On Riemann zeroes, Lognormal Multiplicative Chaos, and Selberg Integral
aa r X i v : . [ m a t h . P R ] N ov On Riemann zeroes, Lognormal Multiplicative Chaos, andSelberg Integral
Dmitry Ostrovsky
Abstract
Rescaled Mellin-type transforms of the exponential functional of the Bourgade-Kuan-Rodgersstatistic of Riemann zeroes are conjecturally related to the distribution of the total mass of thelimit lognormal stochastic measure of Mandelbrot-Bacry-Muzy. The conjecture implies that anon-trivial, log-infinitely divisible probability distribution is associated with Riemann zeroes.For application, integral moments, covariance structure, multiscaling spectrum, and asymptoticsassociated with the exponential functional are computed in closed form using the known mero-morphic extension of the Selberg integral.
Keywords:
Riemann zeroes, multiplicative chaos, Selberg integral, multifractal stochastic measure,gaussian free field, infinite divisibility, double gamma function.
Mathematics Subject Classification (2010):
In this paper we contribute to the literature on the statistical distribution of Riemann zeroes in themesoscopic regime. The study of the values of the Riemann zeta function in the mesoscopic regimewas pioneered by Selberg [57], [58] and then extended to the distribution of zeroes by Fujii [23],Hughes and Rudnick [32], Bourgade [14], and most recently by Bourgade and Kuan [15], Rodgers[53], and Kargin [35]. Assuming the Riemann hypothesis (except for Selberg), these authors rigor-ously established various central limit theorems for the distribution of Riemann zeroes. The principaltechnical tools that were used to obtain these theorems were Selberg’s formula for z ′ / z , explicitformulas of Guinand and Weil, and certain moment calculations. Alternatively, beginning with theseminal work of Montgomery [40], a great deal of progress has been made in formulating preciseconjectures about the statistical distribution of the zeroes. These conjectures are all motivated by theempirical fact that the statistical properties of the zeroes are very close to those of eigenvalues of largeHermitian matrices with independent entries, i.e. the so-called GUE matrices, up to small arithmeticcorrections, and calculations are typically justified by means of semi-classical methods for quantumchaotic systems, Keating-Snaith philosophy of modeling the value distribution of the Riemann zetafunction on the critical line by the characteristic polynomials of certain large random matrices, andconjectural forms of the approximate functional equation for the zeta function. For example, Berry[11] calculated the GUE term and arithmetic corrections for the number variance, Bogomolny andKeating [12] did the same for the pair correlation function, which was later extended to multiple-point correlations by Conrey and Snaith [17] and Bogomolny and Keating [13], Keating and Snaith[36] calculated the moments of the zeta function on the critical line, Conrey et. al. [16] formulated the1atios conjecture for its average values, Farmer et. al. [20] and Fyodorov and Keating [30] estimatedthe magnitude of its extreme values on the critical line, to name a few.In this paper we conjecture a mod-gaussian limit theorem associated with the distribution of Rie-mann zeroes in the mesoscopic regime by combining the approach of Bourgade and Kuan with ourprevious work on the limit lognormal stochastic measure (also known as lognormal multiplicativechaos) and Selberg integral. Bourgade and Kuan and Rodgers independently proved that a class oflinear statistics of Riemann zeroes converge to gaussian vectors and, most importantly, computedthe covariance of the limiting vector explicitly. The starting point of our approach is that this lim-iting gaussian vector approximates the centered gaussian free field when the statistic is based on asmoothed indicator function of subintervals of the unit interval, and it approximates the gaussian freefield plus an independent gaussian random variable when the statistic is based on a smoothed indi-cator function of subintervals of a particular unbounded interval. The limit lognormal measure isdefined as a limit of the exponential functional of the gaussian free field, hence by taking the Mellintransform of the exponential functional of the Bourgade-Kuan-Rodgers statistic in an appropriatelyrescaled limit, we conjecturally obtain the Mellin transform of the total mass of the measure. In a se-ries of papers, cf. [43], [44], [45], [48], we investigated the total mass and made a precise conjectureabout its probability distribution. The positive integral moments of the total mass are known to begiven by the classical Selberg integral. In [45] we rigorously constructed a probability distribution(called the Selberg integral distribution) whose n th moment coincides with the Selberg integral ofdimension n and conjectured that this distribution is the same as the distribution of the total mass.Thus, the main result of this paper is a conjecture that particular rescaled limits of two Mellin-typetransforms of the exponential functional of the Bourgade-Kuan-Rodgers statistic corresponding toa smoothed indicator function of certain bounded or unbounded intervals coincide with the Mellintransform of the Selberg integral distribution. In [48], [49], [50] we established many properties ofthe Mellin transform so that we can make a number of precise statements about our rescaled limit ascorollaries of the main conjecture. The type of rescaling and convergence that we use in this paperis closely related to the rescaling that was used by Keating and Snaith [36] and Nikeghbali and Yor[42], formalized by Jacod et. al. [33] in their theory of mod-gaussian convergence, and significantlyextended in recent publications of Feray et. al. [21] and M´eliot and Nikeghbali [39].The main technical innovation of our work is the explicit use of the gaussian free field, limitlognormal measure, Selberg integral, and Selberg integral distribution in the context of the statisticaldistribution of Riemann zeroes. We note that the Selberg integral and Selberg integral distributionpreviously appeared, respectively, in conjectures of Keating and Snaith [36] about the moments andof Fyodorov and Keating [30] about extreme values of the Riemann zeta function on the critical line.These conjectures are based on the analogy between the value distribution of z ( / + it ) and that ofthe characteristic polynomials of certain large random matrices. Our conjecture deals instead with thezeroes of the zeta function on the critical line and is based on the convergence of particular statisticsof the zeroes to the gaussian free field or its centered version. In particular, we prove that our statisticsor, equivalently, particular integrals of Im log z ( / + it ) along the critical line, exhibit logarithmiccorrelations and calculate the corresponding covariances explicitly. We believe that these calculationsare new. It is more natural to define the Mellin transform as R ¥ x q f ( x ) dx as opposed to the usual R ¥ x q − f ( x ) dx for ourpurposes. The terms “probability distribution”, “law”, and “random variable” are used interchangeably in this paper. The idea that Imlog z ( / + it ) is logarithmically correlated is not new. Keating and Snaith [36] and later Farmer et. al. [20] argued that Im log z ( / + it ) can be modeled by the imaginary part of the logarithm of the characteristic polynomial i.e. the measure is multifractal. The aforementioned problemsin mathematical physics all exhibit multifractal behavior so that the significance of our conjectureextends beyond the distribution of Riemann zeroes per se for it suggests that the phenomenon ofmultifractality might have a number theoretic origin in the sense that the distribution of Riemannzeroes (conjecturally) provides a natural model for such phenomena.We do not have a mathematically rigorous proof of our conjecture and provide instead some exactcalculations (a ”physicist’s proof”) that explain how we arrived at it. If one assumes the conjectureto be true, the resulting corollaries are mathematically rigorous and their proofs can be found in [48]and [50].The plan of this paper is as follows. In Section 2 we give a brief review of the key results ofBourgade and Kuan and Rodgers and then state our conjecture and its implications. This section doesnot require any knowledge of the limit lognormal measure. In Section 3 we review the limit lognormalmeasure and the Selberg integral distribution. In Section 4 we present a heuristic derivation of ourconjecture. Conclusions are given in Section 5. We begin this section with a brief description of the statistic of Riemann zeroes that was introducedby Bourgade and Kuan [15] and Rodgers [53] (henceforth referred to as the BKR statistic), followingthe approach and notations of Bourgade and Kuan. The Riemann zeta function is defined by z ( s ) , ¥ (cid:229) m = ( m + ) − s , Re ( s ) > , (1)and is continued analytically to the complex plane having a simple pole at s = . Its non-trivial zeroesare known to be located in the critical strip 0 < Re ( s ) < ( s ) = / , cf. [59] for details. Assuming the Riemannhypothesis, we write non-trivial zeroes of the Riemann zeta function in the form { / + i g } , g ∈ R . Let l ( t ) be a function of t > ≪ l ( t ) ≪ log t (2) of CUE matrices, and Hughes et. al. [31] proved that the latter is logarithmically correlated, thereby conjecturing thesame about Imlog z ( / + it ) . The novelty of our work is the computation of the logarithmic covariance structure ofImlog z ( / + it ) from first principles. We will use the symbol , to mean that the left-hand side is defined by the right-hand side.
3n the limit t → ¥ , where the number theoretic notation a ( t ) ≪ b ( t ) means a ( t ) = o (cid:0) b ( t ) (cid:1) . Let w denote a uniform random variable over ( , ) , g ( t ) , l ( t )( g − w t ) , and define the statistic S t ( f ) , (cid:229) g f (cid:0) g ( t ) (cid:1) − log t pl ( t ) Z f ( u ) du (3)given a test function f ( x ) . The class of test functions H / that was considered in [15] is primarilydefined by the condition h f , f i < ¥ , where h f , g i , Re Z | w | ˆ f ( w ) ˆ g ( w ) dw , (4) = − p Z f ′ ( x ) g ′ ( y ) log | x − y | dx dy (5)plus some mild conditions on the growth of f ( x ) and its Fourier transform ˆ f ( w ) , / p R f ( x ) e − iwx dx at infinity and a bounded variation condition (that are satisfied by compactly supported C functions,by example). We note that S t ( f ) is centered in the limit t → ¥ as it is well known that the numberof Riemann zeroes in the interval [ t , t ] is asymptotic to t log t / p in this limit. The principal resultsof [15] and [53] and the starting point of our construction are the following theorems. Theorem 2.1 (Convergence to a gaussian vector)
Given test function f , · · · f k in H / , the ran-dom vector (cid:0) S t ( f ) , · · · , S t ( f k ) (cid:1) converges in law in the limit t → ¥ to a centered gaussian vector (cid:0) S ( f ) , · · · S ( f k ) (cid:1) having the covariance Cov (cid:0) S ( f i ) , S ( f j ) (cid:1) = h f i , f j i . (6)The second theorem deals with the case of diverging limiting variance. Define s t ( f ) , log t / l ( t ) Z − log t / l ( t ) | w || ˆ f ( w ) | dw , (7)then, under the assumption that s t ( f ) → ¥ as t → ¥ , Theorem 2.2 (Convergence in the case of diverging variance) S t ( f ) / s t ( f ) inlaw −→ N ( , ) , (8) where N ( , ) denotes the standard gaussian random variable with the zero mean and unit variance. The significance of the condition l ( t ) ≪ log t is that the number of zeroes that are visited by f as t → ¥ goes to infinity, i.e. Theorems 2.1 and 2.2 are mesoscopic central limit theorems.The intuitive meaning of the BKR theorems and the statistic S t ( f ) can be established from theconnection of S t ( f ) with the error term S ( t ) in the zero counting function N ( t ) . Let N ( t ) denote thenumber of Riemann zeroes having their imaginary part (“height”) between zero and t . Let the function S ( t ) be defined by S ( t ) , p arg z ( / + it ) = p Im log z ( / + it ) . (9) Centered means that its expectation is zero. All expectations, covariances, etc in this section are with respect to thedistribution of w . Rodgers considered a more restrictive class of test functions and stated the formula for the variance only. N ( t ) = p (cid:16) Im log G ( + it ) − t p (cid:17) + + S ( t ) , (10) = t p log t p − t p + S ( t ) + + O ( / t ) (11)in the limit t → ¥ . Let u be fixed and l ( t ) be as in (2), then the asymptotic in equation (11) impliesthat the number of zeroes in the random interval [ w t , w t + u / l ( t )] satisfies in the same limit N (cid:16) w t + u l ( t ) (cid:17) − N ( w t ) = u log t pl ( t ) + S (cid:16) w t + u l ( t ) (cid:17) − S ( w t ) + O (cid:0) / l ( t ) (cid:1) . (12)It follows that the expected number of zeroes in [ w t , w t + u / l ( t )] is given by the leading asymptotic u log t / pl ( t ) , whereas S (cid:0) w t + u / l ( t ) (cid:1) − S ( w t ) gives “the error”, i.e. the fluctuation of the numberof zeroes in this interval from its leading asymptotic. Note that the condition in (2) means that thelength of the interval goes to zero, whereas the expected number of zeroes goes to infinity in the limit t → ¥ , i.e. the interval is mesoscopic. Let c u ( x ) denote the indicator function of the interval [ , u ] , then the corresponding BKR statistic satisfies by (3) and (12) S t ( c u ) = N (cid:16) w t + u l ( t ) (cid:17) − N ( w t ) − u log t pl ( t ) , (13) = S (cid:16) w t + u l ( t ) (cid:17) − S ( w t ) + O (cid:0) / l ( t ) (cid:1) . (14)Hence the statistics measures the fluctuation of the error term over the random interval [ w t , w t + u / l ( t )] . It is easy to see from (7) that the corresponding variance has the leading asymptotic s t ( c u ) ≈ log (cid:0) log t / l ( t ) (cid:1) / p so that Theorem 2.2 gives us S (cid:0) w t + u / l ( t ) (cid:1) − S ( w t ) q log (cid:0) log t / l ( t ) (cid:1) / p inlaw −→ N ( , ) . (15)This special case of Theorem 2.2 is known as Fujii’s central limit theorem, cf. [23], [24]. It turns outthat the interpretation of the BKR statistic as a measure of fluctuation of the error term remains truein general. We have the following identity for compactly supported test functions S t ( f ) = − Z f ′ ( x ) S (cid:16) w t + x l ( t ) (cid:17) dx + O (cid:0) / l ( t ) (cid:1) . (16)Clearly, this equation recovers (14) in the case of f = c u since f ′ is simply the difference of the deltafunctions at the endpoints.We now proceed to state our results. Let 0 < u < c ( i ) u ( x ) , i = , , denote the indicatorfunctions of the intervals [ , u ] and [ − / e , u ] for some fixed e > , respectively. Let f ( x ) be asmooth bump function supported on ( − / , / ) , and denote k , − Z f ( x ) f ( y ) log | x − y | dxdy . (17) This equation does not appear in [15] but follows easily from intermediate steps in their derivation of Proposition 3.In all of our applications, cf. (21) and (36) below, the constant in O (cid:0) / l ( t ) (cid:1) is of the order O (cid:0) || f || (cid:1) so that the O (cid:0) / l ( t ) (cid:1) term is negligible for our purposes due to || f || / l ( t ) ≪ . A proof of (16) is given in the appendix. e -rescaled bump function by f e ( x ) , / ef ( x / e ) , and let f ( i ) e , u ( x ) be the smoothed indicatorfunctions of the intervals [ , u ] and [ − / e , u ] given by the convolution of c ( i ) u ( x ) with f e ( x ) f ( i ) e , u ( x ) , ( c ( i ) u ⋆ f e )( x ) = e Z c ( i ) u ( x − y ) f ( y / e ) dy . (18)Clearly, f ( ) e , u ( x ) = x ∈ [ e / , u − e / ] , f ( ) e , u ( x ) = x ∈ [ − / e + e / , u − e / ] , f ( ) e , u ( x ) = x ≥ u + e / x ≤ − e / , and f ( ) e , u ( x ) = x ≥ u + e / x ≤ − / e − e / . Moreover, f ′ ( ) e , u ( x ) = f e ( x ) − f e ( x − u ) , (19) f ′ ( ) e , u ( x ) = f e ( x + / e ) − f e ( x − u ) . (20)Theorem 2.1 applies to f ( i ) e , u ( x ) for all u > e > . Fix e > times f ( i ) e , u ( x ) by S ( i ) t ( m , u , e ) S ( i ) t ( m , u , e ) , p p m h (cid:229) g ( c ( i ) u ⋆ f e ) (cid:0) g ( t ) (cid:1) − log t pl ( t ) Z ( c ( i ) u ⋆ f e )( x ) dx i . (21)The meaning of the first statistic ( i = ) is that it counts zeroes in the interval [ t , t ] over three asymp-totic scales. Specifically, over the interval of length u / l ( t ) − e / l ( t ) , g − u l ( t ) + e l ( t ) < w t < g − e l ( t ) , (22)the zero is counted with the weight 1, whereas it is counted with a diminishing weight that is deter-mined by e and the bump function over the boundary intervals of length e / l ( t ) , g − e l ( t ) < w t < g + e l ( t ) , (23) g − u l ( t ) − e l ( t ) < w t < g − u l ( t ) + e l ( t ) . (24)The second statistic ( i = ) has the same interpretation except that instead of (22) and (23) we have g − u l ( t ) + e l ( t ) < w t < g − e l ( t ) + el ( t ) , (25) g − e l ( t ) + el ( t ) < w t < g + e l ( t ) + el ( t ) , (26)respectively, so that the middle interval has length u / l ( t ) − e / l ( t ) + / el ( t ) , whereas the boundaryintervals have length e / l ( t ) as before. Hence, the three scales are e / l ( t ) , u / l ( t ) , t , and they satisfythe asymptotic condition 1log t ≪ el ( t ) < u l ( t ) ≪ ≪ t . (27)This means that t defines the global scale and u / l ( t ) ≫ average spacing so that u / l ( t ) and e / l ( t ) are on the mesoscopic scale. The choice of the constant p √ m , < m < S ( i ) t ( m , u , e ) can be elucidated further by means of Theorem 2.1 and(16). Given the expressions for the derivatives in (19) and (20), which show that the derivatives aresmooth approximations of the difference of the delta functions at the endpoints, we can write S ( ) t ( m , u , e ) = p p m Z (cid:0) f e ( x − u ) − f e ( x ) (cid:1) S (cid:16) w t + x l ( t ) (cid:17) dx + O (cid:0) / l ( t ) (cid:1) , (28) S ( ) t ( m , u , e ) = p p m Z (cid:0) f e ( x − u ) − f e ( x + / e ) (cid:1) S (cid:16) w t + x l ( t ) (cid:17) dx + O (cid:0) / l ( t ) (cid:1) , (29)so that the statistics are smooth approximations of p p m h S (cid:16) w t + u l ( t ) (cid:17) − S ( w t ) i , (30) p p m h S (cid:16) w t + u l ( t ) (cid:17) − S (cid:16) w t − el ( t ) (cid:17)i , (31)and can be interpreted as fluctuations of the smoothed error term over [ w t , w t + u / l ( t )] and [ w t − / el ( t ) , w t + u / l ( t )] , respectively. Moreover, by Theorem 2.1 and calculations in Corollaries 4.2and 4.4, the processes u → S ( i ) t ( m , u , e ) , u ∈ ( , ) , converge in law in the limit t → ¥ to centeredgaussian fields having the asymptotic covariances, for i = , ( − m (cid:0) log e − k + log | u − v | − log | u | − log | v | (cid:1) + O ( e ) , if | u − v | ≫ e , − m (cid:0) log e − k − log | u | (cid:1) + O ( e ) , if u = v , (32)and, for i = , ( − m (cid:0) e − k + log | u − v | (cid:1) + O ( e ) , if | u − v | ≫ e , − m (cid:0) e − k (cid:1) + O ( e ) , if u = v , (33)thereby exhibiting logarithmic correlations. These covariances will be identified in Section 4 as corre-sponding to the centered gaussian free field and the gaussian free field plus an independent gaussianrandom variable, respectively. We see from (32) and (33) that the asymptotic covariances of bothstatistics diverge as log e in the limit e → . We are primarily interested in the statistical structure ofthe log | u − v | terms, which is hiding behind these divergences. We will remove them so as to revealthe underlying structure by introducing rescaling factors into the Mellin transforms as shown below.It should also be noted that the need for smoothing is necessitated by the singularity of our statis-tics in the limit e → . Indeed, S ( ) t ( m , u , e ) is not even defined in this limit, and Var [ S ( ) t ( m , u , )] (cid:181) log ( log t / l ( t )) in Fujii’s theorem, cf. (15), is asymptotically different from Var [ lim t → ¥ S ( ) t ( m , u , e )] (cid:181) − log ( e ) in (32). This is explained by the difference between the formulas for the variance in (4) and(7), in fact, as shown in Lemma 4.10, the difference between the two for smoothed indicator functionsis of the order O ( l ( t ) / e log t ) and so becomes significant if one takes the e → t limit first and, when the two limits are takensimultaneously, e ( t ) will vary “slowly” enough to achieve he same end.Motivated by the proof of Theorem 2.2 in [15], we are also interested in the same type of statisticsthat are based on t -dependent e ( t ) . Let l ( t ) log t ≪ e ( t ) ≪ , for i = , (34) l ( t ) log t ≪ e ( t ) ≪ l ( t ) ≪ e ( t ) , for i = . (35)7n other words, we replace e with e ( t ) in (21), resulting in the statistics S ( i ) t ( m , u ) , S ( i ) t ( m , u , e ( t )) , S ( i ) t ( m , u ) , p p m h (cid:229) g ( c ( i ) u ⋆ f e ( t ) ) (cid:0) g ( t ) (cid:1) − log t pl ( t ) Z ( c ( i ) u ⋆ f e ( t ) )( x ) dx i . (36)These statistics also count zeroes in the interval [ t , t ] over the three scales e ( t ) / l ( t ) , u / l ( t ) , t , asbefore, except that u / l ( t ) ≫ e ( t ) / l ( t ) ≫ average spacing , t ≪ e ( t ) l ( t ) ≪ u l ( t ) ≪ ≪ t . (37) Remark
We note that the asymptotic conditions in (34) and (35) have a simple interpretation of being“slow” decay conditions that require that the lengths of intervals, over which the zeroes are counted,go to zero, whereas the expected numbers of the zeroes counted go to infinity. The lengths of the mid-dle and boundary intervals of the first statistic satisfy O (cid:0) / l ( t ) (cid:1) and O (cid:0) e ( t ) / l ( t ) (cid:1) , so that the corre-sponding expected numbers of zeroes are O (cid:0) log t / l ( t ) (cid:1) and O (cid:0) e ( t ) log t / l ( t ) (cid:1) by (12), respectively.Thus, the lengths go to zero by (2) and the expected numbers go to infinity by (2) and (34). Similarly,the lengths of the middle and boundary intervals of the second statistic are O (cid:0) / l ( t ) + / e ( t ) l ( t ) (cid:1) and O (cid:0) e ( t ) / l ( t ) (cid:1) so that the above argument remains valid provided e ( t ) l ( t ) ≫ , hence (35). Wefinally note that the interpretation of S ( i ) t ( m , u ) as fluctuations of smoothed error terms and calculationsof limiting covariances in (32) and (33) remain valid provided (34) and (35) are satisfied as shownin Lemmas 4.9 and 4.10, i.e. the limiting fields are gaussian and have e ( t ) -dependent asymptoticcovariance given in (32) and (33) with e = e ( t ) . The statements of our results below require some familiarity with the double gamma functionof Barnes (or the Alexeiewsky-Barnes G − function). We will give a brief summary of its definitionand properties here and refer the reader to [6], [7] for the original construction, to [54] for a moderntreatment, and to [48] and [50] for detailed reviews. One starts with the double zeta function z ( s , w | t ) , ¥ (cid:229) k , k = ( w + k + k t ) − s (38)that is defined for Re ( s ) > , Re ( w ) > , and t > . It can be analytically extended to s ∈ C withsimple poles at s = s = . The double gamma function is then defined as the exponential of the s -derivative of z ( s , w | t ) at s = , G ( w | t ) , exp (cid:16) ¶ s (cid:12)(cid:12) s = z ( s , w | t ) (cid:17) . (39)The resulting function can be analytically extended to w ∈ C having no zeroes and poles at w = − ( k + k t ) , k , k ∈ N . (40)Barnes gave an infinite product formula for G ( w | t ) , which in our normalization takes on the form G − ( w | t ) = e P ( w | t ) w ¥ (cid:213) k , k = ′ (cid:16) + wk + k t (cid:17) exp (cid:18) − wk + k t + w ( k + k t ) (cid:19) , (41) We note that c ( ) u ( x ) = c [ − / e ( t ) , u ] ( x ) becomes t -dependent so that || c ( ) u || = O (cid:0) / e ( t ) (cid:1) and the O (cid:0) / l ( t ) (cid:1) term in(16) is negligible due to e ( t ) l ( t ) ≫ P ( w | t ) is a polynomial in w of degree 2 , and the prime indicates that the product is over allindices except k = k = . The double gamma function satisfies the functional equations G ( z | t ) G ( z + | t ) = t z / t − / √ p G (cid:16) z t (cid:17) , (42) G ( z | t ) G ( z + t | t ) = √ p G (cid:0) z (cid:1) , (43)where G ( z ) denotes Euler’s gamma function. An explicit integral representation of log G ( w | t ) andadditional infinite product representations of G ( w | t ) can be found in [48].From now on, it is always assumed that l ( t ) and e ( t ) satisfy (2) and (34) or (35), respectively,and k is as in (17). Given 0 < m < , henceforth let t , m > . (44)The results given below are stated for the S ( ) t ( m , u , e ) and S ( ) t ( m , u ) statistics to avoid redundancy,for the same formulas apply to S ( ) t ( m , u , e ) provided one simultaneously replaces e S ( ) t ( m , u , e ) ←→ u m q e S ( ) t ( m , u , e ) , (45) e m ( log e − k ) q ( q + ) ←→ e m log e q e m ( log e − k ) q ( q + ) , (46)and, similarly to S ( ) t ( m , u ) , provided one replaces e S ( ) t ( m , u ) ←→ u m q e S ( ) t ( m , u ) and uses (35) insteadof (34). For clarity, this translation is shown explicitly in Conjectures 2.3 and 2.4. Given thesepreliminaries, our results are as follows. Conjecture 2.3 (Mellin-type transforms: weak version)
Let Re ( q ) < t , then lim e → e m ( log e − k ) q ( q + ) h lim t → ¥ E h(cid:16) Z u − m q e S ( ) t ( m , u , e ) du (cid:17) q ii , (47) = lim e → e m log e q e m ( log e − k ) q ( q + ) h lim t → ¥ E h(cid:16) Z e S ( ) t ( m , u , e ) du (cid:17) q ii , (48) = t q t ( p ) q G − q (cid:0) − / t (cid:1) G ( − q + t | t ) G ( + t | t ) G ( − q + t | t ) G ( t | t ) G ( − q + t | t ) G ( − q + t | t ) . (49) Let − ( t + ) / < Re ( q ) < t , then lim e → e m ( log e − k ) q ( q + ) h lim t → ¥ E h(cid:16) Z e S ( ) t ( m , u , e ) du (cid:17) q ii , (50) = lim e → e m log e q e m ( log e − k ) q ( q + ) h lim t → ¥ E h(cid:16) Z u m q e S ( ) t ( m , u , e ) du (cid:17) q ii , (51) = t q t ( p ) q G − q (cid:0) − / t (cid:1) G ( + q + t | t ) G ( + q + t | t ) G ( − q + t | t ) G ( + t | t ) G ( − q + t | t ) G ( t | t ) G ( + q + t | t ) G ( + t | t ) . (52)9 onjecture 2.4 (Mellin-type transforms: strong version) Let Re ( q ) < t , then lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z u − m q e S ( ) t ( m , u ) du (cid:17) q i = lim e → e m ( log e − k ) q ( q + ) ⋆⋆ h lim t → ¥ E h(cid:16) Z u − m q e S ( ) t ( m , u , e ) du (cid:17) q ii , (53)lim t → ¥ e m log e ( t ) q e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z e S ( ) t ( m , u ) du (cid:17) q i = lim e → e m log e q e m ( log e − k ) q ( q + ) ⋆⋆ h lim t → ¥ E h(cid:16) Z e S ( ) t ( m , u , e ) du (cid:17) q ii . (54) Let − ( t + ) / < Re ( q ) < t , then lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z e S ( ) t ( m , u ) du (cid:17) q i = lim e → e m ( log e − k ) q ( q + ) h lim t → ¥ E h(cid:16) Z e S ( ) t ( m , u , e ) du (cid:17) q ii , (55)lim t → ¥ e m log e ( t ) q e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z u m q e S ( ) t ( m , u ) du (cid:17) q i = lim e → e m log e q e m ( log e − k ) q ( q + ) ⋆⋆ h lim t → ¥ E h(cid:16) Z u m q e S ( ) t ( m , u , e ) du (cid:17) q ii . (56)The following corollaries can be formulated with either the double or single limits as in Conjectures2.3 and 2.4. We will state them with the single limit for notational simplicity. All corollaries apply toboth statistics, cf. (45) and (46) for the translation. Note that in Corollaries 2.5, 2.6, and 2.7 we have q = n , − n , N , respectively, so that the multiplier u m q in (45) and the scaling factor in (46) are adjustedaccordingly, when translated for the second statistic. Corollary 2.5 (Positive integral moments and Selberg integral)
Let n ∈ N such that n < t . lim t → ¥ e m ( log e ( t ) − k ) n ( n + ) E h(cid:16) Z u − m n e S ( ) t ( m , u ) du (cid:17) n i = Z [ , ] n n (cid:213) i < j | s i − s j | − m ds · · · ds n , (57) = n − (cid:213) k = G ( − ( k + ) / t ) G ( − / t ) G ( − k / t ) G ( − ( n + k − ) / t ) , (58)lim t → ¥ e m ( log e ( t ) − k ) n ( n + ) E h(cid:16) Z e S ( ) t ( m , u ) du (cid:17) n i = Z [ , ] n n (cid:213) i = s m ni n (cid:213) i < j | s i − s j | − m ds · · · ds n , (59) = n − (cid:213) k = G ( − ( k + ) / t ) G ( − / t ) G ( + n / t − k / t ) G ( − k / t ) G ( + n / t − ( k − ) / t ) . (60) Corollary 2.6 (Negative integral moments)
Let n ∈ N . lim t → ¥ e m ( log e ( t ) − k ) n ( n − ) E h(cid:16) Z u m n e S ( ) t ( m , u ) du (cid:17) − n i = n − (cid:213) k = G (cid:0) + ( n + + k ) / t (cid:1) G (cid:0) − / t (cid:1) G (cid:0) + ( k + ) / t (cid:1) G (cid:0) + k / t (cid:1) . (61)10 et n ∈ N such that n < ( t + ) / . lim t → ¥ e m ( log e ( t ) − k ) n ( n − ) E h(cid:16) Z e S ( ) t ( m , u ) du (cid:17) − n i = n − (cid:213) k = G (cid:0) + ( − n + + k ) / t (cid:1) G (cid:0) − / t (cid:1) G (cid:0) − n / t + ( k + ) / t (cid:1) G (cid:0) + ( k + ) / t (cid:1) G (cid:0) + k / t (cid:1) . (62) Corollary 2.7 (Joint integral moments)
Let n , m ∈ N and denote N , n + m , N < / t . Let I and I be non-overlapping subintervals of the unit interval. Then, lim t → ¥ e m ( log e ( t ) − k ) N ( N + ) E h(cid:16) Z I u − m N e S ( ) t ( m , u ) du (cid:17) n (cid:16) Z I u − m N e S ( ) t ( m , u ) du (cid:17) m i = Z I n × I m N (cid:213) i < j | s i − s j | − m ds · · · ds N , (63)lim t → ¥ e m ( log e ( t ) − k ) N ( N + ) E h(cid:16) Z I e S ( ) t ( m , u ) du (cid:17) n (cid:16) Z I e S ( ) t ( m , u ) du (cid:17) m i = Z I n × I m N (cid:213) i = s m Ni N (cid:213) i < j | s i − s j | − m ds · · · ds N . (64) Corollary 2.8 (Asymptotic expansions)
Given q ∈ C , then in the limit m → we have the asymp-totic expansions lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z u − m q e S ( ) t ( m , u ) du (cid:17) q i ∼ exp (cid:16) ¥ (cid:229) r = (cid:16) m (cid:17) r + r + h z ( r + ) ×× (cid:2) B r + ( q + ) + B r + ( q ) − B r + r + − q (cid:3) + (cid:0) z ( r + ) − (cid:1)(cid:2) B r + ( q − ) − B r + ( q − ) r + (cid:3)i(cid:17) . (65)lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z e S ( ) t ( m , u ) du (cid:17) q i ∼ exp (cid:16) ¥ (cid:229) r = (cid:16) m (cid:17) r + r + h z ( r + ) (cid:2) B r + ( q + ) + B r + ( q ) r + ++ B r + ( − q ) − B r + ( − q ) − B r + r + − q (cid:3) + (cid:0) z ( r + ) − (cid:1)(cid:2) B r + ( − q − ) − B r + ( − ) r + (cid:3)i(cid:17) . (66) Corollary 2.9 (Covariance structure)
Let < s < s < . Then, in the limit D → , lim t → ¥ h m ( log e ( t ) − k ) + Cov (cid:16) log s + D Z s e S ( ) t ( m , u ) du , log s + D Z s e S ( ) t ( m , u ) du (cid:17)i = − m (cid:0) log | s − s | − log | s |−− log | s | (cid:1) + O ( D ) , (67)lim t → ¥ h m ( e ( t ) − k ) + Cov (cid:16) log s + D Z s e S ( ) t ( m , u ) du , log s + D Z s e S ( ) t ( m , u ) du (cid:17)i = − m log | s − s | + O ( D ) . (68)The next two results deal with the probabilistic structure that is underlying the Mellin-type transformin (49). As usual, z ( s ) denotes the Riemann zeta function and B n ( s ) the n th Bernoulli polynomial. Note that z ( ) never entersany of the final formulas as the coefficient it multiplies is identically zero. orollary 2.10 (Non-central limit) The limit in (49) is the Mellin transform of a positive probabilitydistribution. Call it M m . Then, for Re ( q ) < t , lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z u − m q e S ( ) t ( m , u ) du (cid:17) q i = E [ M q m ] . (69)log M m is infinitely divisible on the real line having the L´evy-Khinchine decomposition log E [ M q m ] = q m ( m ) + q s ( m ) + R R \{ } (cid:0) e qu − − qu / ( + u ) (cid:1) d M m ( u ) for some m ( m ) ∈ R and the followinggaussian component and spectral function s ( m ) = t , (70) M m ( u ) = ¥ Z u h (cid:0) e x + + e − x ( + t ) (cid:1)(cid:0) e x − (cid:1) ( e x t − ) − e − x ( + t ) / (cid:0) e x / − (cid:1) ( e x t / − ) i dxx (71) for u > , and M m ( u ) = for u < . M m is a product of a lognormal, Fr´echet and independent Barnesbeta random variables. Corollary 2.11 (Multifractality)
Let < s < . Then, for Re ( q ) < t , the limit lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z s u − m q e S ( ) t ( m , u ) du (cid:17) q i = E (cid:2) M q m ( s ) (cid:3) (72) is the Mellin transform of a probability distribution, call it M m ( s ) , which satisfies the multifractallaw M m ( s ) inlaw = s exp (cid:0) W s (cid:1) M m , (73) where W s is a gaussian random variable with the mean ( m / ) log s and variance − m log s that isindependent of M m and M m is as in Corollary 2.10. This law is understood as the equality of randomvariables in law at fixed s < . In particular, M m ( s ) is also log-infinitely divisible. Corollary 2.12 (Multiscaling)
Let < s < . Then, for < Re ( q ) < t , lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z s u − m q e S ( ) t ( m , u ) du (cid:17) q i = const ( m , q ) s q − m ( q − q ) , (74)lim t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z s e S ( ) t ( m , u ) du (cid:17) q i = const ( m , q ) s q + m ( q + q ) , (75)i.e. these Mellin-type transforms are multiscaling as functions of s . The rationale for considering two separate transforms in Conjectures 2.3 and 2.4 is that they arecomplementary. The transforms in (47) and (53) are not, strictly speaking, rescaled Mellin transformsbecause of the u − m q multiplier. This multiplier is introduced to obtain the Mellin transform of aprobability distribution on the right-hand side of these equations. If we drop this multiplier, we loosethis probabilistic interpretation but gain a bona fide rescaled Mellin transforms of R e S ( ) t ( m , u , e ) du This decomposition is shown in detail in Section 3, cf. Theorem 3.4. Also referred to in the literature as stochastic self-similarity or scale-consistent continuous dilation invariance. Itdetermines the law of M m ( s ) , s < , from the law of M m in (69) . R e S ( ) t ( m , u ) du on the left-hand side so that the limit fits into the general theory of mod-gaussianconvergence, cf. [33]. The interest in introducing the second statistic S ( ) t ( m , u , e ) and S ( ) t ( m , u ) isprecisely to eliminate the need to introduce the u − m q multiplier so that the transforms in (48) and(54) are both bona fide rescaled Mellin transforms and give the Mellin transform of a probabilitydistribution on the right-hand side of these equations. We note that the right-hand sides of both (49)and (52) are special cases of the Mellin transform of the Selberg integral distribution that is given in(117) and (135). The meaning of the conditions Re ( q ) < t and − ( t + ) / < Re ( q ) < t is that theright-hand sides of (49) and (52) are analytic and zero-free over these regions. As will be explainedin Section 4, one can insert the u l ( − u ) l prefactor into the exponential functionals on the left-handsides of (47) – (56) so as to obtain the general Mellin transform of the Selberg integral distribution onthe right-hand side, cf. Theorem 4.5 for details. We chose not to state the most general case here toavoid unnecessary complexity. Remark
A random matrix analogue of the Bourgade-Kuan-Rodgers theorems was established in[27]. The corresponding linear statistic is an appropriately centered log-absolute value of the charac-teristic polynomial of a suitably scaled GUE matrix. The eigenvalues of the matrix replace the rootsand its dimension N replaces log t . It is shown in [27] that the N → ¥ limit of the GUE statistic isgaussian, and the limiting covariance can be shown to be equivalent to the Bourgade-Kuan-Rodgerscovariance. As it will become clear in Section 4, our results apply to any linear statistic that isasymptotically gaussian with the Bourgade-Kuan-Rodgers covariance. Thus, under the correspon-dence N ∼ log t , they apply equally to the GUE statistic. The significance of this observation is thatthe convergence of our statistics to the gaussian free field or its centered version provides a theoreticalexplanation for why our conjecture about the zeroes can be approached directly from the GUE side. Remark
Our conjecture has only GUE terms and no arithmetic corrections, contrary to most of theconjectures that we mentioned in the Introduction. The reason for this is the choice of the mesoscopicscale 1 ≪ l ( t ) . As Fujii [24] shows in his analysis of Berry’s conjecture, which in particular dealswith the second moment of the fluctuation of the error term, cf. (14), the GUE term in Berry’s formuladominates arithmetic corrections precisely under the condition 1 ≪ l ( t ) . As our conjecture involvesessentially the same statistic, except for smoothing, we expect that in our case this condition achievesthe same effect of dominating arithmetic terms. Remark
The interest in the strong version of our conjecture is that it contains information about thestatistical distribution of the zeroes at large but finite t , whereas the weak version only describes thedistribution at t = ¥ . Indeed, as it will be explained in Section 4, the strong version is equivalent to theweak version provided the statistics S ( i ) t ( m , u ) converge to their gaussian limits faster than 1 / | log e ( t ) | , in the sense of the rate of convergence of the variance to its asymptotic value, and this is expectedto be the case due to the asymptotic condition in (34). Moreover, as the strong conjecture fits intothe general framework of mod-gaussian convergence, the results of [21] and [39] and the explicitknowledge of our limiting functions make it possible to quantify the normality zone, i.e. the scale upto which the tails of our exponential functionals are normal, and the breaking of symmetry near theedges of the normality zone thereby quantifying precise deviations at large t . The actual computationof these quantities is left for future research. Nickolas Simm, private communication. A Review of the Limit Lognormal Measure and Selberg Integral Dis-tribution
In this section we will give a self-contained review of the limit lognormal measure of Mandelbrot-Bacry-Muzy on the unit interval and of the Selberg integral probability distribution mainly followingour earlier presentations in [45], [48], and [50]. We will indicate what is known and what is conjec-tured and refer the reader to appropriate original publications for the proofs.The limit lognormal measure (also known as lognormal multiplicative chaos) is defined as theexponential functional of the gaussian free field. Let w m , L , e ( s ) be a stationary gaussian process in s , whose mean and covariance are functions of three parameters m > , L > , and e > . We considerthe random measure M m , L , e ( a , b ) = b Z a e w m , L , e ( s ) ds . (76)The mean and covariance of w m , L , e ( s ) are defined to be, cf. [41], E (cid:2) w m , L , e ( t ) (cid:3) = − m (cid:18) + log L e (cid:19) , (77) Cov (cid:2) w m , L , e ( t ) , w m , L , e ( s ) (cid:3) = m log L | t − s | , e ≤ | t − s | ≤ L , (78) Cov (cid:2) w m , L , e ( t ) , w m , L , e ( s ) (cid:3) = m (cid:18) + log L e − | t − s | e (cid:19) , | t − s | < e , (79)and covariance is zero in the remaining case of | t − s | ≥ L . Thus, e is used as a truncation scale. L is the decorrelation length of the process. m is the intermittency parameter (also known as inversetemperature in the physics literature). The two key properties of this construction are, first, that E (cid:2) w m , L , e ( t ) (cid:3) = − Var (cid:2) w m , L , e ( t ) (cid:3) (80)so that E (cid:2) exp (cid:0) w m , L , e ( s ) (cid:1)(cid:3) = Var (cid:2) w m , L , e ( t ) (cid:3) (cid:181) − log e (82)is logarithmically divergent as e → . The first property is essential for convergence, the second isresponsible for multifractality, and both are originally due to Mandelbrot [37].The interest in the limit lognormal construction stems from the e → e → w m , L , e ( t ) does not exist as a stochastic process (this limiting “process” is known as thegaussian free field). Remarkably, using the theory of T -martingales developed by Kahane [34], thework of Barral and Mandelbrot [9] on log-Poisson cascades, and conical constructions of Rajput andRosinski [51] and Marsan and Schmitt [55], Bacry and Muzy [5] showed that M m , L , e ( dt ) convergesweakly (as a measure on R + ) a.s. to a non-trivial random limit measure M m , L ( dt ) M m , L ( a , b ) = lim e → M m , L , e ( a , b ) , (83)14rovided 0 ≤ m < , and the limit is stationary M m , L ( t , t + t ) inlaw = M m , L ( , t ) and non-degenerate inthe sense that E [ M m , L ( , t )] = t . We will denote the total random mass of [ , L ] by M m , L , M m , L , M m , L ( , L ) . (84)It is shown in [5] that for q > q < m = ⇒ E (cid:2) M q m , L (cid:3) < ¥ and q > m = ⇒ E (cid:2) M q m , L (cid:3) = ¥ . (85)The fundamental property of the limit measure is that it is multifractal. This can be understoodat several levels, for our purposes, this means that its total mass exhibits stochastic self-similarity,also known as continuous dilation invariance, as first established in [3]. Given g < , let W g denote agaussian random variable that is independent of the process w m , L , e ( s ) such that E (cid:2) W g (cid:3) = m log g , (86) Var (cid:2) W g (cid:3) = − m log g . (87)Then, there hold the following invariances W g + w m , L , e ( s ) inlaw = w m , L , ge ( g s ) , (88) W g + w m , L , e ( s ) inlaw = w m , L / g , e ( s ) , (89) W g + w m , L , e ( s ) inlaw = w ( ) m ( + log g ) , L , e ( s ) + w ( ) − m log g , eL , e ( s ) , (90)that are understood as equalities in law of stochastic processes in s on the interval [ , L ] at fixed e , L , and 0 < g < . In (90) the superscripts denote independent copies of the free field, e stands for thebase of the natural logarithm, and e − < g < . The truncation scale invariance in (88) implies themultifractal law of the limit measure for t < LM m , L ( , t ) inlaw = tL e W tL M m , L , (91)which implies that the moments of the total mass obey for 0 < q < / m the multiscaling law E (cid:2) M m , L ( , t ) q (cid:3) (cid:181) (cid:16) tL (cid:17) q − m ( q − q ) (92)as a function of t < L . The decorrelation length invariance in (89) implies that the dependence of thetotal mass on L is trivial, M m , L inlaw = L M m , , (93) The first invariance was discovered in [3] and later generalized in [41]. We discovered the other two invariances in [43]and developed a general theory of such invariances in [46]. It must be emphasized that this equality is strictly in law, that is, W g is not a stochastic process, i.e. W g and W g ′ for g = g ′ are not defined on the same probability space. In particular, (91) determines the distribution of M m , L ( , t ) in termsof the law of the total mass but says nothing about the latter or their joint distribution. This invariance also determines how the law of the total mass behaves under a particular change of probability measure,cf. [46] for details.
15o that we can restrict ourselves to L = L = , (94)and L is dropped from all subsequent formulas. Finally, the significance of the intermittency invari-ance in (90) is that it gives the rule of intermittency differentiation and effectively determines the lawof the total mass, cf. Theorem 3.1 and the discussion following it below.The law of the total mass can be reformulated as a non-central limit problem. Let us break up theunit interval into the subintervals of length e so that s j = j e , w j , w m , e ( s j ) , and N e = . It is shownin [5] that the total mass M m can be approximated as e N − (cid:229) j = e w j → M m a s e → . (95)The essence of this result is that the limit is not affected by one’s truncation of covariance so longas (80) holds. This representation is quite useful in calculations. In particular, it is easy to seethat it implies an important relationship between the moments of the generalized total mass of thelimit measure and a class of (generalized) Selberg integrals, originally due to [3]. Let j ( s ) be anappropriately chosen test function and I a subinterval of the unit interval, then E h(cid:0) Z I j ( s ) M m ( ds ) (cid:1) n i = Z I n n (cid:213) i = j ( s i ) n (cid:213) i < j | s i − s j | − m ds · · · ds n . (96)More generally, the same type of result holds for any finite number of subintervals of the unit interval.For example, the joint ( n , m ) moment is given by a generalized Selberg integral of dimension n + m E h(cid:16) Z I j ( s ) M m ( ds ) (cid:17) n (cid:16) Z I j ( s ) M m ( ds ) (cid:17) m i = Z I n × I m n (cid:213) i = j ( s i ) n + m (cid:213) i = n + j ( s i ) n + m (cid:213) k < p | t p − t k | − m ds · · · ds n + m . (97)It can be shown as a corollary of this equation, cf. [3], that the covariance structure of the total massis logarithmic. Given 0 < s < , Cov (cid:16) log s + D s Z s M m ( du ) , log D s Z M m ( du ) (cid:17) = − m log s + O ( D s ) , (98)which indicates that the mass of non-overlapping subintervals of the unit interval exhibits strongstochastic dependence. In the special case of j ( s ) = s l ( − s ) l , we have an explicit formula formoments of order n < / m , as was first pointed out in [4], that is given by the classical Selbergintegral, cf. Chapter 4 of [22] for a modern treatment and [56] for the original derivation, E h(cid:0) Z s l ( − s ) l M m ( ds ) (cid:1) n i = n − (cid:213) k = G ( − ( k + ) m / ) G ( + l − k m / ) G ( + l − k m / ) G ( − m / ) G ( + l + l − ( n + k − ) m / ) . (99) By a slight abuse of terminology, we refer to any integral of the form R j ( s ) M m ( ds ) as the generalized total mass. R s l ( − s ) l M m ( ds ) is not known rigorously, even for l = l = , i.e. the total mass.It is possible to derive it heuristically so as to formulate a precise conjecture about it as follows.Consider the expectation of a general functional of the limit lognormal measure v ( m , j , f , F ) , E h F (cid:16) Z e m f ( s ) j ( s ) M m ( ds ) (cid:17)i . (100)The integration with respect to M m ( ds ) is understood in the sense of e → v ( m , j , f , F ) = lim e → v e ( m , j , f , F ) and v e ( m , j , f , F ) , E h F (cid:16) R e m f ( s ) j ( s ) M m , e ( ds ) (cid:17)i with M m , e ( ds ) as in (76).Also, let g ( s , s ) be defined by g ( s , s ) , − log | s − s | . (101)Finally, we will use [ , ] k to denote the k − dimensional unit interval [ , ] × · · · × [ , ] . Then, wehave the following rule of intermittency differentiation, cf. [44] and [48] for derivations and [47] foran extension to the joint distribution of the mass of multiple subintervals of the unit interval.
Theorem 3.1 (Intermittency differentiation)
The expectation v ( m , j , f , F ) is invariant under in-termittency differentiation and satisfies ¶¶m v ( m , j , f , F ) = Z [ , ] v (cid:0) m , j , f + g ( · , s ) , F ( ) (cid:1) e m f ( s ) f ( s ) j ( s ) ds ++ Z [ , ] v (cid:0) m , j , f + g ( · , s ) + g ( · , s ) , F ( ) (cid:1) e m (cid:0) f ( s )+ f ( s )+ g ( s , s ) (cid:1) g ( s , s ) j ( s ) j ( s ) ds ds . (102)The mathematical content of (102) is that differentiation with respect to the intermittency parameter m is equivalent to a combination of two functional shifts induced by the g function. This differentiationrule is nonlocal as it involves the entire path of the process s → M m ( , s ) , s ∈ ( , ) . It is clear thatboth terms in (102) are of the same functional form as the original functional in (100) so that Theorem3.1 allows us to compute derivatives of all orders. There results the following formal expansion withsome coefficients H n , k ( j ) that are independent of F . Let S l ( m , j ) , Z [ , ] l l (cid:213) i = j ( s i ) l (cid:213) i < j | s i − s j | − m ds · · · ds l , (103) x , Z j ( s ) ds . (104)Then, we obtain the formal intermittency expansion E h F (cid:16) Z j ( s ) M m ( ds ) (cid:17)i = F ( x ) + ¥ (cid:229) n = m n n ! h n (cid:229) k = F ( k ) ( x ) H n , k ( j ) i . (105)The expansion coefficients H n , k ( j ) are given by the binomial transform of the derivatives of thepositive integral moments H n , k ( j ) = ( − ) k k ! k (cid:229) l = ( − ) l (cid:18) kl (cid:19) x k − l ¶ n S l ¶m n | m = , (106)17nd satisfy the identity H n , k ( j ) = ∀ k > n . (107) Remark
The last equation and Theorem 3.1 say that the intermittency expansion in (105) is anexactly renormalized expansion in the centered moments of R j ( s ) dM m ( s ) . Indeed, we have theidentity ¶ n / ¶m n | m = E h(cid:0) Z j ( s ) M m ( ds ) − x (cid:1) k i = k ! H n , k ( j ) . (108)From now on we will focus on j ( s ) , s l ( − s ) l , l , l > − m , (109)which corresponds to the full Selberg integral. Clearly, we have x = G ( + l ) G ( + l ) G ( + l + l ) . (110)The moments S l ( m , j ) = S l ( m , l , l ) are given by Selberg’s product formula in (99). By expand-ing log S l ( m , l , l ) in powers of m near zero and computing the resulting H n , k ( j ) coefficients, wederived in [45] and [48] the following expansion for the Mellin transform in terms of Bernoulli poly-nomials and values of the Hurwitz zeta function at the integers E h(cid:16) Z s l ( − s ) l M m ( ds ) (cid:17) q i = x q exp (cid:16) ¥ (cid:229) r = m r + r + b r ( q , l , l ) (cid:17) , (111) b r ( q , l , l ) , r + h(cid:0) z ( r + , + l ) + z ( r + , + l ) (cid:1)(cid:16) B r + ( q ) − B r + r + (cid:17) −− z ( r + ) q + z ( r + ) (cid:16) B r + ( q + ) − B r + r + (cid:17) −− z ( r + , + l + l ) (cid:16) B r + ( q − ) − B r + ( q − ) r + (cid:17)i . (112)The series in (111) is generally divergent and interpreted as the asymptotic expansion of the Mellintransform in the limit m → . The intermittency expansion of the Mellin transform implies a similarexpansion of the general transform of log e M m ( l , l ) , where we introduced the normalized distribu-tion e M m ( l , l ) , G ( + l + l ) G ( + l ) G ( + l ) Z s l ( − s ) l M m ( ds ) . (113)Then, given constants a and s and a smooth function F ( s ) , the intermittency expansion is E h F (cid:0) s + a log e M m ( l , l ) (cid:1)i = ¥ (cid:229) n = F n (cid:0) a , s , l , l (cid:1) m n n ! , (114) F (cid:0) a , s , l , l (cid:1) = F ( s ) , (115) F n + (cid:0) a , s , l , l (cid:1) = n (cid:229) r = n ! ( n − r ) ! b r (cid:16) a dds , l , l (cid:17) F n − r (cid:0) a , s , l , l (cid:1) . (116) It is convergent for ranges of integral q as shown in [45]. q with a d / ds in the solution for the Mellin transform in(111). We note parenthetically that the general transform is particularly interesting in the special caseof a purely imaginary a , in which case the operator F ( s ) → E h F (cid:0) s + a log e M m ( l , l ) (cid:1)i is formallyself-adjoint.The calculation of the intermittency expansion of the Mellin transform naturally poses the prob-lem of constructing a positive probability distribution such that its positive integral moments are givenby Selberg’s formula, cf. (99), and the asymptotic expansion of its Mellin transform coincides withthe intermittency expansion in (111). Equivalently, one wants to construct a meromorphic function M ( q | m , l , l ) that (1) recovers Selberg’s formula for positive integral q < / m , (2) is the Mellintransform of a probability distribution for Re ( q ) < / m as long as 0 < m < , and (3) has the asymp-totic expansion in the limit m → l = l = Define M ( q | m , l , l ) , t q t ( p ) q G − q (cid:0) − / t (cid:1) G ( − q + t ( + l ) | t ) G ( + t ( + l ) | t ) G ( − q + t ( + l ) | t ) G ( + t ( + l ) | t ) ⋆⋆ G ( − q + t | t ) G ( t | t ) G ( − q + t ( + l + l ) | t ) G ( − q + t ( + l + l ) | t ) (117) Theorem 3.2 (Existence of the Selberg integral distribution)
Let < m < , l i > − m / , and t = / m . Then, M ( q | m , l , l ) is the Mellin transform of a probability distribution on ( , ¥ ) for Re ( q ) < t , and its moments satisfy M ( n | m , l , l ) = n − (cid:213) k = G ( − ( k + ) / t ) G ( + l − k / t ) G ( + l − k / t ) G ( − / t ) G (cid:0) + l + l − ( n + k − ) / t (cid:1) , ≤ n < Re ( t ) , (118) M ( − n | m , l , l ) = n − (cid:213) k = G (cid:0) + l + l + ( n + + k ) / t (cid:1) G (cid:0) − / t (cid:1) G (cid:0) + l + ( k + ) / t (cid:1) G (cid:0) + l + ( k + ) / t (cid:1) G (cid:0) + k / t (cid:1) , n ∈ N . (119) The function M ( q | m , l , l ) satisfies the functional equations M ( q | m , l , l ) = M ( q − | m , l , l ) G ( − q / t ) G (cid:0) + l + l − ( q − ) / t (cid:1) G ( − / t ) ⋆⋆ G (cid:0) + l − ( q − ) / t (cid:1) G (cid:0) + l − ( q − ) / t (cid:1) G (cid:0) + l + l − ( q − ) / t (cid:1) G (cid:0) + l + l − ( q − ) / t (cid:1) . (120) M ( q | m , l , l ) = M ( q − t | m , l , l ) t ( p ) t − G − t (cid:16) − t (cid:17) G ( t − q ) ⋆⋆ G (cid:0) ( + l ) t − ( q − ) (cid:1) G (cid:0) ( + l ) t − ( q − ) (cid:1) G (cid:0) ( + l + l ) t − ( q − ) (cid:1) G (cid:0) ( + l + l ) t − ( q − ) (cid:1) G (cid:0) ( + l + l ) t − ( q − ) (cid:1) , (121) The general case was first considered by Fyodorov et. al. [28], who gave an equivalent expression for the right-handside of (117) and so recovered the positive integral moments without proving analytically that their formula correspondsto the Mellin transform of a probability distribution or matching the asymptotic expansion, i.e. solved (1) only. Instead,they used Selberg’s formula to deduce the functional equation in (120) for positive integral q , conjectured that it holds forcomplex q , and then found a meromorphic function that satisfies (120). he function log M ( q | m , l , l ) has the asymptotic expansion as m → + M ( q | m , l , l ) ∼ q log (cid:16) G ( + l ) G ( + l ) G ( + l + l ) (cid:17) + ¥ (cid:229) r = (cid:16) m (cid:17) r + r + h − z ( r + ) q ++ (cid:0) z ( r + , + l ) + z ( r + , + l ) (cid:1)(cid:16) B r + ( q ) − B r + r + (cid:17) + z ( r + ) ⋆⋆ (cid:16) B r + ( q + ) − B r + r + (cid:17) − z ( r + , + l + l ) (cid:16) B r + ( q − ) − B r + ( q − ) r + (cid:17)i . (122)The structure of the corresponding probability distribution, which we denote by M ( m , l , l ) , is mostnaturally explained using the theory of Barnes beta distributions that we developed in [48], [49]. Forsimplicity, we restrict ourselves here to the special case of the distribution of type ( , ) and refer thereader to [49], [50] for the general case. Theorem 3.3 (Barnes beta distribution of type ( , ) ) Let b , b , b > and t > . Define h ( q | t , b ) , G ( q + b | t ) G ( b | t ) G ( b + b | t ) G ( q + b + b | t ) G ( b + b | t ) G ( q + b + b | t ) G ( q + b + b + b | t ) G ( b + b + b | t ) , (123) = ¥ (cid:213) n , n = h b + n + n t q + b + n + n t q + b + b + n + n t b + b + n + n t q + b + b + n + n t b + b + n + n t ⋆⋆ b + b + b + n + n t q + b + b + b + n + n t i . (124) Then, h ( q | t , b ) is the Mellin transform of a probability distribution on ( , ) . Denote it by b ( t , b ) . E (cid:2) b ( t , b ) q (cid:3) = h ( q | t , b ) , Re ( q ) > − b . (125) The distribution − log b ( t , b ) is absolutely continuous and infinitely divisible on ( , ¥ ) and has theL´evy-Khinchine decomposition E h exp (cid:0) q log b ( t , b ) (cid:1)i = exp (cid:16) ¥ Z ( e − xq − ) e − b x ( − e − b x )( − e − b x )( − e − x )( − e − t x ) dxx (cid:17) , Re ( q ) > − b . (126)We can now describe the probabilistic structure of the Selberg integral distribution. Let t > L , exp (cid:0) N ( , / t ) (cid:1) , (127)where N ( , / t ) denotes a zero-mean gaussian with variance 4 log 2 / t , and a Fr´echet variable Y having density t y − − t exp (cid:0) − y − t (cid:1) dy , y > , so that its Mellin transform is E (cid:2) Y q (cid:3) = G (cid:16) − q t (cid:17) , Re ( q ) < t , (128)and log Y is infinitely divisible. Given l i > − / t , let X , X , X have the b − ( t , b ) distribution withthe parameters X , b − , (cid:16) t , b = + t + tl , b = t ( l − l ) / , b = t ( l − l ) / (cid:17) , (129) X , b − , (cid:16) t , b = + t + t ( l + l ) / , b = / , b = t / (cid:17) , (130) X , b − , (cid:16) t , b = + t , b = ( + t + tl + tl ) / , b = ( + t + tl + tl ) / (cid:17) . (131) Without loss of generality, l ≥ l . If l = l , then X = , otherwise the parameters of X satisfy Theorem 3.3. heorem 3.4 (Structure of the Selberg integral distribution) Let t = / m . M ( m , l , l ) decomposesinto independent factors,M ( m , l , l ) inlaw = p − (cid:2) ( + t )+ t ( l + l ) (cid:3) / t G (cid:0) − / t (cid:1) − L X X X Y . (132) In particular, log M ( m , l , l ) is absolutely continuous and infinitely divisible. Its L ´ evy-Khinchine de-composition log M ( q | m , l , l ) = q m ( m ) + q s ( m ) + R R \{ } (cid:0) e qu − − qu / ( + u ) (cid:1) d M ( m , l , l ) ( u ) is s ( m ) = t , (133) M ( m , l , l ) ( u ) = − ¥ Z u h (cid:0) e x + e − x tl + e − x tl + e − x ( + t ( + l + l )) (cid:1)(cid:0) e x − (cid:1) ( e x t − ) − e − x ( + t ( + l + l )) / (cid:0) e x / − (cid:1) ( e x t / − ) i dxx (134) for u > , and M ( m , l , l ) ( u ) = for u < , and some constant m ( m ) ∈ R . The Stieltjes momentproblems for both positive, cf. (118) , and negative, cf. (119) , moments of M ( m , l , l ) are indeterminate. We finally note that the Mellin transform of the Selberg integral distribution has a remarkable factor-ization, which extends Selberg’s finite product of gamma factors to an infinite product.
Theorem 3.5 (Factorization of the Mellin transform) M ( q | m , l , l ) = t q G (cid:0) − q / t (cid:1) G (cid:0) − q + t ( + l + l ) (cid:1) G q (cid:0) − / t (cid:1) G (cid:0) − q + t ( + l + l ) (cid:1) ¥ (cid:213) m = ( m t ) q G (cid:0) − q + m t (cid:1) G (cid:0) + m t (cid:1) ⋆⋆ G (cid:0) − q + tl + m t (cid:1) G (cid:0) + tl + m t (cid:1) G (cid:0) − q + tl + m t (cid:1) G (cid:0) + tl + m t (cid:1) G (cid:0) − q + t ( l + l ) + m t (cid:1) G (cid:0) − q + t ( l + l ) + m t (cid:1) . (135)We conjectured in [45] in the special case of l = l = Conjecture 3.6
Let m ∈ ( , ) and l i > − m / , then Z s l ( − s ) l M m ( ds ) inlaw = M ( m , l , l ) . (136)The rationale for this conjecture is that we constructed a family of probability distributions parame-terized by m , l , and l having the properties that (1) its moments match the moments of the (gener-alized) total mass R s l ( − s ) l M m ( ds ) , i.e. the Selberg integral and (2) the asymptotic expansion ofits Mellin transform coincides with the intermittency expansion of the total mass. It is finally worthpointing out that the restriction l i > − m / ( q ) < / m . It is not difficult to see from (91) thatlog M m ( , t ) log t → + m , a.s. (137)in the limit t → R s l ( − s ) l M m ( ds ) is actually defined for l i + m / > − ( q ) < min { / m , + ( + l i ) / m } . Calculations
In this section we will give heuristic derivations of Conjectures 2.3 and 2.4 and their corollariesassuming Conjecture 3.6 to be true. The derivations are based on certain exact calculations, combinedwith a key assumption in the case of Conjecture 2.4 that is explained below.The main idea of the derivation of Conjecture 2.3 is that the S ( ) t ( m , u , e ) and S ( ) t ( m , u , e ) statisticsthat we introduced in (21) in Section 2 converge in law, up to O ( e ) terms, to (modifications of)the processes w m , e ( u ) − w m , e ( ) and w m , e ( u ) , respectively, in the limit t → ¥ , where w m , e ( u ) is theapproximation of the gaussian free field that we defined in (77)-(79) (recall L = Lemma 4.1 (Main lemma for S ( ) t ( m , u , e ) ) Let f ( ) e , u ( x ) be as in (18) , c ( ) u ( x ) = c [ , u ] ( x ) , f ( ) e , u ( x ) , ( c ( ) u ⋆ f e )( x ) = e Z c ( ) u ( x − y ) f ( y / e ) dy , (138) k be as in (17) , and the scalar product be defined as in (5) . Then, given < u , v < , in the limit e → , h f ( ) e , u , f ( ) e , v i = − p (cid:0) log e − k + log | u − v | − log | u | − log | v | (cid:1) + O ( e ) , if | u − v | ≫ e , (139) h f ( ) e , u , f ( ) e , u i = − p (cid:0) log e − k − log | u | (cid:1) + O ( e ) , if u = v . (140) Proof
Recall that f ′ ( ) e , u = f e ( x ) − f e ( x − u ) , where f e ( x ) , / ef ( x / e ) is the rescaled bump function,cf. (19). Then, Z f e ( x ) log | x − z | dx = ( log | z | + O ( e ) , if | z | ≫ e , log e + R f ( x ) log (cid:12)(cid:12) x − z e (cid:12)(cid:12) dx , if | z | < e . (141)It follows Z (cid:0) f e ( x ) − f e ( x − u ) (cid:1) log | x − y | dx = log | y | − log | u − y | + O ( e ) if | y | , | y − u | ≫ e , log e + R f ( x ) log (cid:12)(cid:12) x − y e (cid:12)(cid:12) dx − log | u − y | + O ( e ) if | y | < e , log | y | − log e − R f ( x ) log (cid:12)(cid:12) x − y − u e (cid:12)(cid:12) dx + O ( e ) (142)if | y − u | < e . Now, by the definition of the scalar product, h f ( ) e , u , f ( ) e , v i = − p Z (cid:0) f e ( y ) − f e ( y − v ) (cid:1)(cid:0) f e ( x ) − f e ( x − u ) (cid:1) log | x − y | dxdy , = − p Z f e ( y ) h(cid:0) f e ( x ) − f e ( x − u ) (cid:1) log | x − y | dx i dy + p Z f e ( y ) h(cid:0) f e ( x ) − f e ( x − u ) (cid:1) log | x − ( y + v ) | dx i dy . (143)The resulting integrals are all of the functional form that we treated in (141) and (142) so the resultfollows. 22 orollary 4.2 (Covariance structure of S ( ) t ( m , u , e ) ) Let the statistic S ( ) t ( m , u , e ) be as in (21) ,S ( ) t ( m , u , e ) , p p m h (cid:229) g f ( ) e , u (cid:0) g ( t ) (cid:1) − log t pl ( t ) Z f ( ) e , u ( x ) dx i . (144) Then, in the limit t → ¥ the process u → S ( ) t ( m , u , e ) , u ∈ ( , ) , converges in law to the centeredgaussian field S ( ) ( m , u , e ) having the asymptotic covariance Cov (cid:16) S ( ) ( m , u , e ) , S ( ) ( m , v , e ) (cid:17) = ( − m (cid:0) log e − k + log | u − v | − log | u | − log | v | (cid:1) , if | u − v | ≫ e , − m (cid:0) log e − k − log | u | (cid:1) , if u = v , + O ( e ) . (145) Proof
This follows from Theorem 2.1 and the calculation of the scalar product in Lemma 4.1.
Lemma 4.3 (Main lemma for S ( ) t ( m , u , e ) ) Let f ( ) e , u ( x ) be as in (18) , c ( ) u ( x ) = c [ − / e , u ] ( x ) . Then,given < u , v < , in the limit e → , h f ( ) e , u , f ( ) e , v i = − p (cid:0) e − k + log | u − v | (cid:1) + O ( e ) , if | u − v | ≫ e , (146) h f ( ) e , u , f ( ) e , u i = − p (cid:0) e − k (cid:1) + O ( e ) , if u = v . (147) Proof
The calculation is the same as in the proof of Lemma 4.1 and will be omitted.
Corollary 4.4 (Covariance structure of S ( ) t ( m , u , e ) ) Let the statistic S ( ) t ( m , u , e ) be as in (21) , cor-responding to f ( ) e , u . Then, in the limit t → ¥ the process u → S ( ) t ( m , u , e ) , u ∈ ( , ) , converges in lawto the centered gaussian field S ( ) ( m , u , e ) having the asymptotic covariance Cov (cid:16) S ( ) ( m , u , e ) , S ( ) ( m , v , e ) (cid:17) = ( − m (cid:0) e − k + log | u − v | (cid:1) , if | u − v | ≫ e , − m (cid:0) e − k (cid:1) , if u = v , + O ( e ) . (148) Proof
This follows from Theorem 2.1 and the calculation of the scalar product in Lemma 4.3.On the other hand, it is elementary to see from (77)-(79) that the gaussian free field at scale e satisfies Cov (cid:16) w m , e ( u ) , w m , e ( v ) (cid:17) = ( − m log | u − v | , if | u − v | > e , m (cid:0) − log e (cid:1) , if u = v , (149)so that the centered gaussian free field at scale e , ¯ w m , e ( u ) , w m , e ( u ) − w m , e ( ) , (150)satisfies Cov (cid:16) ¯ w m , e ( u ) , ¯ w m , e ( v ) (cid:17) = ( − m (cid:0) log e − + log | u − v | − log | u | − log | v | (cid:1) , if | u − v | > e , − m (cid:0) log e − − log | u | (cid:1) , if u = v . (151)23s we remarked in our discussion of (95), the choice of truncation, and, in particular, the choice ofthe constant, e − independent term in the covariance of the free field, has no effect on the law of thetotal mass so long as the key normalization condition in (80) holds. Hence, for our purposes, we canre-define w m , e ( u ) to be E (cid:2) w m , e ( u ) (cid:3) = − m ( k − log e ) , (152) Cov (cid:2) w m , e ( u ) , w m , e ( v ) (cid:3) = ( − m log | u − v | , e ≤ | u − v | ≤ , m ( k − log e ) , (153)so that ¯ w m , e ( u ) has asymptotically the same covariance as that of S ( ) ( m , u , e ) in (145). Hence, S ( ) ( m , u , e ) inlaw ≈ ¯ w m , e ( u ) (154)as stochastic processes in the limit e → , up to zero-mean corrections having covariance of the order O ( e ) . Similarly, by comparing covariances of S ( ) ( m , u , e ) in (148) and w m , e ( u ) in (153) and recallingthat S ( ) ( m , u , e ) is centered, while the mean of w m , e ( u ) is given in (152), we obtain, also up to O ( e ) , S ( ) ( m , u , e ) inlaw ≈ w m , e ( u ) + N (cid:16) − m ( log e − k ) , − m ( e − k ) (cid:17) , (155)where N (cid:0) − ( m / )( log e − k ) , − m ( e − k ) (cid:1) is an independent gaussian random variable havingthe mean − ( m / )( log e − k ) and variance − m ( e − k ) . We can now explain the origin of Conjecture 2.3. The basic idea is to form the exponentialfunctional of the statistic S ( i ) t ( m , u , e ) and compute its Mellin transform by analogy with the gaussianfree field so as to obtain the total mass of the limit lognormal measure in the limit. The principalobstacle in the first case is that S ( ) t ( m , u , e ) behaves like ¯ w m , e ( u ) as opposed to w m , e ( u ) so that itsexponential functional does not exist in the limit e → . This obstacle is overcome by appropriatelyrescaling the Mellin transform as shown in the following theorem. To this end, we will first formulatea general proposition and then specialize it to Conjecture 2.3. In what follows j ( u ) can be a generaltest function, however for clarity, we will restrict ourselves to j ( u ) , u l ( − u ) l , l , l > − m . (156) Theorem 4.5 (Rescaled Mellin transforms)
Let < m < and I be a subinterval of the unit inter-val. Then, for Re ( q ) < / m we have lim e → e m ( log e − k ) q ( q + ) E h(cid:16) Z I u − m q j ( u ) e ¯ w m , e ( u ) du (cid:17) q i = E h(cid:16) Z I j ( u ) M m ( du ) (cid:17) q i . (157) For − ( + l ) / m − / < Re ( q ) < / m we have lim e → e m ( log e − k ) q ( q + ) E h(cid:16) Z I j ( u ) e ¯ w m , e ( u ) du (cid:17) q i = E h(cid:16) Z I u m q j ( u ) M m ( du ) (cid:17) q i . (158) Moreover, the positive integral moments of order n < / m satisfy lim e → e m ( log e − k ) n ( n + ) E h Z I u − m n j ( u ) e ¯ w m , e ( u ) du i n = Z I n n (cid:213) i = (cid:2) j ( s i ) (cid:3) n (cid:213) i < j | s i − s j | − m ds · · · ds n , (159)lim e → e m ( log e − k ) n ( n + ) E h Z I j ( u ) e ¯ w m , e ( u ) du i n = Z I n n (cid:213) i = (cid:2) j ( s i ) s m ni (cid:3) n (cid:213) i < j | s i − s j | − m ds · · · ds n . (160)24he proof requires two auxiliary calculations. The first involves a version of Girsanov-type theoremfor gaussian fields, which extends what we used in our original derivation of intermittency differenti-ation, cf. Section 8 of [43]. For concreteness, we state it here for the w m , e ( t ) process. Lemma 4.6 (Girsanov)
Let I be a subinterval of the unit interval. Then, we have for Re ( p ) < / m , lim e → e m ( log e − k ) q ( q + ) E h e − q w m , e ( ) (cid:16) Z I u − m q j ( u ) e w m , e ( u ) du (cid:17) p i = E h(cid:16) Z I j ( u ) M m ( du ) (cid:17) p i , (161) and for Re ( q ) > − ( + l ) / m − / , Re ( p ) < min { / m , / m ( + l ) + + ( q ) } , lim e → e m ( log e − k ) q ( q + ) E h e − q w m , e ( ) (cid:16) Z I j ( u ) e w m , e ( u ) du (cid:17) p i = E h(cid:16) Z I u m q j ( u ) M m ( du ) (cid:17) p i . (162) Proof
The integral R I u m q j ( u ) M m ( du ) is defined for m Re ( q ) + l + m / > − ( p ) < min { / m , / m ( + l ) + + ( q ) } , cf. (137). Next, we will establish theidentities e m ( log e − k ) q ( q + ) E h e − q w m , e ( ) (cid:16) Z I u − m q j ( u ) e w m , e ( u ) du (cid:17) p i = E h(cid:16) Z I j ( u ) e w m , e ( u ) du (cid:17) p i , (163) e m ( log e − k ) q ( q + ) E h e − q w m , e ( ) (cid:16) Z I j ( u ) e w m , e ( u ) du (cid:17) p i = E h(cid:16) Z I u m q j ( u ) e w m , e ( u ) du (cid:17) p i . (164)This result is equivalent to a change of measure. Introduce an equivalent probability measure d Q , e − m ( k − log e ) q ( q + ) e − q w m , e ( ) d P , (165)where P is the original probability measure corresponding to E . In fact, it follows from (152) and(153) that E h e − m ( k − log e ) q ( q + ) e − q w m , e ( ) i = . (166)Then, the law of the process u → w m , e ( u ) − q Cov (cid:0) w m , e ( u ) , w m , e ( ) (cid:1) with respect to P equals thelaw of the original process u → w m , e ( u ) with respect to Q . Indeed, it is easy to show that the twoprocesses have the same finite-dimensional distributions by computing their characteristic functions.The continuity of sample paths can then be used to conclude that the equality of all finite-dimensionaldistributions implies the equality in law. Once this equality is established, then e − m ( k − log e ) q ( q + ) E h e − q w m , e ( ) (cid:16) Z I u − m q j ( u ) e w m , e ( u ) du (cid:17) p i = E (cid:12)(cid:12)(cid:12) Q h(cid:16) Z I u − m q j ( u ) e w m , e ( u ) du (cid:17) p i , = E h(cid:16) Z I j ( u ) e w m , e ( u ) du (cid:17) p i , (167)which completes the proof of (163) and of (161) by taking the limit e → . The argument for (162)is essentially the same except that one needs to restrict p so that the right-hand side is well-defined.Given (117), the conditions are Re ( p ) < / m and 1 − Re ( p ) + ( q ) + ( + l ) / m > . The second step in the proof of Theorem 4.5 entails a key moment calculation, in which wecompute the asymptotic of the rescaled positive integral moments in terms of generalized Selbergintegrals. 25 emma 4.7 (Single moments)
Let n < / m and I be a subinterval of the unit interval. Then, as e → , E h Z I u − m n j ( u ) e ¯ w m , e ( u ) du i n ∼ e m ( − log e + k ) n ( n + ) Z I n n (cid:213) i = (cid:2) j ( s i ) (cid:3) n (cid:213) i < j | s i − s j | − m ds · · · ds n , (168) E h Z I j ( u ) e ¯ w m , e ( u ) du i n ∼ e m ( − log e + k ) n ( n + ) Z I n n (cid:213) i = (cid:2) j ( s i ) s m ni (cid:3) n (cid:213) i < j | s i − s j | − m ds · · · ds n . (169) Proof
We will give the proof of (169). The idea is to discretize as in (95). Let N = | I | / e , ¯ w j , ¯ w m , e ( s j ) , s j = e j , j = · · · N . As ¯ w , · · · , ¯ w N are jointly gaussian with zero mean, we have E (cid:2) e N (cid:229) j = j ( s j ) e ¯ w j (cid:3) n = e n N (cid:229) j ... j n = j ( s j ) · · · j ( s j n ) e Var ( ¯ w j + ... + ¯ w jn ) . (170)Now, using (152) and (153), Var ( ¯ w j + . . . + ¯ w j n ) = n (cid:229) k = Var ¯ w j k + n (cid:229) k < l = Cov (cid:0) ¯ w j k , ¯ w j l (cid:1) = m n (cid:229) k = (cid:0) − log e + k + log s j k (cid:1) ++ m n (cid:229) k < l = (cid:0) − log e + k − log | s j k − s j l | + log | s j k | + log | s j l | (cid:1) , = m ( − log e + k ) n ( n + ) + m n n (cid:229) k = log s j k − m n (cid:229) k < l = log | s j k − s j l | . (171)It follows from (170) that E (cid:2) e N (cid:229) j = j ( s j ) e ¯ w j (cid:3) n ∼ e m ( − log e + k ) n ( n + ) e n N (cid:229) j ... j n = n (cid:213) k = (cid:2) j ( s j k ) s m nj k (cid:3) n (cid:213) k < l = | s j k − s j l | − m . (172)In the limit we have e (cid:229) Nj k = → R I ds k , hence the result. The proof of (168) follows by re-labeling j ( u ) → u − m n j ( u ) . We note that the same type of argument gives a formula for the joint moments.
Lemma 4.8 (Joint moments)
Let N = n + m , N < / m , and I , I be subintervals of [ , ] . Then, as e → , E h(cid:16) Z I j ( u ) e ¯ w m , e ( u ) du (cid:17) n (cid:16) Z I j ( u ) e ¯ w m , e ( u ) du (cid:17) m i ∼ e m ( − log e + k ) N ( N + ) Z I n × I m n (cid:213) i = (cid:2) j ( s i ) s m Ni (cid:3) × N (cid:213) i = n + (cid:2) j ( s i ) s m Ni (cid:3) N (cid:213) i < j | s i − s j | − m ds · · · ds N . (173)We can now give a proof of Theorem 4.5 and explain the origin of Conjecture 2.3.26 roof of Theorem 4.5. Let Re ( p ) < / m . By Lemma 4.6,lim e → e m ( log e − k ) q ( q + ) E h e − q w m , e ( ) (cid:16) Z I u − m q j ( u ) e w m , e ( u ) du (cid:17) p i = E h(cid:16) Z I j ( u ) M m ( du ) (cid:17) p i . (174)In particular, (174) holds for q = p , if Re ( q ) < / m . On the other hand, the left-hand side of (174)can be reduced when q = p as follows. Substituting the definition of ¯ w m , e ( u ) , cf. (150), we obtainlim e → e m ( log e − k ) q ( q + ) E h e − q w m , e ( ) (cid:16) Z I u − m q j ( u ) e w m , e ( u ) du (cid:17) q i , = lim e → e m ( log e − k ) q ( q + ) E h(cid:16) Z I u − m q j ( u ) e ¯ w m , e ( u ) du (cid:17) q i . (175)The argument for (158) is essentially the same except that we use (162) instead of (174). GivenRe ( q ) > − ( + l ) / m − / , Re ( p ) < min { / m , / m ( + l ) + + ( q ) } , if q = p the conditionRe ( p ) < min { / m , / m ( + l ) + + ( q ) } is equivalent to Re ( q ) > − ( + l ) / m − Proof of Conjecture 2.3.
Let j ( u ) = I = [ , ] in Theorem 4.5. Recalling the asymptoticequality of laws of ¯ w m , e ( u ) and S ( ) t ( m , u , e ) at finite e > t → ¥ that we establishedin (154) above, we formally interchange the order of t limit and u integration, take the t limit byCorollary 4.2, then replace S ( ) ( m , u , e ) with ¯ w m , e ( u ) by (154), and, finally take the limit e → l = l = , and l = m q , l = , respectively, cf. Theorem 3.2 above, hence (49) and(52). For example, the argument for (49) is as follows.lim e → e m ( log e − k ) q ( q + ) h lim t → ¥ E h(cid:16) Z u − m q e S ( ) t ( m , u , e ) du (cid:17) q ii , = lim e → e m ( log e − k ) q ( q + ) h E h(cid:16) Z u − m q e S ( ) ( m , u , e ) du (cid:17) q ii , = lim e → e m ( log e − k ) q ( q + ) h E h(cid:16) Z u − m q e ¯ w m , e ( u ) du (cid:17) q ii , = E h(cid:16) Z M m ( du ) (cid:17) q i . (176)The derivation of the weak conjecture for S ( ) t ( m , u , e ) follows the same steps but is simpler asit is based directly on the asymptotic equality of laws of w m , e ( u ) and S ( ) t ( m , u , e ) at finite e > t → ¥ , cf. (155) above. By taking the exponential functional of S ( ) t ( m , u , e ) as in (48) andusing (155), we obtain the Mellin transform of the total mass of the limit lognormal measure timesthe Mellin transform of exp (cid:0) N ( − ( m / )( log e − k ) , − m ( e − k )) (cid:1) . The latter accounts for thescaling factor in (48), and the result then follows from Conjecture 3.6 with l = l = . Similarly,(51) follows in the same way from Conjecture 3.6 with l = m q , l = . We next proceed to the derivation of Conjecture 2.4. The basic idea is to let e ( t ) approach zero“slowly” compared to the growth of t to infinity so that the statistic S ( i ) t ( m , u ) in (36) behaves as thecentered gaussian free field for i = i = e ( t ) . The main technical challenge of quantifying the required rate of decay of e ( t ) is27hat neither Theorem 2.1 nor 2.2 applies to the statistic S ( i ) t ( m , u ) . Theorem 2.1 does not apply as thevariance of S ( i ) t ( m , u ) is divergent in the limit t → ¥ . Theorem 2.2 does not apply as the asymptotic ofthe variance R log t / l ( t ) − log t / l ( t ) | w || b c u ( w ) | dw (cid:181) log ( log t / l ( t )) in (7) is different from the asymptotic of ourvariance ¥ R − ¥ | w || \ ( c u ⋆ f e )( w ) | dw (cid:181) − log e , cf. Lemmas 4.1 and 4.3 above. Instead, we need a slightmodification of Theorem 2.2. Lemma 4.9 shows that for a sufficiently slowly decaying e ( t ) one canobtain a limiting gaussian field having e ( t ) -dependent asymptotic covariance. Lemma 4.10 showsfurther that the limiting covariance can be approximated by the scalar product in Theorem 2.1. Let S t ( f ) be as in (3). For simplicity, we restrict ourselves here to finite linear combinations of indicatorfunctions (cid:229) c k c u k , < u k < . Lemma 4.9 (Modified convergence)
Let l ( t ) satisfy (2) and e ( t ) satisfy (34) for i = or (35) fori = , respectively. Let c ( ) u ( x ) = c [ , u ] ( x ) and c ( ) u ( x ) = c [ − / e ( t ) , u ] ( x ) . Define s ( i ) t , log t / l ( t ) Z − log t / l ( t ) | w | (cid:12)(cid:12)(cid:12) (cid:229) c k \ ( c ( i ) u k ⋆ f e ( t ) )( w ) (cid:12)(cid:12)(cid:12) dw , (177) then, as t → ¥ , S t (cid:16) (cid:229) c k c ( i ) u k ⋆ f e ( t ) (cid:17) inlaw = s ( i ) t Y ( i ) t + o ( ) , Y ( i ) t inlaw → N ( , ) . (178)The proof is sketched in the appendix. Let f ( i ) e , u ( x ) , ( c ( i ) u ⋆ f e )( x ) be as in Lemmas 4.1 and 4.3. Itfollows by linearity that the covariance has the asymptotic form as t → ¥ Cov (cid:16) S t ( f ( i ) e ( t ) , u ) , S t ( f ( i ) e ( t ) , v ) (cid:17) = Re log t / l ( t ) Z − log t / l ( t ) | w | [ f ( i ) e ( t ) , u ( w ) [ f ( i ) e ( t ) , v ( w ) dw + o ( ) . (179)It remains to show that this asymptotic is the same as that of the covariance in (139) and (140) for i = i = . This is shown in the next lemma.
Lemma 4.10 (Limit covariance)
Let l ( t ) satisfy (2) and e ( t ) satisfy (34) for i = or (35) for i = , respectively, and the scalar product be as in (4) . In the limit t → ¥ , Re log t / l ( t ) Z − log t / l ( t ) | w | [ f ( i ) e ( t ) , u ( w ) [ f ( i ) e ( t ) , v ( w ) dw = h f ( i ) e ( t ) , u , f ( i ) e ( t ) , v i + o ( ) . (180) Proof of Lemma 4.10.
Given the equality between the representations of the scalar product in(4) and (5), we need to estimate Re R | w | > log t / l ( t ) | w | [ f ( i ) e ( t ) , u ( w ) [ f ( i ) e ( t ) , v ( w ) dw . Denote a , log t / l ( t ) , f e ( x ) , f ( i ) e , u ( x ) , and g e ( x ) , f ( i ) e , v ( x ) . Then, by substituting the definition of the Fourier transform andintegrating by parts,Re a Z − a | w | ˆ f e ( w ) ˆ g e ( w ) dw = − p a Z dww h ZZ f ′ e ( x ) g ′ e ( y ) (cid:0) cos ( wy ) − cos ( w ( y − x )) (cid:1) dxdy i . (181)28he cos ( wy ) term can be replaced with cos ( w ) when integrating from 0 to a and dropped whenintegrating from a to infinity because R f ′ e ( x ) dx = . Using the Frullani integral in the form ¥ Z dww (cid:0) cos ( w ) − cos ( w ( y − x )) (cid:1) = log | y − x | , (182)we wish to show that the remainder term satisfies the estimate in the limit a → ¥ and e → , ¥ Z a dww h ZZ f ′ e ( x ) g ′ e ( y ) cos ( w ( y − x )) dxdy i = O ( / ae ) . (183)Let i = f ′ e ( x ) = f e ( x ) − f e ( x − u ) , g ′ e ( y ) = f e ( y ) − f e ( y − v ) , and theassumption that f ( x ) is compactly supported so that the only values of x and y that contribute to theseintegrals are concentrated in intervals of size e , it is sufficient to show ¥ Z a dww h Z f e ( x ) cos ( w x ) dx i = ¥ Z ae dww h Z f ( x ) cos ( w x ) dx i = O ( / ae ) , (184)which follows by integrating the inner integral by parts. Remark
One concludes that the statistic S ( ) t ( m , u ) and the centered gaussian free field ¯ w m , e ( t ) ( u ) have asymptotically the same law, as do the statistic S ( ) t ( m , u ) and the gaussian free field plus anindependent gaussian, as in (154) and (155) with the scale e ( t ) . The difference between S ( ) t ( m , u ) and S ( ) t ( m , u ) on the one hand and the corresponding free fields on the other is that these statisticsare only gaussian in the limit t → ¥ , whereas the free fields are gaussian for e > . We will assumethat deviations of these statistics from their gaussian limits can be ignored for sufficiently large t forthe purpose of carrying out calculations. We can quantify this assumption in terms of the behavior of Y ( i ) t near its gaussian limit in (178), by conjecturing the asymptotic of S ( i ) t ( m , u ) to be S ( i ) t ( m , u ) inlaw = s ( i ) t N ( , ) + o ( ) , (185)which means that Y ( i ) t converges to N ( , ) faster (in the sense of variance) than s ( i ) t (cid:181) − log e ( t ) goes to infinity. We can heuristically estimate the rate of convergence of Y ( i ) t to N ( , ) by therelative error in the formula for the variance of our statistics. It follows from (183) in the proof ofLemma 4.10 that the relative error is O (cid:0) l ( t ) / log t e ( t ) (cid:1) / | log e ( t ) | so that the rate of convergence ≪ / | log e ( t ) | by (34) as required for (185) to be true. Obviously, the assumption in (185) is a majorgap between the weak and strong conjectures. Proof of Conjecture 2.4.
The argument is the same as that for Conjecture 2.3 above except thatwe use Lemma 4.9 instead of Theorem 2.1 and Lemma 4.10 instead of Lemmas 4.1 and 4.3, and, inaddition, assume that S ( i ) t ( m , u ) is near-gaussian for large but finite t as remarked above in (185).The proofs of the corollaries are given below. They are the same for either the single or doublelimit on the left-hand side. For concreteness, they are stated for the single limit as in Section 2.29 roof of Corollary 2.5. We computed the positive moments and verified (57) in Lemma 4.7. On theother hand, using (49), (58) follows from the known formula for the positive integral moments of theSelberg integral distribution, cf. (118). The argument for (59) and (60) is the same.
Proof of Corollary 2.6.
This again follows from Conjecture 2.4 and the known formula for thenegative integral moments of the Selberg integral distribution, cf. (119).
Proof of Corollary 2.7.
This is a special case of Lemma 4.8.
Proof of Corollary 2.8. (65) is a special case of (122) corresponding to l = l = . (66) is a specialcase of the following asymptotic expansion in the limit m → , M ( q | m , m z , ) ∼ exp (cid:16) ¥ (cid:229) r = (cid:16) m (cid:17) r + r + h z ( r + ) (cid:2) B r + ( q + ) + B r + ( q ) r + ++ B r + ( q − z ) − B r + ( − z ) − B r + r + − q (cid:3) + (cid:0) z ( r + ) − (cid:1)(cid:2) B r + ( q − − z ) − B r + ( q − − z ) r + (cid:3)i(cid:17) . (186)This expansion coincides with the intermittency expansion of E h(cid:0) R s m z M m ( ds ) (cid:1) q i as one can verifyby the methods of the previous section. Another way of deriving (186) is to follow the argument thatwe used in [48] to derive (122) from (117). Then, (66) corresponds to z = q in (186). Proof of Corollary 2.9.
We will give the argument for the second statistic. We have by (155),
Cov (cid:16) log s + D Z s e S ( ) t ( m , u ) du , log s + D Z s e S ( ) t ( m , u ) du (cid:17) ≈ Cov (cid:16) log s + D Z s e w m , e ( t ) ( u ) du , log s + D Z s e w m , e ( t ) ( u ) du (cid:17) −− m ( e ( t ) − k ) , (187)and the result follows by (98) and the stationarity property of the limit lognormal measure. Theargument for the first statistic is similar but is more involved as it requires a generalization of (98) forthe centered gaussian free field, which follows from (173) in the same way as (98) follows from (97).The details are straightforward and will be omitted. Proof of Corollary 2.10.
Given Conjecture 2.4, these results follow from Theorems 3.2 and 3.4.
Proof of Corollary 2.11.
Given the multifractal law of the limit lognormal measure, cf. (91), it issufficient to show the identity for any 0 < s < t → ¥ e m ( log e ( t ) − k ) q ( q + ) E h(cid:16) Z s u − m q e S ( ) t ( m , u ) du (cid:17) q i = E h(cid:16) Z s M m ( du ) (cid:17) q i . (188)which is a special case of (157) corresponding to j ( u ) = I = [ , s ] (by formally replacing¯ w m , e ( u ) with S ( ) t ( m , u ) as in the derivation of Conjecture 2.4 above).30 roof of Corollary 2.12. Given the multiscaling law of the limit lognormal measure, (74) followsfrom (92). To prove (75), we need to generalize (91) to the following identity in law, Z s u m q M m ( du ) inlaw = s + m q e W s Z u m q M m ( du ) , (189)where W s is as in (86) and (87), which implies the multiscaling law E h(cid:16) Z s u m q M m ( du ) (cid:17) q i (cid:181) E h(cid:16) s + m q e W s (cid:17) q i , (cid:181) s q + m ( q + q ) (190)as a function of s < . Finally, (189) is a simple corollary of (88).
Remark
It should be clear from Theorem 3.2 and Theorem 4.5 that one can obtain general explicitformulas for j ( u ) = u l ( − u ) l . The reason that we restricted ourselves to l = l = We have formulated two versions of a precise conjecture on limits of rescaled Mellin-type transformsof the exponential functional of the Bourgade-Kuan-Rodgers statistic in the mesoscopic regime. Theconjecture is based on our construction of particular Bourgade-Kuan-Rodgers statistics of Riemannzeroes that converge to modifications of the centered gaussian or gaussian free fields. The statisticsare defined by smoothing the indicator function of certain bounded or unbounded subintervals of thereal line. The smoothing is effected by a rescaled bump function. In the weak version of the conjec-ture the asymptotic scale of the bump function e is fixed so that the resulting statistics S ( i ) t ( m , u , e ) , i = , , satisfy the Bourgade-Kuan-Rodgers theorem. In the strong version, this scale e ( t ) is chosento be mesoscopic, l ( t ) / log t ≪ e ( t ) ≪ i = l ( t ) / log t ≪ e ( t ) ≪ e ( t ) ≫ / l ( t ) for i = , so that the statistics S ( i ) t ( m , u ) satisfy an extension of the Bourgade-Kuan-Rodgers theoremthat we formulated in the paper. We have computed the double limit over the scale e and t in the weakcase and the single limit over t in the strong case of rescaled Mellin-type transforms of the exponen-tial functional of both statistics as if the statistics were the centered gaussian free field or the gaussianfree field plus an independent gaussian random variable, respectively. The exponential functional ofthe gaussian free field is an important object in mathematical physics known as the limit lognormalstochastic measure (or lognormal multiplicative chaos). By using a Girsanov-type result, we havefound an appropriate rescaling factor to compute two Mellin-type transforms of the exponential func-tional of the centered field in terms of the Mellin transform of the exponential functional of the freefield itself, i.e. the Mellin transform of the total mass of the limit lognormal measure, resulting inthe conjecture for the first statistic. The conjecture for the second statistic follows directly from itsconvergence to the gaussian free field plus an independent gaussian, the latter being responsible forrescaling. In both cases, the rescaling factors are determined by the asymptotic scale and the choiceof the bump function that effect the smoothing. Finally, our conjectural knowledge of the distributionof the total mass of the measure has allowed us to calculate a number of quantities that are associatedwith the statistics exactly. 31he principal difference between the weak and strong conjectures is their informational content.Conjecture 2.3 alone can be thought of as a number theoretic re-formulation of Conjecture 3.6 onthe equality of the Selberg integral distribution and the law of the total mass of the limit lognormalmeasure. It associates the Selberg integral distribution with the zeroes but does not contain any infor-mation about their distribution that is not already contained in the Bourgade-Kuan-Rodgers theorem.Indeed, the order of the u integral and the t limit can be interchanged at a finite e > , resulting in theMellin transform of the exponential functional of a gaussian field, which converges to the centeredgaussian free field when i = i = e → , so that the weak conjecture only requires the t → ¥ limit ofthe S ( i ) t ( m , u , e ) statistic. On the other hand, Conjecture 2.4 combines the e → t → ¥ limits intoa single limit so that the order of the u integral and the resulting limit can no longer be interchangedbecause the statistic S ( i ) t ( m , u ) , unlike S ( i ) t ( m , u , e ) , converges to the centered gaussian free field when i = i = e ( t ) and so becomes singular as t → ¥ . The strong conjecture assumes that deviations of S ( i ) t ( m , u ) from its gaussian limit are negligible at large but finite t , thereby providing some new informationabout the statistical distribution of the zeroes at finite t . In particular, as the strong conjecture fits intothe framework of mod-gaussian convergence, the normality zone and precise deviations of tails ofour exponential functionals can be computed from the general theory and explicit knowledge of ourlimiting functions.We have provided a self-contained review of some of the key properties of the limit lognormalmeasure and the distribution of its total mass to make our work accessible to a wider audience. Inparticular, we have covered the invariances of the gaussian free field, the multifractal law of the limitmeasure, the derivation of the law of its total mass by exact renormalization, and several character-izations of the Selberg integral distribution, which is believed to describe the law of the total mass.The Selberg integral distribution is a highly non-trivial, log-infinitely divisible probability distributionhaving the property that its positive integral moments are given by the Selberg integral of the samedimension as the order of the moment. We have reviewed both its analytic and probabilistic structuresthat are relevant to out calculations.We have provided a number of calculations that support our conjecture. Our calculations areuniversal as they apply to any asymptotically gaussian linear statistic having the covariance structurethat is given by the Bourgade-Kuan-Rodgers formula. In particular, they apply to the GUE statistics ofFyodorov et. al. [27] and provide a theoretical explanation, via convergence to the gaussian free field,for why our results for the Bourgade-Kuan-Rodgers and GUE statistics are the same. Our argumentsare however not mathematically rigorous as we do not know how our statistics behave near theirgaussian limits. Our assumption that their behavior in the limit can be used to do calculations nearthe limit in the strong case is the principal mathematical gap between the weak and strong conjecturesthat renders our use of the free fields in place of the statistics heuristic. We have quantified that thevariances of our statistics should converge to their asymptotic limits faster than 1 / | log e ( t ) | for thisassumption to be valid and explained heuristically why we expect this to be true.In broad terms, on the one hand, our conjecture relates a limit of a statistic of Riemann zeroeswith the Selberg integral and, more generally, the Selberg integral probability distribution and soassociates a non-trivial, log-infinitely divisible distribution with the zeroes. On the other hand, ourconjecture implies that the limit lognormal measure can be modeled in terms of the zeroes. As thismeasure appears naturally in various contexts that involve multifractality, we can speculate that thereis a number theoretic interpretation of multifractal phenomena. In particular, a proof of (even the32eak) the conjecture might lead to a number theoretic proof of the conjectured equality of the law ofthe total mass of the limit lognormal measure and the Selberg integral distribution.We have interpreted our statistics as fluctuations of the smoothed error term in the zero countingfunction so that our conjecture gives rescaled Mellin transforms of the exponential functional of thesefluctuations. Aside from verification or disproof of our conjecture, it would be quite interesting to seewhat properties of the error term follow from the conjecture and to compute the rate of convergenceof our statistics to their gaussian limits. We believe that our conjecture is only valid for 1 ≪ l ( t ) ≪ log t and breaks down for l ( t ) ∼ z , which we believe will clarify the relationshipbetween their conjecture and ours. Acknowledgements
The author wants to thank Jeffrey Kuan for a helpful correspondence relating to ref. [15]. The authoralso wishes to express gratitude to Nickolas Simm for bringing ref. [27] to our attention, pointing outthe analogy between the Bourgade-Kuan-Rodgers and GUE statistics, and simplifying our proof ofEq. (16). The author is grateful to the referees for many helpful suggestions.
A Appendix
In this section we will give a proof of (16) and a sketch of the proof of Lemma 4.9. The startingpoint is the second equation following Eq. (16) in [15], (cid:229) g Im (cid:16) g ( t ) − ( x + iy ) (cid:17) −
12 log t l ( t ) = l ( t ) Re z ′ z (cid:16) + y l ( t ) + i ( w t + x l ( t ) (cid:17) + l ( t ) O (cid:16) | log ( w + xt l ( t ) | (cid:17) . (A.1)Recalling the identity lim y → Z f ( x ) Im (cid:16) g − ( x + iy ) (cid:17) dx = p f ( g ) , (A.2)we multiply (A.1) by f ( x ) / p , integrate with respect to x , and let y → . We then obtain by (3) S t ( f ) = pl t Z R f ( x ) Re ¶¶ s log z ( s ) | s = + i (cid:0) w t + x l t (cid:1) dx + O (cid:0) / l ( t ) (cid:1) , (A.3) = p Z R f ( x ) ¶¶ x Im h log z (cid:16) + i (cid:0) w t + x l t (cid:1)(cid:17)i dx + O (cid:0) / l ( t ) (cid:1) , (A.4) = − Z R f ′ ( x ) S (cid:16) w t + x l t (cid:17) dx + O (cid:0) / l ( t ) (cid:1) . (A.5)It should be noted that Re ( z ′ / z )( s ) is integrable along the critical line due to Hadamard’s factoriza-tion, which shows that the singular part of ( z ′ / z )( s ) near the zeroes is of the form 1 / ( s − g ) and This proof is due to Nickolas Simm. We originally established this result as a corollary of Eq. (17) in [15]. Uponseeing our derivation, Nickolas Simm found a much simpler proof, which is given here.
33o is purely imaginary along the critical line, hence vanishing upon taking the real part. It is clearfrom (A.1) that the error term is indeed O (cid:0) / l ( t ) (cid:1) provided f ( x ) is compactly supported and thesupport is independent of t . If the support depends on t as in (36), provided its size ≪ t l ( t ) , the erroris O (cid:0) || f || / l ( t ) (cid:1) as indicated in footnote 7.The proof of Lemma 4.9 follows verbatim the proof of Theorem 2.2 given in [15]. We will onlyindicate the main steps here and refer the reader to [15] for all details, including definitions of specialnumber theoretic functions that are needed in the proof. Proof of Lemma 4.9.
Let c u ( x ) = c [ , u ] ( x ) , i.e. i = . Let { c k } and { u k } be fixed and denote f t ( x ) , (cid:16) (cid:229) c k c u k ⋆ f e ( t ) (cid:17) ( x ) . (A.6)Then, f t and f ′ t are bounded in L uniformly in t , and f ′′ t satisfies || f ′′ t || = O ( / e ( t )) . (A.7)The Fourier transform of f t satisfies uniformly in t as w → ¥ w | b f t ( w ) | , (cid:0) w | b f t ( w ) | (cid:1) ′ = O ( / w ) . (A.8)Then, by the approximate explicit formula of Bourgade and Kuan, cf. Proposition 3 in [15], S t ( f t ) = l ( t ) (cid:229) n ≥ L √ t ( n ) √ n (cid:16)b f t (cid:16) log n l ( t ) (cid:17) n i w t + cc (cid:17) + error term , (A.9)where L u ( n ) denotes Selberg’s smoothed von Mangoldt function. In our case, as in the case ofTheorem 2.2, the error term is of the order O ( l ( t ) / log t e ( t )) . Let s t be as in (177) and denote Y t , √ s t l ( t ) (cid:229) primes p L √ t ( p ) √ p b f t (cid:16) log p l ( t ) (cid:17) p i w t . (A.10)Then, by (A.9) and Lemma 4 of [15], S t ( f t ) = s t √ (cid:16) Y t + Y t (cid:17) + O (cid:16) l ( t ) log t e ( t ) (cid:17) + o ( ) (A.11)in the limit t → ¥ . Finally, by Lemma 5 and Proposition 6 of [15], Y t converges to the standardcomplex normal variable Y t → N ( , ) + i N ( , ) √ , (A.12)The result for i = c u ( x ) = c [ − / e ( t ) , u ] ( x ) and f t be as in (A.6). The argument given above still goes throughbut requires somewhat more delicate estimates. We have for n = , , , || f ( n ) t || = O ( / e ( t )) , (A.13) || x log x f ( n ) t || = O ( log ( / e ( t )) / e t ) , (A.14)34o that (A.9) still holds with the error term of the order O ( l ( t ) / log t e ( t )) . To verify (A.11) we needLemma 4 of [15], which requires that || f t || be uniformly bounded, whereas we have (A.13) instead.However, a careful reading of the proof of Lemma 4 indicates that it still holds provided || f t || l ( t ) = o ( ) , (A.15)which in our case translates into 1 e ( t ) l ( t ) = o ( ) . (A.16)Thus, (A.11) holds given the condition in (35). Similarly, Lemma 5 of [15] requires the bounds in(A.8), whereas we have instead w | b f t ( w ) | = O ( / w ) , (A.17) (cid:0) w | b f t ( w ) | (cid:1) ′ = O ( / w e ( t )) . (A.18)Once again, a careful reading of the proof of Lemma 5 indicates that the bound in (A.18) is sufficientprovided 1 / e ( t ) l ( t ) = o ( ) as in (35). The rest of the argument goes though verbatim. References [1] R. Allez, R. Rhodes, V. Vargas (2013), Lognormal ⋆ − scale invariant random measures, Probab.Theory Relat. Fields , 751-788.[2] K. Astala, P. Jones, A. Kupiainen, E. Saksman (2011), Random conformal weldings,
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