α,u ( γ ; T ) = 1log log T log Leb { h ∈ [0 ,
1] : (cid:101) X h > γ log log T } , < γ < γ (cid:63) . The limit of these quantities in probability can be computed following [7].
Lemma 12.
The limit E α,u ( γ ) = lim T →∞ E α,u ( γ ; T ) exists in probability. We have for u < , E α,u ( γ ) = − γ (1 + u ) α + (1 − α ) , and for u ≥ , E α,u ( γ ) = (cid:40) − γ (1+ u ) α +(1 − α ) if γ < γ c − α − ( γ − (1+ u ) α ) (1 − α ) if γ ≥ γ c . for γ c = (1 + u ) α + (1 − α )(1 + u ) . S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 13
Remark 13.
The dichotomy in the log-measure is due to the fact that for h with values beyond γ c log log T , the intermediate values X h ( α ) is restricted by the maximal level α log log T . Moreprecisely, consider(34) M T =Leb { h ∈ [0 ,
1] : (cid:101) X h > γ log log T }M (cid:48) T =Leb { h ∈ [0 ,
1] : (1 + u ) X h ( α ) ≥ λ log log T }M (cid:48)(cid:48) T =Leb { h ∈ [0 ,
1] : (1 + u ) X h ( α ) ≥ λ log log T, X h ( α, ≥ ( γ − λ ) log log T } . Clearly, we must have M (cid:48)(cid:48) T ≤ M T . It turns out that M (cid:48)(cid:48) T and M T are comparable for an optimalchoice of λ given by, when u < λ (cid:63) = γ (1 + u ) α (1 + u ) α + 1 − α , γ < γ (cid:63) , and when u > λ (cid:63) = (cid:40) γ (1+ u ) α (1+ u ) α +1 − α if 0 < γ < γ c , (1 + u ) α if γ c ≤ γ < γ (cid:63) . One can see this at a heuristic level by considering first moments. Since the maximum of X h is wellapproximated by the maximum over lattice points spaced (log T ) − apart, there should be γ -highpoints only if(37) (log T ) · M (cid:48)(cid:48) T ≥ . Moreover, we have that if M (cid:48) T = 0, then M (cid:48)(cid:48) T = 0. And the maximum of X h ( α ) is well approximatedby the maximum over lattice points spaced (log T ) − α apart, so there should be γ -high points onlyif(38) (log T ) α · M (cid:48) T ≥ . Since X h ( α ) and X h ( α,
1) are approximately Gaussian with variance α log log T and (1 − α ) log log T ,the following should hold approximatelylog E [(log T ) α · M (cid:48) T ]log log T = α − λ (1 + u ) α + o(1)log E [(log T ) · M (cid:48)(cid:48) T ]log log T = 1 − λ (1 + u ) α − ( γ − λ ) − α + o(1)Together with conditions (37) and (38), we obtain constraints on the value of λ : α − λ (1 + u ) α ≥ , (39) 1 − λ (1 + u ) α − ( γ − λ ) − α ≥ . (40)By maximizing M (cid:48)(cid:48) T , under the constraints (39) and (40), one gets the values (35) and (36) for λ .With Remark 13 in mind, we are ready to bound the log-measure. Proof of Lemma 12. Upper bound on the log-measure.
For 0 < γ < γ (cid:63) , consider M T as in (34).We need to show that for ε > T →∞ P (cid:16) M T > (log T ) E α,u ( γ )+ ε (cid:17) = 0 . We first prove the easiest cases where u ≥ γ < γ c , as well as u ≤
0. Let ε >
0. And write V = 1 − α + (1 + u ) α for short. Observe that by Markov’s inequality and Fubini’s theorem(42) P ( M T > (log T ) − γ V + ε ) ≤ (log T ) γ V − ε (cid:90) P ( (cid:101) X h > γ log log T ) d h = (log T ) γ V − ε P ( (cid:101) X h > γ log log T ) , where we used the fact that the variables (cid:101) X h , h ∈ [0 , P ( (cid:101) X h >γ log log T ) (cid:28) exp( − γ log log T /V ) by Equation (21), the claim (41) follows.The case u > γ > γ c is more delicate as we need to control the values at scale α . For ε (cid:48) > P (cid:0) Leb { h ∈ [0 ,
1] : X h ( α ) > ( α + ε (cid:48) ) log log T } > (cid:1) ≤ P (cid:0) ∃ h ∈ [0 ,
1] : X h ( α ) > ( α + ε (cid:48) ) log log T (cid:1) → . The same hold by symmetry for − X h ( α ). This implies P (cid:16) M T > (log T ) E α,u ( γ )+ ε (cid:17) = P (cid:16) Leb { h : (cid:101) X h > γ log log T, | X h ( α ) | ≤ ( α + ε (cid:48) ) log log T } > (log T ) E α,u ( γ )+ ε (cid:17) + o(1) . It remains to prove that the first term is o(1). As in the proof of Lemma 11, we consider thepartition of [0 ,
1] by intervals J j , j ≤ (log T ) α , and the sub-partition I jk , k ≤ (log T ) − α . We alsodivide the interval [ − ( α + ε (cid:48) ) log log T, ( α + ε (cid:48) ) log log T ] into intervals [ q, q + 1]. Then by Markov’sinequality and the additivity of the Lebesgue measure(44) P (cid:16) Leb { h : (cid:101) X h > γ log log T, | X h ( α ) | ≤ ( α + ε (cid:48) ) log log T } > (log T ) E α,u ( γ )+ ε (cid:17) ≤ (log T ) −E α,u ( γ ) − ε (cid:88) j,k (cid:88) | q |≤ ( α + ε (cid:48) ) log log T E (cid:104) Leb { h ∈ I jk : (cid:101) X h > γ log log T, X h ( α ) ∈ [ q, q + 1] } (cid:105) ≤ (log T ) −E α,u ( γ ) − ε (cid:88) j,k (cid:88) | q |≤ ( α + ε (cid:48) ) log log T (log T ) − P ( X h ( α, > γ log log T − (1 + u )( q + 1) , X h ( α ) ≥ q } ) . The last line follows from Fubini’s theorem and the fact that Leb( I jk ) = (log T ) − . The probabilitiescan be bounded by the Gaussian bound (20) P ( X h ( α, > γ log log T − (1 + u )( q + 1) , X h ( α ) ≥ q } ) (cid:28) exp (cid:18) − q α log log T (cid:19) · exp − (cid:16) γ log log T − (1 + u )( q + 1) (cid:17) (1 − α ) log log T . It is easily checked that the expression is maximized at q > ( α + ε (cid:48) ) log log T for ε (cid:48) . Moreover, at theoptimal q = ( α + ε (cid:48) ) log log T in the considered range, the probability equals (1 + o(1))(log T ) E α,u ( γ ) .Using this observation to bound the probability for each q in (44), we get P (cid:16) Leb { h : (cid:101) X h > γ log log T, | X h ( α ) | ≤ ( α + ε (cid:48) ) log log T } > (log T ) E α,u ( γ )+ ε (cid:17) (cid:28) (log T ) − ε log log T = o(1) . This finishes the proof of the upper bound.
S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 15
Lower bound on the log-measure.
For ε >
0, the goal is to show(45) P (cid:16) M T > (log T ) E α,u ( γ ) − ε (cid:17) → T → ∞ .This is done using the Paley-Zygmund inequality, which states that for a random variable M ≥ ≤ η T ≤ P ( M > η T E [ M ]) ≥ (1 − η T ) E [ M ] E [ M ] . We will have η T →
0, so the main task will be to demonstrate(47) E (cid:2) M (cid:3) = (1 + o (1)) E [ M ] . This cannot be achieved when M = M T because of the correlations in X h . To overcome thisproblem, we define a modified version of M T by coarse graining the field as described in [26].For K ∈ N (that will depend eventually on ε ), assume without loss of generality that { , K , K , . . . , K − K , } is a partition of [0 ,
1] that is a refinement of { , α, } . Consider λ < λ (cid:63) as defined in (35) and (36),and δ > ε ). Define the events for the K -level coarse increments:(48) J h ( m ) = (cid:110) (1 + u ) X h (cid:0) m − K , mK (cid:1) ≥ (1 + δ ) λ log log TαK (cid:111) if m = 1 , . . . , αK, (cid:110) X h (cid:0) m − K , mK (cid:1) ≥ (1 + δ ) ( γ − λ ) log log T (1 − α ) K (cid:111) if m = αK + 1 , . . . , K. Moreover, define the sets(49) A = { h : J h ( m ) occurs ∀ m = 2 , . . . , K } B = { h : (1 + u ) X h (cid:18) K (cid:19) ≤ − δ T } . Note that if h ∈ A , by adding up the inequalities in J h ( m ), we have for K large enough,(50) (cid:101) X h − (1 + u ) X h (cid:18) K (cid:19) ≥ (1 + δ ) (cid:18) γ − λαK (cid:19) log log T ≥ (cid:18) γ + δ (cid:19) log log T .
Therefore this implies the inclusion
A ⊂ (cid:110) h ∈ [0 ,
1] : (cid:101) X h ≥ γ log log T (cid:111) ∪ B , so that M T ≥ Leb( A ) − Leb( B ). Equation (20) and Fubini’s theorem shows that E [Leb( B )] (cid:28) (log T ) − δ K u )2 . For K large enough, Markov’s inequality then implies P (cid:18) Leb (cid:26) h ∈ [0 ,
1] : (1 + u ) X h (cid:18) K (cid:19) ≤ − δ T (cid:27) ≤ (log T ) E α,u ( γ ) − ε (cid:19) → . The proof of (45) is then reduced to show(51) P (Leb( A ) > T ) E α,u ( γ ) − ε ) = P (Leb( A ) > η T E [Leb( A )]) → , where η T is defined by 2(log T ) E α,u ( γ ) − ε = η T E [Leb( A )].Following (46), we first show η T →
0. By (49), Fubini’s theorem, and independence,(52) E [Leb( A )] = (cid:90) K (cid:89) m =2 P ( J h ( m )) d h = K (cid:89) m =2 P ( J h ( m )) , since the X h ’s are identically distributed. By Proposition 7,(53) P ( J h ( m )) (cid:29) (log T ) − (1+ δ )2 λ α K (1+ u )2 +o(1) when m = 1 , . . . , αK, (log T ) − (1+ δ )2( γ − λ )2(1 − α )2 K +o(1) when m = αK + 1 , . . . , K. Thus, by (52) and (53), we have(54) E [Leb( A )] (cid:29) (log T ) − (1+ δ )2 λ α (1+ u )2 − (1+ δ )2( γ − λ )2(1 − α ) (log T ) (1+ δ )2 λ α K (1+ u )2 +o(1) . We can take λ close enough to λ (cid:63) , δ small enough, and K large enough so that E [Leb( A )] (cid:29) (log T ) − λ(cid:63) α (1+ u )2 − ( γ − λ(cid:63) )2(1 − α ) − ε = (log T ) E α,u ( γ ) − ε , where we replace the value of λ (cid:63) of (35) and (36). This shows that η T →
0. Observe that, we alsohave the reverse inequality(55) E [Leb( A )] (cid:28) (log T ) E α,u ( γ )+ ε , using (20) instead of Proposition 7.It remains to show (47). By independence of increments and Fubini’s theorem, we have(56) E [Leb( A ) ] = (cid:90) (cid:90) K (cid:89) m =2 P ( J h ( m ) ∩ J h (cid:48) ( m )) d h d h (cid:48) . We split the integral into four integrals: I for | h − h (cid:48) | > (log T ) − K , II for (log T ) − K ≤ | h − h (cid:48) | ≤ (log T ) − K , III for (log T ) − rK < | h − h (cid:48) | ≤ (log T ) − ( r − K , r = 2 , . . . K , and IV for | h − h (cid:48) | ≤ (log T ) − .We will show that I = E [Leb( A )] (1 + o(1)) and the others o( E [Leb( A )] ). • For II, note that Leb × { ( h, h (cid:48) ) : (log T ) − K ≤ | h − h (cid:48) | ≤ (log T ) − K } (cid:28) (log T ) − K . Moreover,by (22) and Proposition 7, we have P ( J h ( m ) ∩ J h (cid:48) ( m )) (cid:28) P ( J h ( m )) . This implies II =o( E [Leb( A )] ). • For IV, note that clearly P ( J h ( m ) ∩ J h (cid:48) ( m )) ≤ P ( J h ( m )). Thus, IV (cid:28) (log T ) − E [Leb( A )].Using (55) and the fact that 1 + E α,u ( γ ) > γ < γ (cid:63) , one gets IV = o( E [Leb( A )] ). • For I, note that Leb × { ( h, h (cid:48) ) : | h − h (cid:48) | > (log T ) − K } = 1 + o(1). Moreover, by Proposition7, P ( J h ( m ) ∩ J h (cid:48) ( m )) = (1 + o(1)) P ( J h ( m )) . This implies I = (1 + o(1)) E [Leb( A )] ). • For III, the integral is a sum over r = 2 , . . . , K of integrals of pairs with (log T ) − rK < | h − h (cid:48) | ≤ (log T ) − ( r − K . The measure of this set is (cid:28) (log T ) − ( r − K . For fix r , the integrandis K (cid:89) m =2 P ( J h ( m ) ∩ J h (cid:48) ( m )) ≤ r (cid:89) m =2 P ( J h ( m )) K (cid:89) m = r +1 P ( J h ( m ) ∩ J h (cid:48) ( m )) (cid:28) r (cid:89) m =2 P ( J h ( m )) K (cid:89) m = r +1 P ( J h ( m )) , where the last line follows by (22) and Proposition 7. Putting all this together and factoringthe square of the one-point probabilities, one getsIII (cid:28) E [Leb( A )] K (cid:88) r =2 (log T ) − ( r − K r (cid:89) m =2 (cid:16) P ( J h ( m )) (cid:17) − . We show (cid:81) rm =2 (cid:16) P ( J h ( m )) (cid:17) − < (log T ) ( r − K uniformly in T . This finishes the proof sincethe sum is then the tail of a convergent geometric series. In the case u <
0, since λ < λ (cid:63) , S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 17 and (1 + δ ) γ < γ (cid:63) for δ small, we have by (53), P ( J h ( m )) − (cid:28) (log T ) λ(cid:63) α K (1+ u )2 if m ≤ αK (log T ) ( γ(cid:63) − λ(cid:63) )2(1 − α )2 K if m = αK + 1 , . . . , K .By the definition of λ (cid:63) and γ (cid:63) = V / , this implies r (cid:89) m =2 (cid:16) P ( J h ( m )) (cid:17) − (cid:28) (log T ) (1+ u )2 V r − K if r ≤ αK (log T ) α (1+ u )2 V + V r − αKK if r = αK + 1 , . . . , K .Since u <
0, it is straightforward to check that the exponent is smaller than r − K as claimed.The case u ≥ γ c ≤ γ < γ (cid:63) and 0 < γ < γ c .We omit the proof for conciseness. (cid:3) We now have all the results to finish the proof of Proposition 4 using Laplace’s method.
Proof of Proposition 4.
We first prove the limit in probability. The convergence in L , and inparticular the convergence of the expectation, will be a consequence of Lemma 14 below. For fixed ε > M ∈ N , consider γ j = j (1 + ε ) M γ (cid:63) ≤ j ≤ M , and the event(57) A = M (cid:92) j =1 (cid:110) (log T ) E α,u ( γ j ) − ε ≤ Leb { h : (cid:101) X h > γ j log log T } ≤ (log T ) E α,u ( γ j )+ ε (cid:111)(cid:92) (cid:110) Leb { h : (cid:101) X h > γ M log log T } = 0 (cid:111) . By Lemma 11 and Lemma 12, we have that P ( A c ) → T → ∞ . It remains to prove that thefree energy is close to the claimed expression on the event A . On one hand, the following upperbound holds on A : (cid:90) exp β (cid:101) X h d h ≤ M (cid:88) j =1 (cid:90) exp β (cid:101) X h { (log T ) γj − 0, the optimal γ is βV / γ < γ c , i.e., β < / (1 + u ). If γ > γ c , then the optimal γ is(1 + u ) α + β (1 − α )2 until it equals γ (cid:63) . This happens at β ≥ 2. Putting all this together, we obtainthat max γ ∈ [0 ,γ (cid:63) ] { βγ + E α,u ( γ ) } = β (cid:0) (1+ u ) α +(1 − α ) (cid:1) if β < u ) β (1 + u ) α − α + β (1 − α )4 if u ) ≤ β < β (cid:16) (1 + u ) α + (1 − α ) (cid:17) − β ≥ . This corresponds to the expression in Proposition 4 expressed in terms of (13). (cid:3) Lemma 14. The sequence of random variables (cid:16) T log (cid:90) exp (cid:0) β ( X h + uX h ( α ) (cid:1) d h (cid:17) T > is uniformly integrable. In particular, the convergence in probability of the sequence is equivalentto the convergence in L .Proof. Write for short f T = (log log T ) − log (cid:90) exp β (cid:101) X h d h . We need to show that for any ε > 0, there exists C large enough so that uniformly in T , E [ | f T | {| f T ) | >C } ] < ε . It is easy to check that(58) E [ | f T | {| f T | >C } ] = (cid:90) ∞ C P ( f T > y ) d y + C P ( f T > C ) + (cid:90) − C −∞ P ( f T < y ) d y + C P ( f T < − C ) . Therefore, it remains to get a good control on the right and left tail of f T . For the right tail,observe that by Markov’s inequality P ( f T > y ) = P (cid:16) (cid:90) exp β (cid:101) X h d h > (log T ) y (cid:17) ≤ (log T ) − y E (cid:20)(cid:90) exp β (cid:101) X h d h (cid:21) . Using Proposition 6 and Fubini’s theorem, we get P ( f T > y ) (cid:28) (log T ) ((1+ u ) α +(1 − α )) β − y . S THE RIEMANN ZETA FUNCTION IN A SHORT INTERVAL A 1-RSB SPIN GLASS ? 19 This implies (cid:90) ∞ C P ( f T > y ) d y + C P ( f T > C ) (cid:28) (log T ) ((1+ u ) α +(1 − α )) β − C log log T + C (log T ) ((1+ u ) α +(1 − α )) β − C . It suffices to take C > ((1 + u ) α + (1 − α )) β for this to be uniformly small in T . 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