MAKSYM RADZIWI L L. log log T = e u2 /2

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1 LRGE DEVIIONS IN SELBERG S CENRL LIMI HEOREM MKSYM RDZIWI L L bstract. Following Selberg [0] it is known that as, } {log meas ζ + it) t [ ; ] log log e u / π uniformly in log log log ) / ε. We extend the range of to log log ) /0 ε. We also seculate on the size of the largest for which the above normal aroximation can hold and on the correct aroximation beyond this oint.. Introction. he value-distribution of log ζσ + it) is a classical question in the theory of the Riemann zeta function. When σ > / this distribution is well-understood and is that of an almost surely convergent sequence of random variables see [6], [3], [7] or [3]). he half-line is secial, since for σ = we have a central limit theorem, ) meas log ζ + it) t [ ; ] log log = e u / π + o), originally e to Selberg [0]. Whereas the distribution of large values of log ζσ + it) with σ > / has been consistently studied since the ioneering work of Bohr and Jessen [], the corresonding question on the half-line has only attracted more attention recently. Conditionally on the Riemann Hyothesis, Soundararajan [] obtained Gaussian uer bounds for the left-hand side of ) focusing mostly on the range log log ). s an alication he derived near otimal uer bounds for moments of the Riemann zeta-function. In this aer we will be interested in asymtotic formulas for the left-hand side of ) when as. Previously asymtotic formulae of the form ) meas t [ ; ] log ζ + it) log log e u / π where known only in the range log log log ) / ε as a consequence of Selberg s [5] near-otimal refinement of the error term in ) In this aer we introce a new method that allows us to extend ) to the large-deviations range log log ) α for some small but fixed α > 0. heorem. he asymtotic formulae ) holds for log log ) /0 ε. 99 Mathematics Subject Classification. Primary: M06, Secondary: M50. he author is artially suorted by a NSERC PGS-D award. Conjecturally ) admits an asymtotic exansion in owers of log log ) /. Selberg s obtained an error term of Olog log log ) log log ) / ) in ), which is thus close to otimal in this sense.

2 MKSYM RDZIWI L L Our method is very versatile and allows to extend most of the known distribution results about ζσ + it) with σ = / + o )) to a large deviations setting for examle [] or the joint value distribution of R log ζ + it) and I log ζ + it)). he method also adats to the study of the distribution of large values of additive functions over sets of integers where only moments but not moment generating functions) are available. o rove heorem we use Selberg s work to rece the roblem to a question about Dirichlet olynomials. heorem then follows from the Proosition below, which might be of indeendent interest. Proosition. Let x = /log log ). hen, uniformly in = o log log ), 3) meas t [ ; ] R +it e u / π Previously a result such as Proosition was known only for fairly short Dirichlet olynomials i.e x log ) θ ) see [9] and also [8] for related work). he extension in the length of the Dirichlet olynomial is resonsible for our imrovement in heorem. he key idea in our roof of Proosition is to work on a subset of [ ; ] on which the Dirichlet olynomial does not attain large values. We exlain the idea briefly in more details in subsection.) below. Proosition can be extended to the range log log. In that situation we obtain the following result. /log log ) Proosition. Let x = and meas t [ ; ] R +it with 0 < c k a constant, deending only on k. k > 0. Uniformly in k c k e u / log log, π It is thus aarent that 3) does not ersist for ε log log. Similarly, 4) meas log ζ + it) t [ ; ] log log e u / π cannot be true for ε log log with ε > 0 small. Indeed, one can show on the Riemann Hyothesis) that this would contradict the moment conjectures, 5) ζ + it) k dt C k log ) k ) since it is conjectured that C k for k 0, ) and if 4) holds for ε log log then C k = for k ε the Riemann Hyothesis is used because we aeal to [] to first restrict the range of integration in 5) to those t s at which log ζ + it) k log log ). Nonetheless, by analogy to Proosition we exect 4) to hold for all smaller. Conjecture. If = o log log ) then 4) holds. For larger values of in analogy to Proosition we conjecture that 4) deviates from the truth only by a constant multile.

3 LRGE DEVIIONS IN SELBERG S CENRL LIMI HEOREM 3 Conjecture. Let k > 0. If k log log, then meas log ζ + it) t [ ; ] log log C k with C k the same constant as in 5) e u / π n abelian argument conditional on RH and using [] to truncate the tails) shows that if the left-hand side of 4) is at all asymtotic to a constant κ times a standard Gaussian in the range k log log )/ then ζ + it) k dt κ log ) k, hence κ = C k. hus there is only one reasonable choice for the constant in Conjecture above. For negative values of k Conjecture is likely to be false as soon as k /. Indeed in that range the zeros of ζs) should force a transition to an exonential distribution, see [4] for a recise statement... Remarks. ) here is no difficulty in adating our roof to the study of negative values of log ζ + it). lso, our argument carries over to St) := I log ζ + it) with little to no π changes. ) Conditionally on the Riemann Hyothesis we obtain the better range log log ) /6 ε for St). assuming the Riemann Hyothesis, the analogue of Lemma for St) has an k k instead of k 4k. his is resonsible for the imrovement). 3) ssuming the relevant Riemann Hyothesis our result extends to elements of the Selberg class see [] for a definition). 4) Finally, our method covers the case of the joint distribution of log ζ + it) and I log ζ +it). Similarly, one can recover Bourgade s [] result on the joint distribution of shifts of the Riemann zeta function in a large deviation setting. 5) he term k 4k in Lemma is directly resonsible the range log log ) /0 ε in heorem. For examle, an imrovement of the k 4k to k k in Lemma would give heorem uniformly in log log ) /6 ε... Outline of the roof of Proosition. he key idea in the roof of Proosition is to initially work on a subset of [ ; ] on which the Dirichlet olynomial does not attain large values say c log log ). his allows us to exress the moment generating function, 6) ex z R /+it ) dt, z in terms of only the first log log moments of R / it over t. hese moments can be taken over the full interval [ ; ] since is very close in measure to ) and then they become easy to estimate. On adding u the contribution from the moments we obtain a very recise estimate for 6). From there, by standard robabilistic techniques, we obtain 3) with t restricted to. Since is very close in measure to, we get 3) without the restriction to t. cknowledgments. he author would like to thank his suervisor K. Soundararajan for advice and encouragements. Notation. hroughout ε will denote an arbitrary small but fixed ositive real. We allow ε to differ from line to line. Finally log k denotes the k-th iterated natural logarithm, so that log k := log log k and log = log.

4 4 MKSYM RDZIWI L L. Lemmata Lemma Selberg, [5]). Uniformly in k 0, log ζ + it) R k dt k k 4k + k k k log log log ) k /log log ) where x = and +it > 0 constant. Proof. See sang s thesis [5], age 60. Lemma Hwang, [5]). Let W n be a sequence of distribution functions such that for s ε, ) s e st dw n t) = F s) ex φn) + O )) κ n with F s) a function analytic around s = 0 and equal to at s = 0. hen, uniformly in ominκ n, φn)), W n φn)) Proof. his is a secial case of heorem in [5]. +it e u / π Lemma 3 Soundararajan, []). For x /4k, R k ) k dt k! Proof. his is a secial case of Lemma 3 in []. 3. Proof of Proosition Lemma 4. Let,..., k x be rimes. hen, k cost log l )dt = f... k ) + O k x k ) l= where f is a multilicative function defined by f α ) = α α α/) By convention the binomial coefficient is zero when α/ is not an integer. Proof. Write n =... k = q α... q α r r cost log q i ) α i = = α i with q i mutually distinct rimes. Notice that e it log q i + e ) it log q α i i α i ) αi α i / + α i / l α i α i ) αi e iα i l)t log q i l Note that the leading coefficients in front of every e iαi l) is in absolute value. Furthermore α i l α i. herefore k cost log l )dt = r cost log q i ) α i dt = fn) +...) l= i=

5 LRGE DEVIIONS IN SELBERG S CENRL LIMI HEOREM 5 where inside...) there is a sum of α +)... α r +) α... α r = k terms of the form γe itβ log q +...β r log q r ) with γ and integers β i α i. Since β log q +...+β r log q r x k the integral over t of each of these terms is at most x k. Since there is k of these terms and each has a leading coefficient γ the integral over t of all the terms inside...) contributes at most O k x k ). Lemma 5. Let x = /log log ). hen, R ) k dt = ) z dz I 0 πi z + O x) k) k+ +it where I 0 z) = /π) π 0 ez cos θ dθ = n 0 z/)n /n!) is the modified 0-th order Bessel function. Proof. Given an integer n = α... α l l with Ωn) = k and i x there are k!/α!... α l!) ways in which this integer can be written as a roct of k rimes. We define a multilicative function gn) by g α ) = /α! and g α ) = 0, > x, so that k!gn) is equal to the number of ways in which an integer n with Ωn) = k can be exressed as a roct of k rimes x. By Lemma and the above observation, R +it ) k dt = = k!,..., k x Ωn)=k f... k )... k + O x) k) fn) n gn) + O x) k) We detect the condition Ωn) = k by using Cauchy s integral formula. hus the above is equal to k! fn) gn)z Ωn) dz πi n n z + O x) k) k+ Since the functions f, g and z Ωn) are multilicative, the above sum over n does factor into an Euler roct, + l as desired. ) l z l)! We are now ready to rove Proosition. Proof of Proosition. By Lemma 3, R +it k ) ) l = ) z I 0 l dt k! ) k herefore the set of those t [ ; ] for which R it log has measure k/ log ) k. Choosing k = log /e, this measure is log ) δ where δ = /e, which is negligible also, the exact value of δ is unimortant). hus, we can restrict our attention to the set { := t [ ; ] : R } log +it

6 6 MKSYM RDZIWI L L We denote its comlement in [ ; ] by c. For comlex z ε < 7) ex zr +it ) dt = k 3V z k R k! +it 00, ) k dt + O log ) 3 ) where V = log log. he error term arises from bounding the terms with k 3V: each contributes at most ε log log ) k /k! e k since the integral over the set is less than log log ) k by definition of. o comute the moments with k 3V we use Cauchy s inequality and notice that 8) R ) k dt R ) k dt +it meas c ) R +it +it k dt log ) δ/ k! log log ) k/ by Lemma 3. he integral over t is readily available through Lemma 5. hus, R ) k dt = k! ) w dw I 0 πi w + O ) k! log log ) k/ k+ log ) δ/ +it We choose the contour to be a circle of radius r = say) around the origin. Summing the above over k 3V we conclude that 7) is equal to w) z k I 0 πi w dw + O log ) δ/+ε) k+ k 3V Since I 0w/ ) log ) on w =, and k 3V z/w k log ) 3 log +ε we can comlete the sum over k 3V to all k, making an error of log ) δ/5. hus the above and hence 7)) is equal to 9) πi ) w dw I 0 w z + O log ) δ/5) = ) z I 0 + O log ) δ/5) by Cauchy s formula. But, ) I z 0 z = F z) ex ) +O/ log )) with F z) analytic around z = 0 and equal to at z = 0. Since z ε it is clear that the error term in 9) can be made relative at a small cost in the error term. We conclude that ex z R ) ) z dt = F z) ex + Olog ) δ/0 ) +it uniformly in z ε. hus Lemma is alicable, and it follows that meas) meas t R +it meas) e u / π

7 LRGE DEVIIONS IN SELBERG S CENRL LIMI HEOREM 7 Now meas c ) log ) δ while / dt e u π log ) o) because = o log log ). Hence the receding equation becomes meas t [ ; ] his establishes our claim R +it 4. Proof of heorem. e u / π Proof of heorem. Let x = /log log ). Let /4 > δ > 0 to be fixed later. By Lemma the set of those t [ ; ] for which log ζ + it) R L := log log ) 4 +δ +it has measure L k k k 4k log 3 ) k for all k log 3. Choosing k = L / /e we obtain a measure of ex cl / ) for some constant c > 0. hus excet for a set of measure ex cl / ), log ζ + it) = R + θl +it with a θ deending on t. he measure of those t [ ; ] for which 0) R log log + θl +it is since θ ) at least the measure of those t [ ; ] for which R log log + L +it and this measure is at least, ) + o)) +L/ log log e u / π by Proosition. Similarly using this time θ ) the measure of those t [ ; ] for which 0) holds is at most ) + o)) L/ log log e u / π. When L = olog log ) / ) both ) and ) are equal to / e u π + o)). We conclude that, meas t [ ; ] log ζ + it) log log = + o)) π e u / + O e cl/) rovided that L = olog log ) /. he main term dominates the error term as long as L /4 ε. Both conditions L /4 ε and L = olog log ) / are met if we choose δ = 3/0 and require that log log ) /6+δ/4 ε = log log ) /0 ε.

8 8 MKSYM RDZIWI L L We will only give a brief sketch. 5. Proof of Proosition Lemma 6. Let C > 0 be given. hen, uniformly in 0 Rz C, and 4C Iz x /8, ) z I 0 ex ciz) ) for some constant c > 0. Proof. For z x we write ) z I 0 = z I 0 ) z z I 0 z ) e z/ ) ex z z Using the bound I 0 z) ex Iz ) the first term is ex c z 3/ / log z ) Using the bound I 0 z)e z/) ex z 4 ) we get that the contribution of the second term is exc z / log z ). Finally the third term is a Gaussian and thus contributes ex/)rz) Iz) )loglog x/ log z )). Under our assumtions on z and x the last bound dominates and the claim follows. Proof of Proosition. By a small modification of the roof of Proosition, for any fixed δ > 0, uniformly in z, ex z R ) dt = ) z I 0 + O log ) k ) +it with the set and the function F ) as in the roof of Proosition, excet that now is the set of those t s at which the Dirichlet olynomial R / it C log log with some C large enough but fixed. he constant C is choosen so large so as to guarantee an error of log ) k above and meas c ) log ) k. Let Hz; x) = I 0z/ ) ex zσx)) where σx) = )/. By enenbaum s formula see equation 0) and above in [4]), 3) meas t R +it + O = meas) πi c+i π c i c+i ds Hs; x) c i ss + ) + ) Hs; x) ds s + )s + ) + O log ) k ) where = x /6 and c = σx)) /. By the revious Lemma used to handle the range Is < x /8 and the fast decay of /ss + ) used to handle Is > x /8 ) we can truncate the above integral at Is ψx)/σx) with ψx) very slowly. his inces an negligible error term of o / e u ). Furthermore, since is so large, the O ) term in the above equation is also negligible. Write I 0z/ ) = F z) ex z σx) ) with F z) analytic. hen, on comleting the square and changing variables the first integral above truncated at Is ψx)/σx)) can be rewritten as / meas) e σx) π ψx) ψx) F c + iv ) e v / dv σx) c + iv/σx)

9 LRGE DEVIIONS IN SELBERG S CENRL LIMI HEOREM 9 Let Gz) = F z)/z. hen Gc + iv/σx)) = Gc) + ivg c)/σx)ov /σx) ). Plugging this into the integral the middle term vanishes while the last term contributes a negligible amount. hus, we end u with F c) e / π + o)) = F c) e u / + o)) π as a main term for 3). o conclude it suffices to notice that F c) F k) as, since c k and F is analytic. Furthermore as in the roof of Proosition the restriction to t in 3) can be relaced by t [ ; ] because is very close in measure to that is, meas c ) log ) k ). References [] H. Bohr and B. Jessen. Uber die wertverteilung der Riemannschen zetafunktion, erste mitteilung. cta. Math., 50: 35, 930. [] P. Bourgade. Mesoscoic fluctuations of the zeta zeros. Probab. heory Related Fields, 48 no. 3-4: , 00. [3]. Hattori and K. Matsumoto. limit theorem for bohr-jessens robability measures of the riemann zeta-function. J. Reine ngew. Math., 507:93, 999. [4] C. P. Hughes. On the characteristic olynomial of a random unitary matrix and the Riemann zeta function. PhD thesis, University of Bristol, 00. [5] H. H. Hwang. Large deviations for combinatorial distributions. I. Central limit theorems. nn. l. Probab., 6 ):97 39, 996. [6] Y. Lamzouri. On the distribution of extreme values of zeta and L-functions in the stri / < σ <. Internat. Math. Res. Notices, to aear. [7]. Laurinchikas. Limit theorems for the Riemann Zeta-function. Kluwer, Dordrecht. [8]. Laurinchikas. Distribution of trigonometric olynomials. Litovsk. Mat. Sb.,, no. :7 35, 98. [9]. Laurinchikas. Large deviations of trigonometric olynomials. Litovsk. Mat. Sb.,, no. 3:5 6, 98. [0]. Selberg. Contributions to the theory of the Riemann zeta-function. rch. Math. Naturvid., 48, no. 5:89 55, 946. []. Selberg. Old and new conjectures and results about a class of Dirichlet series. Proceedings of the malfi Conference on nalytic Number heory, ages , 99. [] K. Soundararajan. Moments of the Riemann zeta function. nnals of Math., 70 ):98 993, 009. [3] J. Steuding. Value distribution of L-functions. Sringer-Verlag, 007. [4] G. enenbaum. Sur la distribution conjointe de deux fonctions nombre de facteurs remiers. eq. Math., 35:55 68, 988. [5] K. M. sang. he distribution of the values of the Riemann zeta function. PhD thesis, Princeton University, Princeton, NJ, 984. Deartment of Mathematics, Stanford University, 450 Serra Mall, Bldg. 380, Stanford, C address: maksym@stanford.e