Mod-Gaussian convergence and the value distribution of ζ(1/2+it) and related quantities
aa r X i v : . [ m a t h . N T ] M a y MOD-GAUSSIAN CONVERGENCE AND THE VALUE DISTRIBUTION OF ζ (1 / it ) AND RELATED QUANTITIES
E. KOWALSKI AND A. NIKEGHBALI
Abstract.
In the context of mod-Gaussian convergence, as defined previously in our workwith J. Jacod, we obtain asymptotic formulas and lower bounds for local probabilities for asequence of random vectors which are approximately Gaussian in this sense, with increasingcovariance matrix. This is motivated by the conjecture concerning the density of the set ofvalues of the Riemann zeta function on the critical line. We obtain evidence for this fact, andderive unconditional results for random matrices in compact classical groups, as well as forcertain families of L -functions over finite fields. Introduction
It is well-known (see, e.g., [30, Th. 11.9]) that, for 1 / < σ <
1, the set of values ζ ( σ + it ), t ∈ R , is dense in the complex plane. In fact, much more is true: it was proved by Bohr andJessen that there exists a Borel probability measure µ σ on C , such that the support of µ σ isthe whole complex plane, and such that the convergence in law12 T Z T − T f (log ζ ( σ + it )) dt → Z C f ( z ) dµ σ ( z ) , holds for f : C → C continuous and bounded.The corresponding density question for σ = 1 / ζ (1 / it ), | t | T ,do not have a limiting distribution, as evidenced already by the Hardy-Littlewood asymptotic1 T Z T | ζ (1 / it ) | dt ∼ (log T ) , as T → + ∞ , or by Selberg’s result that log | ζ (1 / it ) | , | t | T , is asymptotically normal with variance growing to infinity (see also the work of Ghosh [12] for the imaginary part, and [4, §
5] for arecent proof). In other words, “most” values of ζ (1 / it ) are rather large, though the zetafunction is zero increasingly often as the imaginary part grows.In this paper, we show (Corollary 9) how the density of values of zeta on the critical linewould follow rather directly from a suitable version of the Keating-Snaith moment conjectures,which we viewed in our previous work with J. Jacod [15] as a refined version of the Gaussianmodel. In fact, under suitable assumptions, we could prove a quantitative result, bounding fromabove the smallest t > ζ (1 / it ) lies in a given open disc in C . This argument isbased on a very general probabilistic estimate proved in Section 2, which throws some light onthe nature of the mod-Gaussian convergence that we defined in [15]. We hope that this resultwill be of further use. In another paper (jointly with F. Delbaen, see [8]), it will be seen thatone can weaken considerably the assumption needed in order to prove the density of values of ζ (1 / it ) (but without quantitative information).As applications of the general result, we will also prove the following theorems in Section 3(the precise versions are given there). Mathematics Subject Classification.
Key words and phrases.
Zeta function, L -functions, mod-Gaussian convergence, equidistribution, randommatrix theory, characteristic polynomials, Euler products, monodromy. heorem 1. Let z ∈ C × be arbitrary, ε > such that ε | z | . There exists N ( z , ε ) , whichcan be bounded explicitly, such that µ N ( { g ∈ U ( N ) | | det(1 − g ) − z | < ε } ) ≫ (cid:16) ε | z | (cid:17) N (1) provided N > N , where µ N denotes probability Haar measure on the unitary group U ( N ) ⊂ GL ( N, C ) , and the implied constant is absolute. Theorem 2.
Define P N ( t ) = Y p N (1 − p − / − it ) − (2) for N > and t ∈ R . Let z ∈ C × be arbitrary, ε > such that ε | z | . There exists N ( z , ε ) ,explicitly bounded, such that lim inf T → + ∞ T λ ( { t T | P N ( t ) ∈ V } ) ≫ (cid:16) ε | z | (cid:17) N , for all N > N , where λ is the Lebesgue measure and the implied constant is absolute. In a different direction, we obtain some evidence for the density of ζ (1 / it ) by lookingat special values of families of L -functions over finite fields. In doing so, we also consider theanalogue of Theorem 1 for symplectic and orthogonal matrices. We refer to Section 4 for precisestatements and definitions, and only state here one appealing (qualitative) corollary: Theorem 3.
The set of central values of the L -functions attached to non-trivial primitiveDirichlet characters of F p [ X ] , where p ranges over primes, is dense in C . Notation.
As usual, | X | denotes the cardinality of a set. By f ≪ g for x ∈ X , or f = O ( g )for x ∈ X , where X is an arbitrary set on which f is defined, we mean synonymously that thereexists a constant C > | f ( x ) | Cg ( x ) for all x ∈ X . The “implied constant” refersto any value of C for which this holds. It may depend on the set X , which is usually specifiedexplicitly, or clearly determined by the context. Similarly, f ≍ g for x ∈ X means f ≪ g and g ≪ f , both for x ∈ X . We write ( x ) j = x ( x + 1) · · · ( x + j −
1) the Pochhammer symbol.
Acknowledgments.
The first version of this paper was written while the first author wason sabbatical leave at the Institute for Advanced Study (Princeton, NJ); many thanks are dueto this institution for its support. This material is based upon work supported by the NationalScience Foundation under agreement No. DMS-0635607.The second author was partially supported by SNF Schweizerischer Nationalfonds ProjekteNr. 200021 119970/1.Thanks to K. Soundararajan for pointing out that the density conjecture for ζ (1 / it ) mightbe a good problem to study using complex moments of the zeta function, and to R. Heath-Brownfor explaining us the history of the question. Many thanks also to N. Katz for discussions andexplanations surrounding issues of monodromy computations. Many thanks also to F. Delbaenfor useful discussions concerning the underlying probabilistic framework.The graphs were produced using Sage 4.2 [28], relying on the Barnes function routines inthe mpmath package.2.
Mod-Gaussian convergence and local probabilities
In this section, which is purely probabilistic, we present two versions of “local” boundsfor probabilities in the case of sufficiently uniform mod-Gaussian convergence of sequences ofrandom vectors. This may be compared with the local central limit theorem (see, e.g., [2, § e first introduce the definition, generalizing [15] to random vectors. Fix some integer m >
1, and let ( X N ) be a sequence of R m -valued random variables defined on a probabilityspace (Ω , Σ , P ) (as is the case for convergence in law, we could work without change withrandom variables defined on different probability spaces). Let Q N ( t ) = Q N ( t , t , . . . , t m )be a sequence of non-negative quadratic forms on R m . The sequence ( X N ) is then said to beconvergent in the mod-Gaussian sense with covariance Q N and limiting function Φ iflim N → + ∞ exp( Q N ( t ) / E ( e it · X N ) = Φ( t ) (3)locally uniformly for t ∈ R m ; Φ is then a function continuous at 0 and Φ(0) = 1. Here, · denotesthe standard inner product on R m .The intuitive meaning is that, in some sense, X N is “close” to a (centered) Gaussian vector G N with covariance matrix Q N . As in [15], this notion is of most interest if the covariance“goes to infinity”. However, in contrast with the case of m = 1, this can mean different thingsbecause there is more than a single variance parameter involved.To discuss this, we diagonalize Q N in an orthonormal basis in the form Q N ( t ) = δ ,N u + · · · + δ m,N u m , δ ,N δ ,N · · · δ m,N where u = H N ( t ) is the necessary (orthogonal) change of variable. Then “ Q N goes to infinity,”in the weakest sense, means that the largest eigenvalue δ m,N goes to + ∞ as N → + ∞ .We are interested in the distribution of values of X N as N grows; clearly, if (say) the Q N are already diagonalized in the canonical basis and δ ,N is constant, these values will have firstcoordinate much less spread out than the last ones. To simplify our discussion, and becausethis is the situation in our applications, we will assume this behavior does not occur and thatin fact the smallest eigenvalue goes to infinity. For simplicity, we will assume in fact that forsome fixed µ >
0, we have δ m,N δ µ ,N , δ m,N → + ∞ , (4)so that also δ ,N → + ∞ (we say that the convergence is balanced ). In our main applications,this will be the case with µ = 1. Note in particular that it follows that the discriminant σ N = δ ,N · · · δ m,N > δ m ,N goes to infinity as N → + ∞ , and moreover σ N δ mµ ,N . (5)Our question is now the following: given ( X N ), ( Q N ), as above, with this type of mod-Gaussian convergence, can we bound from below the probability P ( X N ∈ U ) , where U is a fixed open set in R m ?Denoting by ˜ Q N ( x ) the dual quadratic form, the Gaussian model suggests that, if U isrelatively compact (e.g., some non-empty open ball), we could expect P ( X N ∈ U ) ≈ P ( G N ∈ U ) = 1(2 π ) m/ √ σ N Z U e − ˜ Q N ( x ) / dx ∼ Vol( U )(2 π ) m/ √ σ N , (6)as N → + ∞ , since δ ,N → + ∞ implies that ˜ Q N ( x ) → x ∈ R m . We will confirm thatthis holds in certain conditions at least. We strive especially for lower bounds on P ( X N ∈ U ),which we wish to be quantitative, so that we can determine some N (depending explicitly on U ) for which P ( X N ∈ U ) > . Note that such a quantitative result must depend on the location of the open set U , whereasthe limit itself only depends on the volume, as seen above. With respect to the standard inner product on R m . he specific hypothesis we use may seem somewhat arbitrary, but they turn out to be satisfied(with room to spare) in the later applications. Theorem 4.
Let m > be fixed and let ( X N ) be a sequence of R m -valued random variablesdefined on (Ω , Σ , P ) , such that ( X N ) converges in the mod-Gaussian sense with covariance ( Q N ) , and that the convergence is µ -balanced with µ > , with σ N > for all N . Let ( G N ) beGaussian random variables with covariance matrices given by ( Q N ) , so that exp( − Q N ( t ) /
2) = E ( e it · G N ) . Assume moreover the following three conditions: (1)
There exist constants a > , α > and C > such that, for any N > and t ∈ R m suchthat k t k σ aN , we have E ( e it · X N ) = Φ( t ) exp( − Q N ( t ) / n O (cid:16) ασ CN ) (cid:17)o . (7)(2) The function Φ is of class C on {k t k < } . (3) For some A > and β > , we have | Φ( t ) | ≪ exp( β k t k A ) , (8) for t ∈ R m .Let D > be any number such that D > m + 1 + max { a − , A/C, m ( m + 1) µA } ) . (9) Then, for any fixed non-empty open box U = { x ∈ R m | k x − x k ∞ < ε } ⊂ R m , with width ε such that < ε , we have P ( X N ∈ U ) = P ( G N ∈ U ) + O (cid:16) σ / /DN + ε − m σ N (cid:17) , (10) for N > , where the implied constant depends only on ( m, Φ , a, α, C ) and the implied constantin (7) .In particular, for any fixed non-empty open set U ⊂ R m , we have P ( X N ∈ U ) ≫ √ σ N provided N > N , where N and the implied constant depend on U and the same data as above. Note the following elementary lower bound, valid if ε P ( G N ∈ U ) ≫ ε m √ σ N exp (cid:16) − ˜ Q N ( x )2 (cid:17) (11)where the implied constant depends only on m ; this is where the location of U enters, since theerror term will only be smaller than this, roughly, when ˜ Q N ( x ) ≍ Remark . The growth condition (8) is in fact a consequence of the uniform mod-Gaussianconvergence, at least provided the sequence ( σ N ) does not grow too fast. For instance, if σ N +1 M σ BN (12)for all N >
1, for some constants M > B >
0, we can obtain (8) with A = 2 + B/a . Indeed,we can write | Φ( t ) | = (cid:12)(cid:12)(cid:12) Φ N ( t ) e Q N ( t ) / (cid:16) O (cid:16) ασ CN ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) e Q N ( t ) / if N is large enough and k t k σ aN . Note then that Q N ( t ) δ m,N k t k δ m ,N k t k σ N k t k . e now fix N > σ N − k t k /a σ N , and if this value of N is large enough, we get σ N Cσ BN − M k t k B/a , and hence | Φ( t ) | σ N k t k ) M k t k B/a ) , as desired. On the other hand, if this chosen N is too small, k t k is bounded, and the desiredestimate is trivial. Proof of Theorem 4.
Let δ N = δ ,N be the smallest eigenvalue of Q N , so that Q N ( t ) > δ N k t k for all t ∈ R m and N >
1. For simplicity, we denote also γ N = exp( ασ CN ) , (13)as in (7).We now first fix w such that 0 < w <
1, and then fix a smooth, compactly supported function g on R such that 0 g ,g ( x ) = 0 for | x | > , g ( x ) = 1 for | x | w, | g ( j )0 ( x ) | ≪ j ∆ j , for j > , x ∈ R , where ∆ = (1 − w ) − and the implied constant depends only on j (we will define w to be afunction of N at the end, and hence must be careful to have estimates uniform in terms of w ;this is provided by using only the above properties of g ; the maximal value of j used will alsobe bounded only in terms of m , Φ and the data in (7)). It is classical that such a function exists(examples are constructed in [13, § f ( x ) = f ( x , . . . , x m ) = Y j m g ( x j )for x ∈ R m . It follows that 0 f ,f ( x ) = 0 for k x k ∞ > , f ( x ) = 1 for k x k ∞ w. Next, we define f ( x ) = f (cid:16) x − x ε (cid:17) , and we start our argument with the obvious inequality P ( X N ∈ U ) > E ( f ( X N )) = Z R m f ( x ) dν N ( x )where ν N is the law of X N . Applying the Plancherel formula, we get P ( X N ∈ U ) > Z R m f ( x ) dν N ( x ) = Z R m ˆ f ( t )Φ N ( t ) dt where Φ N ( t ) = E ( e it · X N ) is the characteristic function of X N andˆ f ( t ) = 1(2 π ) m/ Z R m f ( x ) e − it · x dx denotes the Fourier transform of f (the smoothness of f guarantees that ˆ f is in L , so thePlancherel formula is valid by a simple Fubini argument).We have ˆ f ( t ) = ε m e − it · x ˆ f ( εt ) = ε m e − it · x Y j m ˆ g ( εt j ) , t ∈ R m . Since ˆ g ( t ) = 1 √ π it ) j Z R g ( j )0 ( x ) e − itx dx or t = 0 and j > | ˆ g ( t ) | ≪ min(1 , ∆ j | t | − j ) , the implied constant depending on j .Using the formula for ˆ f ( t ), selecting for given t an index j so that k t k ∞ = | t j | and applyingthe second upper bound above for this index only, if | t j | >
1, we derive | ˆ f ( t ) | ≪ min( ε m , ∆ B + m ε − B k t k − B − m ∞ ) , t ∈ R m , (14)for any fixed B >
1, where the implied constant depends only on m and B . In particular, for k t k ∞
1, we will use simply the upper bound | ˆ f ( t ) | ε m k ˆ f k ∞ ε m .We now proceed to approximate. First of all, for any radius R N >
1, we can estimate thecontribution of those t with k t k > R N using the estimate above with a value of B > k t k ∞ > R N , we obtain Z k t k >R N ˆ f ( t )Φ N ( t ) dt ≪ ε − B ∆ m + B R − BN for any R N >
1. After selecting R N = σ /BN >
1, we obtain Z k t k >R N ˆ f ( t )Φ N ( t ) dt ≪ ε − B ∆ m + B σ − N , (15)for N >
1, the implied constant depending only on f .On the other hand, provided B > /a , we use (7) and (8) to get Z k t k R N ˆ f ( t )Φ N ( t ) dt = Z k t k R N ˆ f ( t )Φ( t ) exp( − Q N ( t ) / n O (cid:16) γ N (cid:17)o dt = Z k t k R N ˆ f ( t )Φ( t ) exp( − Q N ( t ) / dt + O (cid:16) γ − N Z k t k R N | Φ( t ) ˆ f ( t ) | dt (cid:17) = Z k t k R N ˆ f ( t )Φ( t ) exp( − Q N ( t ) / dt + O ( ε m γ − N exp( βR AN ))for N >
1, using again the definition of f , the implied constant depending on Φ.By (13), the last term can be bounded by ε m γ − N exp( βR AN ) ≪ ε m exp( βσ A/BN − ασ CN ) ≪ ε m σ − N for N > B > A/C , the implied constant depending on ( α, A, B, C ).We then split the first term further in two parts, namely where Q N ( t ) κ N , and where Q N ( t ) > κ N . The parameter κ N will be chosen later, in such a way that the region {k t k } (which is inside {k t k R N } ) contains the region Q N ( t ) κ N (which is a neighborhood of 0that contracts to 0 as N → + ∞ , if κ N does not grow too fast, since it is an ellipsoid withlongest axis κ N / p δ m,N ).The second part of the integral is bounded by Z k t k R N , Q N ( t ) >κ N ˆ f ( t )Φ( t ) e − Q N ( t ) / dt ≪ ε m R mN exp( βR AN − κ N / ε m exp (cid:16) mB log σ N + βσ A/BN − κ N (cid:17) , and in the first part, we use the approximationΦ( t ) = 1 + O ( k t k ) or {k t k } , coming from the C assumption on Φ, to get Z Q N ( t ) κ N ˆ f ( t )Φ( t ) e − Q N ( t ) / dt = Z Q N ( t ) κ N ˆ f ( t ) e − Q N ( t ) / dt + O (cid:16)Z Q N ( t ) κ N | ˆ f ( t ) |k t k dt (cid:17) = Z Q N ( t ) κ N ˆ f ( t ) e − Q N ( t ) / dt + O (cid:16) ε m κ m +1 N √ δ N σ N (cid:17) , (16)the implied constant depending on Φ, where the last integral was estimated using k t k δ N Q N ( t ) κ N δ N , | t i | κ N p δ i,N . We can rewind the computation for the first term, with the Gaussian G N instead of X N : for N >
1, we have Z Q N ( t ) κ N ˆ f ( t ) e − Q N ( t ) / dt = E ( f ( G N )) + O ( ε m e − κ N / )where the implied constant is absolute, and then we write E ( f ( G N )) = P ( k G N − x k ∞ < ε ) + O ( P ( wε k G N − x k ∞ < ε ))= P ( k G N − x k ∞ < ε ) + O ( ε m (1 − w ) σ − / N )for N >
1, where the implied constant depends only on m (the last step is obtained using thedensity of the Gaussian G N ).Summarizing, we have found P ( X N ∈ U ) > P ( G N ∈ U ) + O (cid:16) ε m (1 − w ) σ / N + ∆ m + B ε m σ N + ε m exp (cid:16) mB log σ N + βσ A/BN − κ N (cid:17) + ε m κ m +1 N √ σ N δ N (cid:17) (where we recall that ∆ − = 1 − w ).Now if A/B < / ( mµ ), and κ N = σ A/BN , we have first (see (5)) the condition { Q N ( t ) κ N } ⊂ {k t k κ N δ N } ⊂ {k t k } , and moreover the third error term is absorbed in the second one, while the last becomes ε m κ m +1 N √ σ N δ N ε m σ − − mµ + ( m +1) ABN . Thus if we select any
B > max( a − , A/C, m ( m + 1) Aµ ), the previous conditions on B hold,and we find that P ( X N ∈ U ) > P ( G N ∈ U ) + O (cid:16) ε m (1 − w ) σ / N + ∆ m + B ε m σ N + ε m σ / / (6 mµ ) N (cid:17) . (17)Now, we attempt to select w to equalize the error terms involving it, i.e., so that ε m (1 − w ) √ σ N = ∆ m + B ε m σ N , which translates to ∆ = (1 − w ) − = ( ε m σ / N ) / ( m + B +1) and two cases arise: i) If σ / N > ε − m , we have ∆ > w in this way), and we obtainfrom the above that P ( X N ∈ U ) > P ( G N ∈ U ) + O (cid:16) σ / /DN (cid:17) , where D = 2( m + 1+ B ) (note that, since A is assumed to be >
1, we have 2( m + B + 1) > mµ .)(ii) If we have σ / N ε − m , we take simply w = 1 / ε m (1 − w ) √ σ N + ∆ m + B ε m σ N ≪ ε − m σ N , where the implied constant depends on m and B .The combination of the two cases leads to the lower-bound in (10). The upper bound isproved similarly, using instead of g a function which is = 1 for | x | | x | > w for some suitable w >
0; we leave the details to the reader. (cid:3)
Remark . If we are interested in obtaining a lower bound only (which is the most interestingaspect in a number of applications), it is simpler and more efficient to fix, e.g., w = 1 /
2, fromthe beginning of the proof. For U = { x ∈ R m | k x − x k ∞ < ε } , this leads for instance to P ( X N ∈ U ) > P ( G N ∈ U − ) + O (cid:16) σ / / (2 mµ ) − γN (cid:17) (18)for any γ > B large enough depending on γ ), where U − = { x ∈ R m | k x − x k ∞ < ε/ } and the implied constant depends on Φ, ( m, a, α, C ) and γ . Remark . From the probabilistic point of view, this proposition gives one answer, quantita-tively, to the following type of question: given a sequence ( X N ) of (real-valued) random variablesand parameters σ N → + ∞ such that X N /σ N converges in law to a standard centered Gaussianvariable (with variance 1), to what extent is X N itself distributed like a Gaussian with variance σ N ?Here, we perform the comparison by looking at P ( X N ∈ U ), U a fixed open set. And thisshows clear limits to the Gaussian model: for example, any integer-valued random variable X N will have P ( X N ∈ U ) = 0 for any open set not intersecting Z , and yet may satisfy a CentralLimit Theorem (e.g., the N -th step of a symmetric random walk on Z ). Even more precisely,denoting d K ( X, Y ) = sup x ∈ R | P ( X x ) − P ( Y x ) | the Kolmogorov distance, there exist integer-valued random variables X N with d K ( X N , G N ) ≪ N − / , (19)where G N is a centered Gaussian with variance N , which indicates a close proximity – andyet, again, P ( X N ∈ U ) = 0 if U ∩ Z = ∅ . For instance, take X N with distribution function F N ( t ) = P ( X N t ) given by F N ( t ) = P ( G N k ) for k t < k + 1 , k ∈ Z . In particular, it is also impossible to prove results like Theorem 4 using only this type ofassumption on the Kolmogorov distance. Of course, if we assume that d K ( X N , G N ) ≪ N − / φ ( N ) − , (20)with φ ( N ) → + ∞ (arbitrarily slowly), we get P ( X N ∈ U ) > P ( G N ∈ U ) − d K ( X N , G N ) ≫ N − / , or U = [ α, β ], α < β . But such an assumption is unrealistic in practice. For instance, if oneassumes that ( X N ) converges in the mod-Gaussian sense with covariance Q N ( t ) = σ N t , and ifthe limiting function Φ is C and the convergence holds in C topology, one can straightforwardlyestimate the Kolmogorov distance by d K ( X N , G N ) = sup x ∈ R | P ( X N x ) − P ( G N x ) | ≪ σ − / N , which is comparable to (19), but one can also check that this can not be improved in generalto something like (20).In Example 4 in Section 3, we will also describe a much deeper and more illuminating situationconcerning the limits of what can be hoped, even with something like mod-Gaussian convergence. Remark . Other variants could easily be obtained. In particular, it is clear from the proof thatif Φ has sub-gaussian growth, i.e., we can take A = 2 in (8), the results can be substantiallyimproved. However, in our main applications, this condition fails. Also, one could use testfunctions f which decay at infinity faster than polynomials to weaken the uniformity requirementin the convergence condition (7) (for instance, for m = 1, it is possible to find f which is smooth,non-negative and compactly supported in any fixed open interval and satisfiesˆ f ( t ) ≪ exp( −| t | − ε )for any ε >
0, as constructed, e.g., in [13, Th. 1.3.5] or [14]). Again, for our main unconditionalapplications, our conditions hold with room to spare, so we avoided this additional complexity.
Remark . We can also introduce a linear term (corresponding roughly to the expectation of X N ) in addition to the covariance terms in the definition of mod-Gaussian convergence (asin [15] for m = 1), but this amounts to saying that ( X N ) converges in the mod-Gaussian sensewith covariance ( Q N ) and mean ( ξ N ), ξ N ∈ R m , if the sequence ( X N − ξ N ) converges in ouroriginal sense above. But note that the interpretation of a lower bound for P ( X N − ξ N ∈ U ),as given by Theorem 4 for a fixed U ⊂ R m , is quite different, if ξ N is itself “large”. Maybeone should see the statements in that case as giving natural examples of sets U N for which oneknows that P ( X N ∈ U N ) has the specific decay behavior σ − / N as N → + ∞ .In particular, lower bounds for P ( X N − ξ N ∈ U ) do not give control of P ( X N ∈ U ), andindeed this may be zero for all N large enough (see, for instance the example in Section 4.2below of values at 1 of characteristic polynomials of unitary symplectic matrices, which is always > Random unitary matrices and the zeta function
We present now some applications of Theorem 4 (in particular, proving Theorems 1 and 2).We also give an example that illustrates the limitations of such results, suggesting strongly thatone can not replace mod-Gaussian convergence with the existence of the limits (3) only for t ina neighborhood of the origin. Example 1.
One of the canonical motivating examples of mod-Gaussian convergence is due toKeating and Snaith [22]. Let X N = log det(1 − T N ) , where T N is a Haar-distributed random unitary matrix in the compact group U ( N ); we viewthese random variables as R -valued (via the real and imaginary parts), so if t = ( t , t ), wehave t · X N = t Re( X N ) + t Im( X N ) . (21)We first clarify the choice of the branch of logarithm: X N is defined almost everywhere (when1 is not an eigenvalue of T N ), and for g ∈ U ( N ) with det(1 − g ) = 0, such thatdet(1 − T g ) = Y j N (1 − α j T ) , | α j | = 1 , Using the Berry-Esseen inequality, see e.g. [29, § e define log det(1 − g ) = lim r → r< log det(1 − rg ) = X j N lim r → r< log(1 − rα j ) , where the last logarithms are given by the Taylor expansion around 0. This is the same con-vention as in [22, par. after (7)].Keating and Snaith show that ( X N ) satisfies E ( e it · X N ) = Y j N Γ( j )Γ( j + it )Γ( j + ( it + t ))Γ( j + ( it − t )) (22)for t = ( t , t ) ∈ R (note the asymmetry between it and t ), see [22, eq. (71)], taking intoaccount a slightly different normalization: their t is our it and their s is our t . It is alsouseful to observe that, in the connection with T¨oplitz determinants (see [3]), this characteristicfunction is the N -th T¨oplitz determinant corresponding to the symbol which is a pure Fisher-Hartwig singularity of type b ( e iθ ) = (2 − θ ) it / exp( i ( θ − π ) t / , < θ < π (this is denoted t t / ( e iθ ) u it / ( e iθ ) = ξ ( it − t ) / ( e iθ ) η ( it + t ) / ( e iθ )in [10], where the formula (22) is stated as Eq. (41); see [6] for two elementary computationsof the corresponding T¨oplitz determinants.)We rewrite this in terms of the Barnes function G ( z ), as is customary. We recall that G isan entire function of order 2, such that G (1) = 1 and G ( z + 1) = Γ( z ) G ( z ) for all z , and thatits zeros are located at the negative integers. In particular, it satisfies N Y j =1 Γ( j + θ ) = G (1 + N + θ ) G (1 + θ ) (23)for all N > θ ∈ C .Thus, we have E ( e it · X N ) = Y j N Γ( j )Γ( j + it )Γ( j + it + t )Γ( j + it − t ) = G (1 + it − t ) G (1 + it + t ) G (1 + it ) × G (1 + it + N ) G (1 + N ) G (1 + it − t + N ) G (1 + it + t + N )(see, e.g., [10, eq. (41)]). We now get from [10, Cor. 3.2] that E ( e it · X N ) ∼ N ( it − t )( it + t ) / G (1 + it − t ) G (1 + it + t ) G (1 + it )= exp( − Q N ( t ) / G (1 + it − t ) G (1 + it + t ) G (1 + it ) , where Q N ( t , t ) = δ N ( t + t ) , δ N = log N, hence we have complex mod-Gaussian convergence with limiting functionΦ g ( t , t ) = G (1 + it − t ) G (1 + it + t ) G (1 + it ) . (24)Here we can take µ = 1 in (4). Remark . Note that this is not the product of the two individual limiting functions for mod-Gaussian convergence of the real and imaginary parts of X N separately (which are Φ g ( t ,
0) andΦ g (0 , t )), although after normalizing, one obtains convergence in law of(Re( X N ) / p δ N , Im( X N ) / p δ N ) o independent standard Gaussian variables, as noted by Keating and Snaith.We now check that Theorem 4 can be applied to the sequence of random variables ( X N ). Thefact that Φ g is of class C on R is clear in view of the analytic properties of the Barnes function.Condition (3) is also obvious. A uniformity estimate like (7) is not found in [22] or [10], thoughit is proved for t in a fixed compact region of R in [10, Cor. 3.2]. In Proposition 17 in theAppendix, we prove Φ N ( t ) = Φ( t ) e − Q N ( t ) / (cid:16) O (cid:16) k t k N (cid:17)(cid:17) for k t k N / . In view of σ N = (log N ) /
4, this is compatible with (7) and (13), with a arbitrarily large, C = and A (defined by (8)) can be taken to be any A >
2. Thus theconstant D in (10) can be any number D > , . Moreover, since ˜ Q N ( x , x ) = x + x log N , we see by using Remark 2 (namely, (18)) and (11) that we have the following corollary:
Corollary 5.
For < ε < we have P ( | X N − z | < ε ) ≫ ε (log N ) − (25) for all N with N ≫ max (cid:8) exp( | z | ) , exp( Cε − ) (cid:9) where both implied constants are absolute. (The first condition on N ensures the main term is ≫ ε (log N ) − , while the second ensuresthat the error term is smaller; we have taken 2 / / (2 mµ ) − γ = 1 / − /
36 for definiteness;any number > Lemma 6.
Let z ∈ C × and ε > , and denote w = log | z | + iθ , θ = Arg( z ) ∈ ] − π, π ] . Then, provided ε | z | , we have | e w − z | < ε for all w ∈ C such that | w − w | < ε | z | . For given z ∈ C , non-zero, we get from this and (25), applied to log | z | + i Arg( z ) and to ε/ | z | instead of ε , the following explicit form of Theorem 1: Theorem 7.
Let z ∈ C × be arbitrary, ε > such that ε | z | . We have P ( | det(1 − T N ) − z | < ε ) ≫ (cid:16) ε | z | (cid:17) N for N > N ( z , ε ) , where N ( z , ε ) ≪ max n exp (cid:0) (log | z | ) (cid:1) , exp (cid:16) C (cid:16) ε | z | (cid:17) − (cid:17)o where C and the implied constants are absolute.Remark . It follows from asymptotic formulas for the Barnes function, e.g. [11], that we have k t k log | Φ g ( t ) | ≍ log(2 k t k ) , which is illustrated in Figure 1. This super-gaussian behavior is the main cause of difficulty inthe proof of Theorem 4. Figure 1.
Graph of t log | Φ g (1 , t ) | , 1 t Remark . Note that if one only wants to say that det(1 − T N ), for N growing, has dense imagein C , much simpler topological arguments suffice. Example 2.
Another conspicuous example of mod-Gaussian convergence is the arithmeticEuler factor in the moment conjecture for ζ (1 / it ). In [15, § | ζ ( + it ) | only, and we first generalize this as in the previous section.Consider a sequence ( X p ) of independent random variables uniformly distributed on the unitcircle and indexed by prime numbers, and let L N = − X p N log(1 − p − / X p )where the logarithm is given here by the Taylor expansion around 0. For each individual term E p = − log(1 − p − / X p ), we have E ( e it · E p ) = 12 π Z π (1 − ae iθ ) −
12 ( t + it ) (1 − ae − iθ )
12 ( t − it ) dθ with a = p − / . Expanding by the binomial theorem and picking up the constant term in theexpansion in Fourier series, we obtain E ( e it · E p ) = X j > a j (cid:18) − ( t + it ) j (cid:19)(cid:18) ( t − it ) j (cid:19) = X j > ( ( t + it )) j ( ( it − t )) j ( j !) a j = F ( ( it + t ) , ( it − t ); 1; a )in terms of the Gauss hypergeometric function.Arguing as in [15], we see now that E ( e it · L N ) = Y p N F ( ( it + t ) , ( it − t ); 1; p − )= Y p N (cid:16) − t + t p + O (cid:16) p (cid:17)(cid:17) , nd hence, denoting δ N = − X p N log(1 − p − ) ∼
12 log log
N, Q N ( t ) = δ N ( t + t ) = δ N k t k , we get E ( e it · L N ) ∼ exp( − Q N ( t ) / a ( t ) , as N → + ∞ , with limiting functionΦ a ( t ) = Y p (cid:16) − p (cid:17) −k t k / F ( ( it + t ) , ( it − t ); 1; p − ) . (26)We have here also µ = 1 in (4). Now, to check the uniformity required in (7), we write E ( e it · L N ) = exp( − Q N ( t ) / a ( t ) R N ( t )with R N ( t ) = Y p>N (cid:16) − p (cid:17) −k t k / F ( ( it + t ) , ( it − t ); 1; p − ) . If we expand the p -factor using the binomial theorem, we obtain1 + X j > p j X a + b = j ( ( it + t )) a ( ( it − t )) a ( a !) (cid:18) −k t k / b (cid:19) , and if we assume that k t k A with A >
1, crude bounds show that this p -factor is1 + O (cid:16)X j > jA j p j (cid:17) , where the implied constant is absolute, so that if k t k N / , for instance, we get R N ( t ) = Y p>N (cid:16) O (cid:16)X j > jp − j/ (cid:17)(cid:17) = 1 + O ( N − / ) . Although this is crude, it already gives much more than (7), both in terms of range ofuniformity and sharpness of approximation.Since Condition (3) is also obviously valid here, Theorem 4 (or rather (18)) applies with A any real number > a and C arbitrarily large, and shows that P ( | L N − z | < ε ) ≫ ε (log log N ) − for any z ∈ C and ε <
1, provided N ≫ max (cid:16) exp(exp( | z | )) , exp(exp( Cε − )) (cid:17) for some large constant C > P N ( t ) be given by (2). For fixed N , it is well-known that the random variables t P N ( t ) onthe probability spaces ([0 , T ] , T − λ ) converge in law, as T → + ∞ , to˜ P N = Y p N (1 − p − / X p ) − = exp( L N ) , where X p are as above (independent and uniformly distributed on the unit circle; the inde-pendence is due to the fundamental theorem of arithmetic). For any open set V , it followsthat lim inf T → + ∞ T λ ( { t T | P N ( t ) ∈ V } ) > P ( ˜ P N ∈ V ) . Because we do not know if the probability that ˜ P N is in the boundary of V is zero or not, we do not claim– or need to claim – an equality; see, e.g., [1, Th. 2.1, (iv)]. pplying Lemma 6 as in the proof of Theorem 7, we obtain Theorem 2 with N ( z , ε ) ≪ max n exp (cid:16) exp (cid:16) (log | z | ) (cid:17)(cid:17) , exp (cid:16) exp (cid:16) C (cid:16) ε | z | (cid:17) − (cid:17)(cid:17)o for some absolute constant C . Remark . Again, the density of values of P N ( t ) for N > t ∈ R (or of ˜ P N , which amountsto the same thing) is an easier matter that can be dealt with using topological tools. Example 3.
The two previous examples are of course motivated by their conjectural relationwith the behavior of the Riemann zeta function on the critical line (this is the arithmetic essenceof [22]). Indeed, Keating and Snaith conjecture that:
Conjecture 8.
Define log ζ (1 / iu ) , when u ∈ R is not the ordinate of a non-trivial zero of ζ ( s ) , by continuation along the horizontal line Im( s ) = u , with limit when Re( s ) → + ∞ .For any t = ( t , t ) ∈ R , we have T Z T e it · log ζ (1 / iu ) du = Φ a ( t )Φ g ( t ) exp (cid:16) − t T ) (cid:17) (1 + o (1)) as T → + ∞ , where · is the inner product on R as in (21) . Hence, we see in particular that the following holds:
Corollary 9.
Assume there exist α > , δ > and θ > such that Conjecture holds with theerror term o (1) replaced by exp( − α (log log T ) δ ) uniformly for k t k (log log 6 T ) θ . Then the set of values ζ (1 / it ) is dense in the complexplane. In fact, there exists C > , D > , such that, for any z ∈ C × and ε | z | , there exists t with t ≪ max n exp (cid:0) exp (cid:0) (log | z | ) (cid:1)(cid:1) , exp (cid:0) exp (cid:16) C (cid:16) ε | z | (cid:17) − D (cid:17)(cid:1)o , such that | ζ ( + it ) − z | < ε. Of course, such a strong conjecture concerning the imaginary moments of ζ (1 / it ) looksquite hopeless at the current time: there is no known non-trivial result available, even assumingthe Riemann Hypothesis. But Example 4 below suggests that (with this approach) it is indeednecessary to require that the characteristic function converge uniformly for t in a region growingwith T . In [8], jointly with F. Delbaen, we will explain how the weaker qualitative statement1 T λ ( { u ∈ [0 , T ] | ζ ( + iu ) ∈ V } ) ≫ T for a fixed open set V and for T large enough (which of course suffices to give a positive answerto Ramachandra’s question) can be proved under much weaker assumptions than a uniformversion of Conjecture 8. Remark.
Another remark concerning Conjecture 8 has to do with the factored form of thelimiting function Φ a ( t )Φ g ( t ), which seems to imply some asymptotic independence property.Recall that the real and imaginary parts of Φ a and Φ g are themselves asymptotically independent after renormalization , but are not products of the limiting functions for the two parts separately.So Conjecture 8, if correct, is evidence of quite particular probabilistic behavior. The fourth moment of ζ (1 / it ) and a few other results do provide evidence of a factored limiting function,with “random matrix” term split from the Euler factor. The mod-Poisson analogy is also consistent with this,in the case of the number of prime factors of an integer, as discussed in detail in [27, §
4, 5, 6]. xample 4. Our assumptions in Theorem 4 are probably not optimal. We now describe anilluminating (counter)-example in the direction of understanding when a result like this couldbe true.We again look at random matrices T N in the compact group U ( N ) (as in Example 1), butthis time we consider the random variables counting the number of eigenvalues in certain fixedarcs of the unit circle: fix γ ∈ ]0 , / I = { e iπθ | | θ | γ } ⊂ C . Then let X N be the number of eigenvalues ϑ of T N such that ϑ ∈ I . Note that X N is aninteger-valued random variable. It was proved by Costin and Lebowitz that X N − γNπ − √ log N converges in law to a standard normal random variable. Wieand [31] gave a proof based onasymptotics of T¨oplitz determinants with discontinuous symbols; as noted by Basor, this givesthe asymptotic E ( e it ( X N − γN ) ) ∼ exp (cid:16) − t π log N (cid:17) (2 − πγ ) t π G (cid:16) − t π (cid:17) G (cid:16) t π (cid:17) as N → + ∞ , for all t with | t | < π (see, e.g., [10, Th. 5.47], applied with N = 2, α = α = 0, β = t π , β = − t π , and the condition on t is equation (5.79) in loc. cit., or [3, p. 331]).This asymptotic is of course of the form (3) for these values of t , but the restriction | t | < π isnecessary, since the characteristic function of X N is 2 π -periodic for all N . The convergence issufficiently uniform for t close to 0 to allow the deduction of the renormalized normal behavior(as Wieand did, using the Laplace transform instead of the characteristic function), but when γ is rational, the set of possible values of X N − γN for N > R . Distribution of central values of L -functions over finite fields We now consider examples related to L -functions over finite fields. Our main input will bedeep results of Deligne and Katz, and we are of course motivated by the philosophy of Katzand Sarnak [21].The goal is to make statements about the distribution of values at the central point 1 / L -functions over finite fields. The appealing aspect is that these form discrete sets, henceproving that they are dense in C (as in Theorem 3), for instance, is obviously interesting andmeaningful. We consider examples of our results for the three basic symmetry types in turn:unitary, symplectic, and orthogonal. For the last two, this means first obtaining a suitableanalogue of Example 1. The corresponding limiting functions have already been studied in somerespect by Keating-Snaith [23] and Conrey-Farmer [7], though our expressions seem somewhatmore natural.4.1. Unitary symmetry.
Let F q be a finite field with q elements. Unitary symmetry arises(among other cases) for certain types of one-variable exponential sums over finite fields, whichare associated to Dirichlet characters of F q [ X ], which we now describe; these will lead to a proofof Theorem 3.Let F q be a finite field with q elements of characteristic p = 0. A Dirichlet character modulo g ∈ F q [ X ] is a map η : F q [ X ] → C , Including a more general result concerning the joint distribution of the number of eigenvalues in more thanone interval. For what it’s worth, one may mention that the density of values X N − γN is true for irrational γ , byDirichlet’s approximation theorem, and by the existence of matrices in U ( N ) where the number of eigenvalues in I takes any value between 0 and N . efined by η ( f ) = ( f and g are not coprime η ( f ) otherwise , where η is a group homomorphism η : ( F q [ X ] /g F q [ X ]) × −→ C × . This character is non-trivial if η = 1, and primitive if it can not be defined (in the obviousway) modulo a proper divisor of g . The associated L -function is defined by the Euler product L ( s, η ) = Y π (1 − η ( π ) | π | − s ) − , for s ∈ C , where the product ranges over monic irreducible polynomials in F q [ X ] and | π | = q deg( π ) . One shows quite easily that this is in fact a polynomial (which we denote Z ( η, T )) inthe variable T = q − s of degree deg( g ) −
1, if η is primitive modulo g and non-trivial.The examples used in proving Theorem 3 arise from the following well-known construction.For any integer d >
1, with p ∤ d , any non-trivial multiplicative character χ : F q → C × such that χ d = 1, and any squarefree polynomial g ∈ F q [ X ] of degree d , we let S ( χ, g ) = X x ∈ F q χ ( g ( x )) , where χ (0) is defined to be 0. These are multiplicative exponential sums, and have beenstudied intensively, due in part to their many applications to analytic number theory (for theirgeneralizations to multiple variables, see the paper [17] of Katz).It is also well-known that one can construct a non-trivial Dirichlet character η = η ( g, χ ),primitive modulo g , such that Z ( η, T ) = exp (cid:16) X m > S m ( χ, g ) m T m (cid:17) , where S m ( χ, g ) denotes the “companion” sums over extensions of F q , namely S m ( χ, g ) = X x ∈ F qm χ ( N F qm / F q ( g ( x ))) , where N F qm / F q is the norm map. We will denote L ( s, g, χ ) the corresponding L -function.Moreover, we have the Riemann Hypothesis for these L -functions (due to A. Weil), whichgives the link with random unitary matrices: there exists a unique conjugacy class θ χ,g ( q ) inthe unitary group U ( d −
1) such that L ( s + , g, χ ) = det(1 − q − s θ χ,g ( q )) , (so that, in particular, we recover the Weil bound | S ( χ, g ) | ( d − q / , by looking at the trace of θ χ,g ( q )). For all this, one can see, for instance, [26, § Theorem 10.
For d > and t ∈ Z , let g d,t = X d − dX − t ∈ Z [ X ] . For p prime, let X ( p ) denote the set of pairs ( χ, t ) where χ is non-trivial character of F p and t ∈ F p .Let z ∈ C × and ε > with ε | z | be given. For all integers d > d ( z , ε ) , we have lim inf p → + ∞ |{ ( χ, t ) ∈ X ( p ) | χ d = 1 , | L ( , g d,t , χ ) − z | < ε }|| X ( p ) | ≫ (cid:16) ε | z | (cid:17) d , here d ( z , ε ) ≪ max n exp (cid:0) (log | z | ) (cid:1) , exp (cid:16) C (cid:16) ε | z | (cid:17) − (cid:17)o , where C > and the implied constants are absolute. This result depends on the mod-Gaussian convergence for characteristic polynomials on U ( N )(i.e., on Example 1). Indeed, denoting by U ( N ) ♯ the space of conjugacy classes in U ( N ), wehave the following: Theorem 11.
For any integer d > , any odd prime p with p ∤ d ( d − , the conjugacy classes { θ χ,g d,t ( p ) | χ (mod p ) , χ = 1 , and t ∈ F p with t d − − (1 − d ) d − = 0 (mod p ) } become equidistributed in U ( d − ♯ as p → + ∞ , with respect to Haar measure.Proof. This is an easy consequence of results of Katz (see [18, Th. 5.13]), the only “twist”being the extra averaging over all non-trivial Dirichlet characters to obtain unitary instead ofspecial unitary (or similar) equidistribution.First of all, it is easy to check that if p ∤ d ( d −
1) and t ∈ F p is such that t d − = (1 − d ) d − ,the polynomial g d,t = X d − dX − t ∈ F p [ X ] is a strong Deligne polynomial in one variable (inthe language of [18], these are called “weakly-supermorse” polynomials). Hence the conjugacyclasses in the statement are well-defined.For simplicity, denote U the open subset of the affine t -line where t d − = (1 − d ) d − .Now, according to the Weyl equidistribution criterion, we must show thatlim p → + ∞ p − X ∗ χ (mod p ) | U ( F p ) | X t ∈ U ( F p ) Tr Λ( θ χ,g d,t ( p )) = 0 . for any (fixed) non-trivial irreducible unitary representation Λ of the compact group U ( d − χ where χ d = 1: there are at most 2 d of them.For any other character χ (mod p ), the inner sum over t ∈ U ( F p ) is of the type handled bythe Deligne Equidistribution Theorem. Let k = k ( χ ) be the order of the Dirichlet character χχ , where χ is the real character modulo p . By [18, Th. 5.13, (2)] (the restriction χ d = 1ensures the assumptions hold), provided p ∤ d ( d − p > d ( d − GL k ( d −
1) = { g ∈ GL ( d − | det( g ) k = 1 } , with maximal compact subgroup U k ( d −
1) = { g ∈ U ( d − | det( g ) k = 1 } . For simplicity, we write U = U ( d − U k = U k ( d − U k is afinite sum of irreducible representations of this group (possibly including trivial components).Applying [21, Th. 9.2.6, (5)] to each of the non-trivial ones (and the obvious identity for thetrivial components), we find that1 | U ( F p ) | X t ∈ U ( F p ) Tr Λ( θ χ,g d,t ( p )) = h Λ | U k , i + O ((dim Λ) dp − / )where the implied constant is absolute (this is because we have a one-parameter family, so wecan apply [21, 9.2.5] and the fact proved in [18, 5.12] that the relevant sheaf is everywhere tame,so the Swan-conductor contribution is zero; the parameter curve U has d points at infinity, whichgive the factor d above).Using Frobenius reciprocity or direct integration (using, e.g, [21, Lemma AD.7.1]), we findthat the multiplicity of the trivial representation in Λ (restricted to U k ) satisfies h Λ | U k , i = X h ∈ Z h Λ , det( · ) hk i = ( · ) hk for some h ∈ Z − { } , . or a given Λ, this is equal to 1 only if dim Λ = 1, so Λ = det( · ) r for some r ∈ Z − { } , and if χ is such that k ( χ ) | r . The number of such characters χ is therefore ≪
1, the implied constantdepending on Λ. Hence we find, after adding back the characters with χ d = 1, that1 p − X ∗ χ (mod p ) | U ( F p ) | X t ∈ U ( F p ) Tr Λ( θ χ,g d,t ( p )) ≪ dp + (dim Λ) dp / , where the implied constant depends on Λ. This confirms the claimed equidistribution. (cid:3) Proof of Theorem 10.
This an easy consequence of Theorem 11: for any open set V ⊂ C , wehave first that | U ( F p ) | ∼ p as p goes to infinity, and then we can writelim inf p → + ∞ | X ( p ) | |{ ( χ, t ) ∈ X ( p ) | χ d = 1 , L ( , g d,t , χ ) ∈ V }| > µ d − ( { g ∈ U ( d − | det(1 − g ) ∈ V } ) , and then we apply Theorem 7. (cid:3) Remark . In Theorem 11, we performed the average over χ , because for “standard” familiesof exponential sums (those parametrized by points of algebraic varieties), the (connected com-ponent of the) geometric monodromy group is always semisimple, so its center is finite and itsmaximal compact subgroup can never be U ( N ). However, one may expect that t could be fixedin the example above, and that (for instance) the conjugacy classes { θ χ ( p ) = θ χ,X d − dX − | χ (mod p ) non-trivial } corresponding to the exponential sums S ( χ ) = X x ∈ F p χ ( x d − dx − d > p ∤ d ( d − U ( d − ♯ as p → + ∞ .Very recently, N. Katz [20] has indeed shown that for such families there is always an a-priori“Sato-Tate law”, i.e., that the conjugacy classes become equidistributed in K ♯ for some compactgroup K ⊂ U ( d − t in the vertical direction where one looks at characters of F q m for fixed q and m → + ∞ (see [20, Th. 7.2, Th. 17.6, Remark 17.7]). It seems quite possible that the “horizontal”direction we are interested in will also follow from these new techniques.Similarly, it is likely that families of hyper-Kloosterman sums of certain types exhibit fullunitary monodromy, e.g., for any integer n >
1, any additive character ψ : F q → C × , anymultiplicative character χ : F × q → C × , one can consider the sums X x ,...,x n ∈ F × q x ··· x n =1 χ ( x ) ψ ( x + · · · + x n − + x n ) , for which the basic theory (due to Deligne [9, § L -function (unitarilynormalized so the central point is s = 0) isdet(1 − q − s θ ψ,χ ( q ))for some unique conjugacy class θ ψ,χ ( q ) ∈ U ( n − ψ for F p (e.g., ψ ( x ) = e ( x/p )) and define ψ m ( x ) = ψ (Tr F pm / F p ( x )) for all m >
1, the sets ofconjugacy classes { θ ψ m ,χ ( p m ) | χ : F × p m → C × } become equidistributed in U ( n − ♯ as m → + ∞ . .2. Symplectic symmetry.
A typical example of symplectic symmetry involves families of L -functions of algebraic curves over finite fields. For simplicity, we will consider one of thesimplest ones, but we first start by proving distribution results for characteristic polynomialsof symplectic matrices, which are of independent interest.We first remark that for A ∈ U Sp (2 g, C ), the characteristic polynomial can be expressed inthe form det(1 − T A ) = Y j g (1 − e iθ j T )(1 − e − iθ j T )for some eigenangles θ j , 1 j g , and it follows thatdet(1 − A ) = Y j g | (1 − e iθ j ) | > . This positivity is reflected in a shift in expectation in the mod-Gaussian convergence (it alsomeans that the argument is not an interesting quantity here). We obtain:
Proposition 12.
For g > , let X g = log det(1 − T g ) − log( πg ) , where T g is a Haar-distributed random matrix in the unitary symplectic group U Sp (2 g, C ) . Then X g converges in mod-Gaussian sense with Q g ( t ) = (log g ) t and limiting function Φ Sp ( t ) = G ( ) G ( + it ) . (27) Indeed, we have E ( e itX g ) = exp( − (log g ) t / Sp ( t ) (cid:16) O (cid:16) | t | g (cid:17)(cid:17) for | t | g / , where the implied constant is absolute. Figure 2 is a graph illustrating the logarithmic growth of t log | Φ Sp ( t ) | .
20 40 60 80 100 120 1400.511.5
Figure 2.
Graph of t log | Φ Sp ( t ) | , 1 t The expressions in [23, (32), (67)] and [7, Cor. 4.2] are rather more complicated, but of course they areequivalent. roof. (Compare [15, Prop. 4.9]) Keating-Snaith [23, (10)] compute that E ( e it log det(1 − T g ) ) = 2 git g Y j =1 Γ(1 + g + j )Γ( + it + j )Γ( + j )Γ(1 + it + g + j ) , which, together with the formula (23), gives E ( e itX g ) = (cid:16) πg (cid:17) − it/ G ( ) G ( + it ) × git G ( + it + g ) G (2 + 2 g ) G (2 + it + g ) G ( + g ) G (2 + g ) G (2 + it + 2 g ) . By applying Proposition 17, (3) in the Appendix, we get E ( e itX g ) = (cid:16) g (cid:17) − t / Φ Sp ( t ) (cid:16) O (cid:16) | t | g (cid:17)(cid:17) , as claimed. (cid:3) In particular, the Central Limit Theorem for det(1 − T g ) takes the form of the convergencein law log det(1 − T g ) − log πg (log( g/ / ⇒ (standard Gaussian) , so, for any a < b , we have P (cid:16)(cid:16) πg (cid:17) / e a √ log( g/ < det(1 − T g ) < (cid:16) πg (cid:17) / e b √ log( g/ (cid:17) → √ π Z ba e − t / dt. On the other hand, by applying Theorem 4, as we can according to the previous proposition,we can control the probability of the values of det(1 − T g ) in much smaller (dyadic or similar)intervals: Corollary 13.
Let U =] a, b [ with a < b real numbers. We have P (cid:16) e a (cid:16) πg (cid:17) / < det(1 − T g ) < e b (cid:16) πg (cid:17) / (cid:17) = 1 q π log g Z ba exp (cid:16) − t g (cid:17) dt + O (cid:16) max( b − a, ( b − a ) − )log g + max(1 , b − a )(log g ) / / (cid:17) for g > , where the implied constant is absolute. In particular P (cid:16)(cid:16) πg (cid:17) / < det(1 − T g ) < (cid:16) πg (cid:17) / (cid:17) = 1 q π log g Z log 20 exp (cid:16) − t g (cid:17) dt + O (cid:16) g ) / / (cid:17) . Proof. If b − a
1, we apply Theorem 4, with the constants µ = 1, A arbitrarily close to 2, a arbitrarily large and C = 1 /
2, so that D can be any number with D > , and in particular D = 29 is valid. If b − a >
1, we split the interval ] a, b [ into 2 ⌈ b − a ⌉ intervalsof length 14 b − a ⌈ b − a ⌉ , and apply the previous case to the interior of those intervals. Since the joint distribution ofeigenvalues of T g is absolutely continuous with respect to Lebesgue measure, the probabilityof falling on one of the missing endpoints is zero, and summing over these intervals gives theresult. (cid:3) e now deduce an arithmetic corollary, using families of hyperelliptic curves over finite fields.For any odd q , any integer g > f ∈ F q [ X ] of degree2 g + 1, let C f be the smooth projective model of the affine hyperelliptic curve C f : y = f ( x ) . The number of F q m -rational points on C f satisfies | C f ( F q m ) | = q m + 1 − X x ∈ F qm χ ( N F qm / F q ( f ( x ))) = q m + 1 − S m ( χ , f )where χ is the quadratic character of F × q and the notation is as in Section 4.1. The associated L -function (the numerator of the zeta function) is defined by L ( C f , s ) = L ( s, f, χ ) , or, in other words, we have L ( C f , s ) = Z ( C f , q − s ) , Z ( C f , T ) = exp (cid:16) X m > S m ( χ , f ) m T m (cid:17) . Weil proved that Z ( C f , T ) is a polynomial in Z [ T ], of degree 2 g , all roots of which havemodulus √ q , and which is symplectic : there is a unique conjugacy class θ f ( q ) in U Sp (2 g, C )such that L ( C f , s + ) = det(1 − q − s θ f ( q )) . Theorem 14.
Let H g ( F q ) be the set of squarefree, monic, polynomials of degree g + 1 in F q [ X ] . Fix a non-empty open interval ] α, β [ ⊂ ]0 , + ∞ [ . For all g large enough, we have lim inf q → + ∞ | H g ( F q ) | (cid:12)(cid:12)(cid:12)n f ∈ H g ( F p ) | L ( C f , / p πg/ ∈ ] α, β [ o(cid:12)(cid:12)(cid:12) ≫ √ log g . (28)(Note that this is in fact a very weak version of what we can prove). Proof.
Let first H ∗ g ( F q ) be the set of f ∈ H g ( F q ) for which L ( C f , / = 0. In [15, Prop. 4.9],we showed, using the relevant equidistribution computation in [21, 10.8.2] that the (real-valued)random variables L g = log det(1 − θ F ( q )) − log( πg ) , on H ∗ g ( F q ) (with counting measure) converges in law to X g = log det(1 − T g ) − log( πg ) , where T g is a random matrix in the unitary symplectic group U Sp (2 g, C ), distributed accordingto Haar measure. The previous proposition shows that Theorem 4 is applicable to X g withcovariance Q g ( t ) = (log g ) t and limiting function Φ Sp ( t ). Letting q → + ∞ as in the previoussection, we getlim inf q → + ∞ | H ∗ g ( F q ) | (cid:12)(cid:12)(cid:12)n f ∈ H ∗ g ( F p ) | log L ( C f , / − log( πg ) ∈ ] α, β [ o(cid:12)(cid:12)(cid:12) ≫ √ log g for g large enough. Since | H ∗ g ( F q ) | = | H g ( F q ) | (1 + o (1)) = q g +1 (1 + o (1))for fixed g and q → + ∞ (by an easy application of the equidistribution, see [15, Prop. 4.9]),we get the result stated by exponentiating. (cid:3) Remark . The lower bound (28) is good enough to combine with various other statementsproving arithmetic properties of L -functions which hold for “most” hyperelliptic curves. Forinstance, from [25, Prop. 1.1] (adapted straightforwardly to all hyperelliptic curves instead of pecial one-parameter families), it follows that if we denote by ˜ H g ( F q ) the set of f ∈ H g ( F q )such that the eigenvalues of θ f ( q ) satisfy no non-trivial multiplicative relation, then we have |{ f ∈ H g ( F q ) | f / ∈ ˜ H g ( F q ) }| ≪ g q − γ for some γ = γ ( g ) >
0, and hence we getlim inf q → + ∞ | H g ( F q ) | (cid:12)(cid:12)(cid:12)n f ∈ ˜ H g ( F p ) | L ( C f , / p πg/ ∈ ] α, β [ o(cid:12)(cid:12)(cid:12) ≫ √ log g for g large enough.4.3. Orthogonal symmetry.
Orthogonal symmetry, in number theory, features prominentlyin families of elliptic curves. In contrast with symplectic groups, there are a number of “flavors”involved, due to the “functional equation” T N det(1 − T − A ) = det( − A ) det(1 − T A )for an orthogonal matrix A ∈ O ( N, R ) (the standard maximal compact subgroup of the or-thogonal group O ( N, C )), which implies that det(1 − A ) is zero for “trivial” reasons if N iseven and det( A ) = − N is odd and det( A ) = 1. When this happens, it is of great interestto investigate the distribution of the first derivative at T = 1 of the reversed characteristicpolynomial. For simplicity, however, we restrict our attention here to N even and matrices withdeterminant 1, i.e., to the subgroup SO (2 N, R ) of O (2 N, R ), where N >
1. In that case, it isalso true that eigenangles come in pairs of inverses, and therefore we have det(1 − A ) > Proposition 15.
For N > , let X N = log det(1 − T N ) − log( πN ) , where T N is a Haar-distributed random matrix in the special orthogonal group SO (2 N, R ) . Then X N converges in mod-Gaussian sense with Q N ( t ) = (log N ) t and limiting function Φ SO ( t ) = G ( ) G ( + it ) . (29) Indeed, we have E ( e itX N ) = exp( − (log N ) t / SO ( t ) (cid:16) O (cid:16) | t | N (cid:17)(cid:17) for | t | N / , where the implied constant is absolute.Proof. Using [23, (56)] and (23), we get E ( e it log det(1 − T N ) ) = 2 Nit N Y j =1 Γ( N + j − it + j − )Γ( j − )Γ( it + N + j − , = G ( ) G ( + it ) × Nit G ( + it + N ) G (2 N ) G ( it + N ) G ( + N ) G ( N ) G ( it + 2 N ) , and by applying Proposition 17, (4) in the Appendix, we get the desired formula E ( e itX N ) = (cid:16) N (cid:17) − t / Φ SO ( t ) (cid:16) O (cid:16) | t | N (cid:17)(cid:17) . (cid:3) Remark . If we compare with the symplectic case, we observe the (already well-established)phenomenon that the value det(1 − A ), for A ∈ SO (2 N, R ) tend to be small, whereas they tendto be large for symplectic matrices in U Sp (2 g, C ). An analogue of the hypothetical statement of Q -linear independence of the ordinates of zeros of the Riemannzeta function; non-trivial refers to a relation that can not be deduced from the fact that, if e iθ is an eigenvalue,so is its inverse e − iθ . ur arithmetic corollary is based on families of quadratic twists of elliptic curves over functionfields, and we select a specific example for concreteness (see [24, § q >
3, any integer N >
1, we consider the elliptic curves over thefunctional field F q ( T ) given by the Weierstrass equations E z : Y = ( T N − N T − − z ) X ( X + 1)( X + T ) , where z ∈ F q is a parameter such that z is not a critical value of T N − N T − L -function (which is now defined by the “standard” Eulerproduct over primes in F q [ T ], with suitable ramified factors) is of the form L ( E z , s + 1) = det(1 − θ z ( q ) q − s )where θ z ( q ) is a unique conjugacy class in O (2 N, R ). Theorem 16.
Fix a non-empty open interval ] α, β [ ⊂ ]0 , + ∞ [ . For all N large enough, we have lim inf p → + ∞ ( p − ,N − p (cid:12)(cid:12)(cid:12)n z ∈ F p | (cid:16) N π (cid:17) / L ( E z , / ∈ ] α, β [ o(cid:12)(cid:12)(cid:12) ≫ √ log N .
Proof.
As recalled in [24, Cor. 4.4 and before], for all N >
146 and primes p with p ∤ N ( N − N + 1) and ( p − , N −
1) = 1, the conjugacy classes θ z ( p ), for z ∈ F p not a critical value,become equidistributed in O (2 N, R ) ♯ for the image of Haar measure (precisely, this is statedfor the “vertical direction” where p is fixed and finite fields of characteristic p and increasingdegree are used; however, because the parameter variety is a curve with N + 1 points at infinityand the relevant sheaf is tame, we can recover the horizontal statement as in the proof ofTheorem 11). In particular, there is a subset V p ⊂ F p with | V p | ∼ p/ θ z ( p )) = 1 andthose restricted conjugacy classes become equidistributed in SO (2 N, R ) ♯ . Hence, for N largeenough, we getlim inf p → + ∞ ( p − ,N − | V p | (cid:12)(cid:12)(cid:12)n z ∈ V p | (cid:16) N π (cid:17) / L ( E z , / ∈ ] α, β [ o(cid:12)(cid:12)(cid:12) > µ SO (2 N, R ) ( { A | log det(1 − A ) ∈ ] log α, log β [ } ) ≫ √ log N , as desired. (cid:3)
Remark . Obviously, this result (or its generalizations to other families of quadratic twistsover function fields) has interesting consequences concerning the problem of the distribution ofthe order of Tate-Shafarevich groups of the associated elliptic curves, through the Birch andSwinnerton-Dyer conjecture (which is known to be valid in its strong form for many ellipticcurves over function fields over a finite field with analytic rank 0 or 1). We hope to come backto this question, and its conjectural analogue over number fields, in another work.
Appendix: estimates for the Barnes function
We present in this appendix some uniform analytic estimate for the Barnes function, whichare needed to verify the strong convergence assumption (7) for sequences of random matricesin compact classical groups. Note that we did not try to optimize the results.
Proposition 17. (1)
For all z ∈ C and n > with | z | n / , we have G (1 + z + n ) G (1 + n ) = (2 π ) z/ e − ( n +1) z (1 + n ) z / nz (cid:16) O (cid:16) z + z n (cid:17)(cid:17) . (30)(2) For all N > and all t = ( t , t ) ∈ R with k t k N / we have G (1 + it + N ) G (1 + N ) G (1 + it − t + N ) G (1 + it + t + N ) = N − ( t + t ) / (cid:16) O (cid:16) k t k N (cid:17)(cid:17) . For all g > and all t ∈ R with | t | g / we have git G ( + it + g ) G (2 + 2 g ) G (2 + it + g ) G ( + g ) G (2 + g ) G (2 + it + 2 g ) = (cid:16) g (cid:17) − t / (cid:16)r πg (cid:17) it (cid:16) O (cid:16) | t | g (cid:17)(cid:17) . (4) For all N > and all t ∈ R with | t | N / we have Nit G ( + it + N ) G (2 N ) G ( it + N ) G ( + N ) G ( g ) G ( it + 2 N ) = (cid:16) N (cid:17) − t / (cid:16)r πN (cid:17) it (cid:16) O (cid:16) | t | N (cid:17)(cid:17) . In all estimates, the implied constants are absolute.Proof.
One can use the asymptotic expansions in [11], but we follow instead the nice arrange-ment of the Barnes function in [10, Cor. 3.2], which leads to a quicker and cleaner proof.(1) First, the ratio of Barnes function is well-defined since n >
1. We now use the formula G (1 + z + n ) G (1 + n ) = (2 π ) z/ e − ( n +1) z (1 + n ) z / nz S n ( z ) , (31)where S n ( z ) = e − z ( z − / Y k > n +1 (cid:16) zk (cid:17) k − n (cid:16) k (cid:17) z / nz e − z , which is valid for z ∈ C , n > w ) at the origin,we have log(1 + w ) = w − w O ( w − )for | w | /
2, with an absolute implied constant. Hence we obtainlog S n ( z ) = − z ( z − X k>n (cid:16) k log (cid:16) zk (cid:17) + z (cid:16) k (cid:17) − z (cid:17) + X k>n n (cid:16) z log (cid:16) k (cid:17) − log (cid:16) zk (cid:17)(cid:17) = − z ( z − − z X k>n k + n z ( z − X k>n k + O (cid:16)X k>n (cid:16) z k + z k (cid:17)(cid:17) with an absolute implied constant, for n > | z | n/
2, hence for | z | n / we getlog S n ( z ) = O (cid:16) z + z n (cid:17) , since X k>n k = 1 n + O (cid:16) n (cid:17) , X k>n k = 12 n + O (cid:16) n (cid:17) , for n >
1, with absolute implied constants. Hence, we have S n ( z ) = 1 + O (cid:16) z + z n (cid:17) for | z | n / , for some absolute implied constant, and we get the stated formula G (1 + z + n ) G (1 + n ) = (2 π ) z/ e − ( n +1) z (1 + n ) z / nz (cid:16) O (cid:16) z + z n (cid:17)(cid:17) . (2) Note first that the conditions N > k t k N / ensure that the values of the Barnesfunction in the denominator are non-zero. Next, let u = ( it − t ) / v = ( it + t ) /
2; we canexpress the ratio of Barnes function as G (1 + u + v + N ) G (1 + N ) G (1 + u + N ) G (1 + v + N ) = G (1 + u + v + N ) G (1 + N ) G (1 + N ) G (1 + u + N ) G (1 + N ) G (1 + v + N ) , nd k t k N / gives | u | , | v | N / , allowing us to apply (30) three times. The exponentialterms cancel out, leading to G (1 + u + v + N ) G (1 + N ) G (1 + u + N ) G (1 + v + N ) = (1 + N ) −k t k / (cid:16) O (cid:16) | t | + | t | N (cid:17)(cid:17) which gives the first part of the proposition.(3) We use a similar computation, applying (30) six times with the parameters ( n, z )(2 g, , ( g, it ) , ( g, + it ) , ( g, , (2 g, it ) , ( g, ) , leading, after an easy calculation, to a main term(2 π ) it/ e − it (1 + g ) − t +3 it/ igt (1 + 2 g ) t / − it − igt for the ratio of Barnes functions, and some further computation leads to the stated result (eachparameter z has | z | | t | + 1, so the error term is also as given).(4) We argue exactly as in the previous case, with parameters ( n, z ) given now by(2 N − , , ( N − , , ( N − , it + ) , ( N − , it ) , (2 N − , it ) , ( N − , ) , and we get a main term(2 π ) it/ N − t − it/ iNt (2 N ) t / it − itN = 2 t / it − itN (2 π ) it/ N − t / − it/ , which leads to the conclusion. (cid:3) References [1] P. Billingsley:
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