Random Matrices and the Riemann zeta function

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1 Random Matrices and the Riemann zeta function Paul Bourgade, Marc Yor To cite this version: Paul Bourgade, Marc Yor. Random Matrices and the Riemann zeta function <hal-00940> HAL Id: hal Submitted on 0 Dec 006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Random Matrices and the Riemann zeta function Paul Bourgade ) and Marc Yor ),) ) Laboratoire de Probabilités et Modèles Aléatoires, Universités Paris VI et VII, 4 Place Jussieu - Case 88, F-755 Paris Cedex 05 ) Institut Universitaire de France Abstract. These notes are based on a talk given at the Institut de Mathématiques Élie Cartan de Nancy in June 006. Their purpose is to introduce the reader to some links between two fields of mathematics : analytic number theory and random matrices. After some historical overview of these connections, we expose a conjecture about the moments of the Riemann zeta function, formulated by Keating and Snaith in [8]. Last, we give some probabilistic interpretations of their corresponding results about unitary random matrices. Introduction. The Riemann zeta function : definition and some conjectures Following Patterson in [0] the Riemann zeta function can be defined, for σ := Rs) >, as a Dirichlet series or an Euler product : ζs) := n s =, n= p P p s where P is the set of all prime numbers. This function admits a meromorphic continuation to C, and is regular except at the single pole s =, with residue. Moreover, the classical functional equation holds : ) s s π s/ Γ ζs) = π ) s)/ Γ ζ s). ) The zeta function admits trivial zeros at s =, 4, 6,... corresponding to the poles of Γs/). From equation ) one can check that all non-trivial zeros are confined to the "critical strip" 0 σ, and they are symmetrically positioned about the real axis and the "critical line" σ = /. The Riemann hypothesis states that all these zeros lie on the critical line. The importance of the Riemann hypothesis in many mathematical fields justifies the intensive studies which continue to be made about the behavior of ζ/ + it) for t 0. In particular, there is the following

3 Paul Bourgade and Marc Yor famous conjecture, which is implied by the Riemann hypothesis. The Lindelöf hypothesis. For every ε > 0, as t, ) ζ + it = Ot ε ). ) This conjecture can be shown to be equivalent to another one relative to the moments of ζ on the critical line. The moments hypothesis. For all ε > 0 and k N, as T, I k T) := T T 0 ) ds k ζ + is = OT ε ). 3) Equation 3) may be easier to study than ), because it deals with means of ζ and no values at specific points. By modeling the zeta function with the characteristic polynomial of a random unitary matrix, Keating and Snaith [8] even got a conjecture for the exact equivalent of I k T). This will be exposed in part 3 of these notes, and urges us to define precisely some families of random matrices for compact groups.. Random matrices : classical compact groups and Haar measure Let us first define the following classical compact groups : On) orthogonal group) is the set of n n matrices O such as OO T = I n ; SOn) is the subgroup of On) of matrices A with deta) = ; Un) unitary group) is the set of n n matrices U such as UU = I n, with U the conjugate of U T ; ) 0 In USpn) the symplectic group) is the subgroup of Un) of matrices U commuting with. I n 0 General theory about topological groups implies the following well-known result. Existence and uniqueness of a Haar measure. Let Γ = On), SOn), Un) or USpn). There exists a unique probability measure µ Γ on Γ such as : µ Γ A) > 0 for all nonempty open sets A Γ; µ Γ ga) = µ Γ A) for all g Γ and nonempty open sets A Γ. This measure µ Γ is called the Haar measure of Γ. There is no generic way to choose an element of a group endowed with the Haar measure, but for the special cases considered here, we mention the following results, with no proofs. Let G be a random n n matrix, with all G jk s independent standard normal variables. Then, if O is the Gram-Schmidt transform of G, its distribution is the Haar measure µ On). The same method for generating an element of Un) applies, with this time G jk = a jk +ib jk, and all a jk s and b jk s standard independent normal variables. A decomposition of an orthogonal matrix into reflections leads to the following generator of µ On), as proven in [9]. Let e be the unit vector corresponding to the first coordinate of R n, and v be a unit vector chosen uniformly on the sphere S n. Let u := v + e v + e

4 Random matrices and the Riemann Zeta Function 3 be the unit vector along the bisector of v and e u is well-defined with probability ). The reflection with respect to the hyperplane orthogonal to u is Hv) = I n uu T. If O n is distributed with the Haar measure µ On, and v is uniformly distributed on S n, independently of O n, then ) 0 O n := Hv) 4) 0 O n is distributed with the Haar measure µ On. Similar decompositions for Un) and USpn) exist. A decomposition like 4) can be useful to compute the of some functionals of random matrices. Let us now consider specially the unitary group Un). The Haar measure induces a measure on the eigenvalues e iθ,...,e iθn ) of a unitary matrix. More precisely, the following formula holds. by Weyl s integration formula. With the previous notations, the joint distribution of θ,..., θ n ) is given µ Un) dθ,...,dθ n ) = π) n n! j<m n e iθ j e iθm dθ... dθ n. 5) Similar formulas exist for other compact groups, but we shall not need them in the following. The zeta function and random unitary matrices : probabilistic links. The encounter of Montgomery and Dyson Let us assume the Riemann hypothesis and write all non-trivial roots of the zeta function / ± it n with 0 < t t t n... After making the normalization w n = tn tn π log π, it has been shown that NW) W, W where NW) = n : w n < W. This is a key step for an analytic proof of the prime number theorem, shown independently by Hadamard and de la Vallée Poussin in 896. A further step is the study of the pair correlation w n, w m ) [0, W] : α w n w m β, W and more generally the operator R f, W) = W j,k NW),j k fw j w k ). The following theorem establishes the asymptotics of R f, W) for a large class of functions f. Theorem Montgomery, 973). If the Fourier transform of f ˆfτ) := dxfx)eiπxτ ) has a compact support included in [, ], then with R f, W) W ) sinπx R x) :=. πx dxfx)r x) 6) A conjecture due to the number theorist H. L. Montgomery states that the result remains true even if ˆf has no compact support, and this problem is still open...

5 4 Paul Bourgade and Marc Yor Is there a connection with Random Matrix Theory? In 97, at the Princeton Institute for Advanced Studies, Montgomery and some colleagues stopped working for afternoon tea, an important ritual for most mathematics departments. There, he was introduced to the quantum physicist Freeman Dyson by Indian number theorist Daman Chowla. Then Montgomery described his model for the pair correlation of zeros of the zeta function, and Dyson recognized the pair correlation of eigenvalues of a unitary matrix... The connection was born! More precisely, let e iθ,..., e iθn ) be the eigenvalues of an element A UN). We write φ n := N π θ n for the normalized angles θ n is considered modulo π, but not φ n ). Then it can be shown that β ) ) sin πx dµ UN) A) n, m) : α < φ n φ m < β dx, N UN) N α πx and, more generally, for a continue function f with compact support, the asymptotic of the pair correlation for eigenvalues in UN) is given by N UN) j k fφ j φ k )dµ UN) A) dxfx)r x). 7) N The similarity between formulas 6) and 7) suggests a strong connection between the zeros of the Riemann zeta function and the spectra of the random unitary matrices. This link is reinforced by the Polya-Hilbert program : if one writes the non trivial zeros of the zeta function as / ± it n n ), showing that the t n s are the eigenvalues of an hermitian operator would imply the Riemann hypothesis.. Two central limit theorems The of a unitary matrix determinant gives another example of similarity between Random Matrix Theory and the zeta function. First, we recall the following famous result due to Selberg. Theorem. For any regular Borel set Γ C T dt log ζ + it) Γ T T log log T T here, and in the following, log is a continuous determination of the complex logarithm). Γ dxdy x +y π e 8) In other terms, if our probability space is the interval [, ], fitted with Lebesgue measure du), and if L T u) := log ζ + iut), log log T then, with N standing for a standard normal variable on C, L T T N. Here, regular means that its boundary has zero Lebesgue measure.

6 Random matrices and the Riemann Zeta Function 5 This theorem admits a "cousin", concerning the characteristic polynomial of a generic unitary matrix A UN). Precisely, let us define ZA, θ) := det I N Ae iθ) = where e iθ,...,e iθn are the eigenvalues of A. n= Theorem. For any regular Borel set Γ C dµ UN) A) log ZA, θ) Γ UN) log N e iθn θ)), 9) N Γ dxdy x +y π e 0) Note that the limit does not depend on θ : by definition, the Haar measure is invariant under action of the unitary group especially under action of e iθ I N ), so the of ZA, θ) is the same as the of ZA, 0). Remark : about the interval [T,T] in Selberg s theorem. One may ask what happens if instead of integrating between T and T one does so on the interval [0, T]. A simple dominated convergence argument shows that there is no difference in doing so. Here is the argument. Denote γt, T) := log ζ + it) Γ. log log T Then, write T dt γt, T) = T 0 n+ dt γt, T). n+) T n+) T }{{} =:u nt) n=0 n T As log log n+) T) log log T, every u n T) converges to dxdy T Γ π e x +y. Moreover, the u n T) s are T uniformly bounded by sup T>0 T T dt γt, T) < ) as T, so from the dominated convergence argument one can exchange the sum and the limit, which gives the result. Remark : a "true" multidimensional generalization of Selberg s theorem. In order to transform the "static" limit in 8) into a "dynamic" one ie : showing the convergence in of a sequence of processes towards another process), it is somewhat natural to replace T by N λ with λ > 0. For a fixed λ, we define L λ u, N) := log ζ + iunλ). log N Then, if u is uniformly distributed on [, ], Selberg s theorem immediately implies that where L λ, N) N N λ, PN λ dxdy) = x +y πλ e λ dxdy. ) As λ varies, there is the following multidimensional extension of 8) and ).

7 6 Paul Bourgade and Marc Yor Theorem. [7]) For 0 < λ < λ < < λ k, dy L λ u, N) Γ,..., L λk u, N) Γ k ) N k PN λi Γ i ), a result which may be interpreted as the convergence of the finite dimensional marginals of {L λ, N), λ > 0} towards those of {N λ, λ > 0}, which denotes the totally disordered Gaussian process : its components are all independent, N λ being distributed as indicated in ). This theorem suggests some comments. The process N λ, λ > 0) does not admit a measurable version. Assuming the contrary we would obtain by Fubini : for every a < b, b a dλ N λ = 0. Thus N λ = 0, dλ a. s., which is absurd. In the asymptotic study of the occupation measure of planar Brownian motion Z t, t 0), where for f L T C, dxdy) there is the convergence in of log T 0 dtfz t) towards an exponential variable, the change of scale T = N λ was also performed, but the resulting limiting process f e λ, λ > 0) is not totally disordered but is in fact an "honest" measurable process. A totally disordered Gaussian process Xθ) + iy θ)) has also been encountered by Hugues, Keating and O Connell in [6]. They have proven that, with the notation 9), log Z N A, θ) log N converges weakly towards a totally disordered process with covariance 3. The moments of zeta i= EXθ )Xθ )) = EY θ )Y θ )) = δθ θ ). 3 The Keating-Snaith conjecture In the following we keep the notation from 3) for the moments of the zeta function. The mean I k T) has been extensively studied, but its equivalent is well known in only two cases : I T) log T Hardy-Littlewood, 98); T I T) T πt log T) Ingham, 96). Keating and Snaith [8] have formulated the following conjecture about the asymptotics of I k T). The Keating-Snaith conjecture. For every k N with the following notations : the "arithmetic" factor I k T) T H Mat k)h P k)log T) k H P k) := p P ) k ) ) Γk + m) p m!γk) p m ; m=0 the "matrix" factor H Mat k) := k j=0 j! j + k)!. )

8 Random matrices and the Riemann Zeta Function 7 Here are some first steps to understand the origins of this conjecture. Imagine that all prime numbers were "independent", in such a way that one might approximate I k T) = E ζs) k ) = E ) k p P p by ) s p P E k p the expectations are taken s with respect to Lebesgue measure ds/t on the segment [/, /+ it]). Then one would get exactly H P k)log T) k for the equivalent of I k T). So the coefficient H Mat k) appears as a constant correction compared to this "independent" model. Another model used to approximate the zeta function is the determinant of a unitary matrix : all previous analogies can comfort us in the accuracy of such an approximation. Thanks to Selberg s integral formula [], Keating and Snaith [8] have calculated the generating function for the determinant of a random unitary matrix with respect to the Haar measure : P N s, t) := E Z N A, θ) t e is arg ZNA,θ)) = Thanks to this closed form, they showed in particular that E Z N A, θ) k) j= Γj)Γt + j) Γj + t+s H Mat k)n k, N )Γj + t s ). 3) where H Mat k) is defined by ). This led them to introduce this "matrix" factor in the conjectured asymptotic of I k T). However, these two explanations are not sufficient to understand really how these "arithmetic" and "matrix" factors must be combined to get the Keating-Snaith conjecture. A better comprehension of this point is the purpose of the following paragraph. 3. Understanding the conjecture : the hybrid GHK model In [4], Gonek, Hughes and Keating were able to give a partial justification for the Keating-Snaith conjecture based on a particular factorization of the zeta function. More precisely, let us first introduce some notations. Let s = σ + it with σ 0 and t, let X be a real parameter, and let K be any fixed positive integer. Let ux) be a nonnegative C function of mass, supported on [e /X, e], and set Uz) := 0 ux)e z log x)dx, where E z) is the exponential integral z e w /w) dw. Let also P X s) := exp Λn) n s log n n X where Λ is Van Mangoldt s function Λn) = log p if n is an integral power of a prime p, 0 otherwise), and ) Z X s) := exp ρ n U s ρ n )log X) where ρ n, n 0) are the imaginary parts of the zeros of ζ. Then the following result, independent of any conjecture, was proved in [4]. Theorem. With the previous notations, X K+ ζs) = P X s)z X s) + O s log X) K where the constants before the O only depend on the function u and K. ) + O X σ log X )),

9 8 Paul Bourgade and Marc Yor With this decomposition, the P X term leads to the "arithmetic" factor of the Keating-Snaith conjecture, while the Z X term leads to the "matrix" factor. More precisely, this decomposition suggests a proof for the Keating-Snaith conjecture along the following steps. First, for a value of the parameter X chosen such as X = OlogT) ε ) here and in the sequel, the following conjecture A suggests that the moments of zeta are well approximated by the product of the moments of P X and Z X they are sufficiently "independent"). Conjecture A : the splitting hypothesis. With the previous notations T T 0 ) ds k ζ + is T T T 0 ds P X ) ) k + is T T 0 ds Z X ) ) k + is Assuming that conjecture A is true, we then need to approximate the moments of P X and Z X. The moments of P X were evaluated in [4], where the following result is proven. Theorem : the moments of P X.With the previous notations, T T 0 ds P X ) k )) + is = H P k) e γ log X) k + O. log X Finally, a conjecture about the moments of Z X, if proven, would be the last step. Conjecture B : the moments of Z X.With the previous notations, T T 0 ds Z X ) k + is H Matk) t ) k log T e γ. log X The reasoning which leads to conjecture B is the following. First of all, the function Z X is not as complicated as it seems, because as X tends to, the function u tends to the Dirac measure at point e, so Z X + it ) ρ n it ρ n )e γ log X). The ordinates ρ n where ζ is zero) are supposed to have many statistical properties identical to those of the eigenangles of a random element of UN). In order to make an adequate choice for N, we recall that the γ n are spaced π/ log T on average, whereas the eigenangles have average spacing π/n : thus N should be chosen to be the greatest integer less than or equal to log T. Then the calculation for the moments of this model leads to conjecture B. 4 Probabilistic interpretations for some Keating-Snaith formulas The main point of this section is to give a simple representation for Z N A, θ), thanks to formula 3). We first recall some results about gamma and beta variables. 4. Some formulas about the beta-gamma algebra A gamma random variable γ a with coefficient a > 0 has density on [0, [ given by P γ a dt) dt = ta Γa) e t.

10 Random matrices and the Riemann Zeta Function 9 Its Mellin transform is s > 0) Eγa s Γa + s) ) =. 4) Γa) A beta random variable β a,b with strictly positive coefficients a and b has density on [0, ] given by P β a,b dt) dt = Γa + b) Γa)Γb) ta t) b. Its Mellin transform is s > 0) E βa,b s ) Γa + s) Γa + b) = Γa) Γa + b + s). These formulas will also be useful in the following for justifications, the reader could see [3]). The algebra property, where all variables are independent : β a,b γ a+b = γa. 5) The duplication formula for the gamma function, with all variables independent : γ j = γ j γ j+. 6) 4. Decomposition of Z N into independent random variables Let us recall formula 3) : P N s, t) := E Z N A, θ) t e isarg ZNA,θ)) = j= Γj)Γt + j) Γj + t+s )Γj + t s ). Remark. As we already mentioned, the result does not depend on θ, which is a consequence of the definition of the Haar measure. So in the following we will write EfZ N )) for EfZ N A, θ))). Moreover, this Mellin-Fourier transform implies EZN t ) = for all t > 0. A somewhat similar, but more trivial identity is ) E e sn+in ), where N and N are independent, standard, real Gaussian variables : one can show it directly with the independence of N and N, or using the fact that e αzt is a martingale, with Z t, t 0) a complex planar Brownian motion. There does not seem to be a trivial explanation for EZN t ) =, but one can give another proof of this result. Let g be a function on the unitary group UN), stable on the conjugation classes : gu) can be written gθ,...,θ N ), where the θ i s are the eigenangles of U. Then Weyl s formula 5) gives the existence of a density function f invariant by translation fθ,..., θ N ) = fθ + θ,..., θ n + θ)) such as EgU)) = gθ,...,θ N )fθ,..., θ N ) dθ... dθ N, [0,π] N where the expectation is with respect to the Haar measure.

11 0 Paul Bourgade and Marc Yor For the function g : U deti N U) t ) = N k= e iθ k ) t, we get E ZN t ) = fθ,..., θ N ) dθ...dθ N [0,π] N = = [0,π] [0,π] dθ π dθ π ) e iθ k t k= ) fθ,..., θ N ) dθ...dθ N e iθ k t [0,π] N k= θ + θ,..., θ N + θ) N d θ... d θ N ) e i θ t k +θ) [0,π] N f E ZN t ) = f θ,..., θ N ) d θ...d θ N [0,π] N k= ) e i θ t k +θ). dθ [0,π] π k= }{{} ) When t is an integer, ) can be expanded with Newton s binomial formula, and we get ) = dθ [0,π] π + ) Nt l= a le ilθ =, so EZN t ) =. This reasoning can be generalized to the case t > 0, but we need to use the definition u t = e tlog u, with exp and log defined by their Taylor series ; the justification of the exchange between integral and sum then becomes tricky. We now come back to the general Mellin-Fourier transform. It implies the following proposition. Proposition. The modulus Z N of the characteristic polynomial Z N satisfies both identities in j= γ j Z N = ZN = N j= j= γ j γ j, 7) β j, j j= β j+, j, 8) where all random variables are independent and we make the convention β a,0 =. Proof. First, formula 3) with s = 0 can be written, for all t > 0, j= Γj + t) Γj) which translates thanks to formula 4)) as j= γ j = γ j γ j+ = E Z N t) N )) Γ j + t, Γj) γ j = ZN j= j= γ j γ j, because the Mellin transform is injective. To deduce 8) from 7), we successively use formulas 6) and 5) : ), which gives the desired result. ) = β j, j γ j β j+, j γ j As a nice consequence of 7), we show how to recover partially the central limit theorem 0).

12 Random matrices and the Riemann Zeta Function Corollary. The following central limit theorem holds : log Z N log N as N tends to, with N a standard normal variable. N Proof. Take logarithms on both sides of 7), and subtract expectations : with obvious notations. After division by follows easily. L N = log ZN + L N + L N ), log N the LHS converges in towards N0, /3). The result We gave a probabilistic interpretation for the of Z N, but we can do more with the joint of R log Z N, I log Z N ). Indeed, thanks to the Mellin-Fourier transform 3), after some calculation one can check that N R log Z N, I log Z N ) = β j,j cosw j, with all β s and W s independent, and W j with density on [ π/, π/] PW j dx) dx j= = c j cosx) j, where c j is the normalization constant c j = Γ + j/) /Γ + j)). Another way to express this result is the following proposition. Proposition. There is the representation Z N j= W j N = β j,j e i W j ) 9) where the beta and W variables are independent. Furthermore, Wj is distributed on [0, π] as 4.3 Mellin type limit theorems j= P W j dw) = c j ) sin w ) j ) dw. This paragraph shows how the Mellin-Fourier transform 3) implies the central limit theorem 0) and a generalization. Let us first recall some results about the Barnes G-function. It is defined on the half plane R + z) > 0 by G + z) := π) z e +γ)z +z n= where γ is Euler s constant. Its logarithm can be written + z ) n e z+ z n, n log G + z) = log π ) z + γ)z + ) n ζn ) zn n. 0) n=3

13 Paul Bourgade and Marc Yor As mentioned by Jacod in [5], the Barnes G-function appears naturally in some limit theorems. In N ) ) λ particular, let ψ := Γ /Γ denote the usual dilogarithm function and Γ N λ) := E = N j= Γj+λ) Γj). Then Γ N λ) N λ e λ P N j= ψj) N π e ) λ j= γ j G + λ). ) Keating and Snaith [8] are able to recover 0) after writing the series expansion of P N s, t). For simplicity, let us consider the case s = 0. Then one can write ) P N 0, t) = E Z N t ) = exp α N) t m m, m! that is : the coefficients α N) m m=0 m = 0,,...) are the cumulants of log Z N. From 3), one obtains α N) 0 = α N) = 0 α N) = log N + γ + ) + O N ) α N) m = ) m m ) Γm)ζm ) + O. ) ) N, m 3 m Thus, it follows that E exp t log Z N = E Z N log N t log N ) e t N, which yields to the central limit theorem for log Z N. However, the previous analysis shows more than 0) : from the expression ) of the cumulants and formula 0), Keating and Snaith immediately get the following theorem. Mellin-type limit theorem. For λ < /, N λ E Z N λ) N G + λ) G + λ). Remark. Rather than using directly the information about the cumulants given in ), one can show the previous theorem with the duplication identity 7) and the limit ). Moreover, one can write 3) as P N s, t) = Γ N t+s Γ N t) ) ΓN t s ), with the notations of ). Then, using the limit ), we get the following generalization of the Mellin-type limit theorem. Proposition. If t s, N t s 4 G P N s, t) N + t+s ) ) G + t s G + t).

14 Random matrices and the Riemann Zeta Function Further works Formula 9) can be written deti N A N ) = X N deti N A N ), where A N resp A N ) is distributed with the Haar measure µ UN ) resp µ UN) ), and X N is a random variable independent of A N. Can we directly ie : without the Selberg integrals) find such a formula? A recursive decomposition of the Haar measure, presented in [] will give a positive answer to this question, not only for the unitary group but also for On), SOn), USpn)... Moreover, further developments about the characteristic polynomials of random matrices linked to these notes might be the following : Is it possible to extend formula 9) for the characteristic polynomial deti N xa), and not only for its value at x =? For distinct angles θ and θ, what is the asymptotic of Z N A, θ )Z N A, θ )? References [] G. E. Andrews, R. A. Askey, R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, 7, Cambridge University Press, Cambridge, 999. [] P. Bourgade, C. P. Hughes, A. Nikeghbali, M. Yor, The characteristic polynomial of a unitary matrix : the probabilistic approach, in preparation. [3] L. Chaumont, M. Yor, Exercices in probability, vol 3 in Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 003. [4] S. M. Gonek, C. P. Hughes and J. P. Keating, A hybrid Euler-Hadamard product formula for the Riemann zeta function, preprint, 005. [5] J. Jacod, personal communication, 006. [6] C. P. Hughes, J. P. Keating and N. O Connell, On the characteristic polynomial of a random unitary matrix, Comm. Math. Phys. 0, n, p 49-45, 00. [7] C. P. Hughes, A. Nikeghbali, M. Yor, in preparation April 006). [8] J. P. Keating and N. C. Snaith, Random Matrix Theory and ζ + it), Comm. Math. Phys. 4, p 57-89, 000. [9] F. Mezzadri, How to generate random matrices from the classical compact groups, preprint, September 006. [0] S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Studies in Advanced Mathematics, New Edition, 995. [] M. Yor, Interpretations in terms of Brownian and Bessel meanders of the distribution of a subordinator perpetuity, p In : Lévy processes - Theory and Applications, eds : O. Barndorff-Nielsen, T. Mikosch, S. Resnick. Birkhäuser, 00.