SELBERG S CENTRAL LIMIT THEOREM FOR log ζ( it)

SELBERG S CENRAL LIMI HEOREM FOR log ζ + it MAKSYM RADZIWI L L AND K SOUNDARARAJAN Introduction In this paper we give a new and simple proof of Selberg s influential theorem [8, 9] that log ζ + it has an approximately normal distribution with mean zero and variance log log t Apart from some basic facts about the Riemann zeta function, we have tried to make our proof self-contained heorem Let V be a fixed real number hen for all large, { } meas t : log ζ + it V log log e u / du π We outline the steps of the proof he first step is to show that log ζ + it is usually close to log ζσ + it for suitable σ near Proposition Let be large, and suppose t hen for any σ > / we have t+ log ζ dy + iy log ζσ + iy σ log t he proof of Proposition is the only place where we will briefly need to mention the zeros of ζs From now on, we set σ 0 = + W, for a suitable parameter W 3 to be chosen log later From Proposition, and in view of the heorem that we set out to prove, we see that if W = o log log then we may typically approximate log ζ + it by log ζσ 0 + it hus we may from now on focus on the distribution of log ζσ 0 + it here is considerable latitude in choosing parameters such as W, but to fix ideas we select W = log log log 4, X = /log log log, and Y = /log log Here X and Y are two parameters that will appear shortly Put Ps = Ps; X = Λn n s log n n X By computing moments, it is not hard to determine the distribution of Ps Proposition As t varies in t, the distribution of RePσ 0 +it is approximately normal with mean 0 and variance log log Date: September, 05 he first author was partially supported by NSF grant DMS-855 he second author is partially supported by NSF grant DMS-50037, and a Simons Investigator award from the Simons Foundation V

MAKSYM RADZIWI L L AND K SOUNDARARAJAN Our goal is now to connect RePσ 0 + it with log ζσ 0 + it for most values of t his is done in two stages First we introduce a Dirichlet polynomial Ms which we show is typically close to exp Ps Define an = if n is composed of only primes below X, and it has at most 00 log log primes below Y, and at most 00 log log log primes between Y and X; set an = 0 in all other cases Put 3 Ms = n µnan n s Note that an = 0 unless n Y 00 log log 00 log log log X < Dirichlet polynomial Proposition 3 With notations as above, we have for t except perhaps on a subset of measure o Mσ 0 + it = + o exp Pσ 0 + it, ɛ, and so Ms is a short he final step of the proof shows that ζσ 0 + itmσ 0 + it is typically close to Proposition 4 With notations as above, so that for t we have except perhaps on a set of measure o ζσ 0 + itmσ 0 + it dt = o, ζσ 0 + itmσ 0 + it = + o, Proof of heorem o recapitulate the argument, Proposition 4 shows that typically ζσ 0 + it Mσ 0 +it, which by Proposition 3 is exppσ 0 +it, and therefore by Proposition we may conclude that log ζσ 0 + it is normally distributed Finally by Proposition we deduce from this the normal distribution of log ζ + it his completes the proof of heorem After developing the proofs of the propositions, in Section 7 we compare and contrast our approach with previous proofs, and also discuss possible extensions of this technique Proof of Proposition Put Gs = ss π s/ Γs/ and ξs = Gsζs denote the completed ζ-function If t is large and t y t +, then by Stirling s formula log Gσ + iy/g/ + iy σ / log t, and so it is enough to prove that t+ log ξ + iy dy σ log ξσ + iy t

SELBERG S CENRAL LIMI HEOREM FOR log ζ + it 3 Recall Hadamard s factorization formula ξs = e A+Bs ρ s e s/ρ, ρ where A and B are constants with B = ρ Re /ρ hus assuming that y is not the ordinate of a zero of ζs log ξ + iy = log + iy ρ ξσ + iy σ + iy ρ ρ Integrating the above over y t, t + we get t+ 4 log ξ + iy dy t ξσ + iy ρ t+ t log + iy ρ dy σ + iy ρ Suppose ρ = β + iγ is a zero of ζs If t γ then we check readily that t+ log + iy ρ dy σ σ + iy ρ t γ t In the range t γ we use t+ log + iy ρ dy σ + iy ρ t hus in either case t+ t log β + x dx = πσ β σ + x log + iy ρ dy σ σ + iy ρ + t γ Inserting this in 4, and noting that there are logt + k zeros with k t γ < k +, the proposition follows 3 Proof of Proposition We begin by showing that we may restrict the sum in Ps just to primes he contribution of cubes and higher powers of primes is clearly O, and we need only discard the contribution of squares of primes By integrating out, it is easy to see that p p σ 0+it X dt p,p X min, logp /p herefore, the measure of the set t [, ] with the contribution of prime squares being larger than L say is at most /L In view of this, to establish Proposition, it is enough to prove that P 0 σ 0 + it := Re p σ 0+it p X

4 MAKSYM RADZIWI L L AND K SOUNDARARAJAN has an approximately Gaussian distribution with mean 0 and variance log log We establish this by computing moments Lemma Suppose that k and l are non-negative integers with X k+l hen, if k l, while if k = l we have P 0 σ 0 + it k P 0 σ 0 it l dt, P 0 σ 0 + it k dt = k! log log k + O k log log k +ɛ Proof Write Ps k = n a knn s, where a k n = 0 unless n has the prime factorization n = p α p αr r where p,, p r are distinct primes below X, and α + + α r = k, in which case a k n = k!/α! α r! herefore, expanding out the integral, we obtain P 0 σ 0 + it k P 0 σ 0 it l dt = n a k na l n n σ 0 + O m n a k ma l n mn σ 0 logm/n If m n, then logm/n / mn and so the off-diagonal terms above contribute m n a kma l n X k+l Note that if k l then a k na l n is always zero, and the first statement of the lemma follows It remains in the case k = l to discuss the diagonal term n a kn /n σ 0 he terms with n not being square-free are easily seen to contribute O k log log k Finally the square-free terms n give k! p,,p k X p j distinct p p k = k! σ 0 p σ 0 p X k + Ok log log k, and the lemma follows From Lemma we see that if X k then for odd k Re P 0 σ 0 + it k dt, while if k is even then k Re P 0 σ 0 + it k dt = k k/!log log k/ + O k log log k +ɛ k/ hese moments match the moments of a Gaussian random variable with mean zero and variance log log, and since the Gaussian is determined by its moments, our proposition follows

SELBERG S CENRAL LIMI HEOREM FOR log ζ + it 5 4 Proof of Proposition 3 Let us decompose Ps as P s + P s, where Put M s = P s = 0 k 00 log log n Y Lemma For t we have Λn n s log n, and P s = k P s k, and M s = k! Y <n X Λn n s log n 0 k 00 log log log 5 P σ 0 + it log log, and P σ 0 + it log log log, k P s k k! except perhaps for a set of measure / log log log When the bounds 5 hold, we have 6 M σ 0 + it = exp P σ 0 + it + Olog 99, and 7 M σ 0 + it = exp P σ 0 + it + Olog log 99 Proof Note that P σ 0 + it dt and similarly n,n Y Λn Λn min, log log, n n σ 0 log n log n logn /n P σ 0 + it dt log log log he first assertion 5 follows If z K then using Stirling s formula it is straightforward to check that e z z k e 99K, k! 0 k 00K and therefore the estimates 6 and 7 hold Put a n = if n is composed of at most 00 log log primes all below Y, and zero otherwise Put a n = if n is composed of at most 00 log log log primes all between Y and X, and zero otherwise hen Ms = M sm s with M s = n µna n n s and M s = n µna n n s

6 MAKSYM RADZIWI L L AND K SOUNDARARAJAN Lemma 3 With notations as above, we have and M σ 0 + it M σ 0 + it dt log 60, M σ 0 + it M σ 0 + it dt log log 60 Proof We establish the first estimate, and the second follows similarly If we expand M s into a Dirichlet series n bnn s, then we may see that bn for all n, bn = 0 unless n Y 00 log log is composed only of primes below Y, and bn = µna n if Ωn 00 log log It is the presence of prime powers in Ps that prevents M s from simply being M s hus, putting cn = bn µna n temporarily, we see that M σ 0 + it M σ 0 + it dt n,n cn cn n n σ 0 min, logn /n he terms with n n contribute since logn /n / n n in that case ɛ n n Y 00 log log he diagonal terms n = n contribute, for any < r <, n r 00 log log p Y p n = p Y Ωn>00 log log + r p + r p + Choosing r = e /3, say, the above is log 60 Proof of Proposition 3 From Lemma 3 it follows that except on a set of measure o, one has M σ 0 + it = M σ 0 + it + Olog 5 Moreover, from 6 except on a set of measure o we note that M σ 0 + it = exp P σ 0 + it + Olog 99, and by 5 that log M σ 0 + it log herefore, we may conclude that, except on a set of measure o, M σ 0 + it = M σ 0 + it + Olog 5 = exp P σ 0 + it + Olog 0 Similarly, except on a set of measure o, we have M σ 0 + it = M σ 0 + it + Olog log 5 = exp P σ 0 + it + Olog log 0 Multiplying these estimates we obtain completing our proof Mσ 0 + it = exp Pσ 0 + it + Olog log 0,

SELBERG S CENRAL LIMI HEOREM FOR log ζ + it 7 5 Proof of Proposition 4 For t, one has ζσ 0 + it = n n σ 0 it + O, and so ζσ 0 + itmσ 0 + itdt = n m amµm mn σ herefore, expanding the square, we see that mn it dt + O +ɛ = + O +ɛ 8 ζσ 0 + itmσ 0 + it dt = ζσ 0 + itmσ 0 + it dt + O +ɛ It remains to evaluate the integral above, and to do this we shall use the following familiar lemma see for example Lemma 6 of Selberg [8] For completeness we include a quick proof of the lemma in the next section, and we note that we give only a version sufficient for our purposes and not the sharpest known result Lemma 4 Let h and k be non-negative integers, with h, k hen, for any σ >, h k it ζσ + it dt = ζσ h, k hk + O σ+ɛ minh, k σ + t π σζ σ h, k hk σ dt Assuming Lemma 4, we now complete the proof of Proposition 4 In view of 8 we must show that 9 h,k µhµkahak hk σ 0 h k it ζσ0 + it, and to do this we appeal to Lemma 4 he error terms are easily seen to be o, and we now focus on the main terms arising from Lemma 4, beginning with the first main term his contributes 0 ζσ 0 h,k µhµkahak h, k σ 0 hk σ 0 Write h = h h where h is composed only of primes below Y, and h of primes between Y and X, and then ah = a h a h in the notation of section 3 Writing similarly ak = a k a k, we see that the quantity in 0 factors as µh µk a h a k ζσ 0 h h k σ, k σ µh 0 µk a h a k h 0 h k σ, k σ 0 0 h,k h,k

8 MAKSYM RADZIWI L L AND K SOUNDARARAJAN Consider the first factor in If we ignore the condition that h and k must have at most 00 log log prime factors, then the resulting sum is simply µh µk h k h, k σ σ 0 = 0 p σ 0 p Y h,k p h k = p Y In approximating the first factor by the product above, we incur an error term which is at most by symmetry we may suppose that h has many prime factors µh µk h h k σ, k σ 0 0 h,k p h k = p Y Ωh >00 log log 00 log log e log 00 p Y h,k p h k = p Y + + e p µh µk h h k σ, k σ 0 e Ωh 0 log 90 Similarly one obtains that the second factor in is + Olog log 90 p σ 0 Y <p X Using these in, we obtain that the first main term is ζσ 0 = p σ 0 p σ 0 p X p>x since σ 0 = olog / log X In the same way we see that the second main term arising from Lemma 4 is t σ0dt µhµkahak ζ σ 0 h, k σ 0 π hk h,k t π σ0dt ζ σ 0 p X his completes our proof of 9, and hence of Proposition 4 6 Proof of Lemma 4 p + = o p σ 0 Put Gs = π s/ ss Γs/, so that ξs = Gsζs = ξ s is the completed zeta function Define for any given s C Is = Is = dz z ξz + sξz + se πi z, c

SELBERG S CENRAL LIMI HEOREM FOR log ζ + it 9 where the integral is over the line from c i to c + i for any c > 0 By moving the line of integration to the left, and using the functional equation ξz + sξz + s = ξ z + sξ z + s we obtain that ζs = Is + I s Gs From now on suppose that s = σ + it with t, and σ If z is a complex number with real part c = σ + / log, then an application of Stirling s formula gives herefore, we see that Is Gs = πi Gz + sgz + s Gs = σ+/ log t π z + O z t zζz + sζz + se z dz π z + O σ+ɛ Since we are in the region of absolute convergence of ζz + s and ζz + s, we obtain h it Is k Gs dt = e z hm it t zdt dz πi z mn z+σ kn π 3 σ+/ log + O σ+ɛ m,n= In the integral in 3, we distinguish the diagonal terms hm = kn from the off-diagonal terms hm kn he diagonal terms hm = kn may be parametrized as m = Nk/h, k and n = Nh/h, k, and therefore these terms contribute 4 πi σ+/ log m,n= hm kn e z h, k z+σ z ζz + σ hk t zdt dz π As for the off-diagonal terms, the inner integral over t may be bounded by σ min, / loghm/kn, and therefore these contribute 5 σ mn min, minh, k σ+ɛ +/ log loghm/kn he final estimate above follows by first discarding terms with hm/kn > or < /, and for the remaining terms assume that k h noting that for a given m the sum over values n may be bounded by k ɛ here it may be useful to distinguish the cases hm > and hm < From 3, 4 and 5, we conclude that 6 h k it Is Gs dt = πi σ+/ log + Ominh, k σ+ɛ e z h, k z+σ z ζz + σ hk t zdt dz π

0 MAKSYM RADZIWI L L AND K SOUNDARARAJAN A similar argument gives h it I s k Gs dt = O σ+ɛ minh, k + e z h, k z+ σ t z+ σdt 7 πi z ζz + σ dz hk π σ+/ log With a suitable change of variables, we can combine the main terms in 6 and 7 as h, k z t z σdt e z σ ζz πi hk π z σ + ez +σ dz, z + σ +/ log and moving the line of integration to the left we obtain the main term of the lemma as the residues of the poles at z = σ and z = σ note that the potential pole at z = / from ζz is canceled by a zero of e z σ /z σ + e z +σ /z + σ there his completes our proof of Lemma 4 7 Discussion In common with Selberg s proof of heorem our proof relies on the Gaussian distribution of short sums over primes, as in Proposition In contrast with Selberg s proof, we do not need to invoke zero density estimates for ζs, the easier mean-value theorem in Proposition 4 provides for us a sufficient substitute Selberg s original proof also used an intricate argument expressing log ζs in terms of primes and zeros; an elegant alternative approach was given by Bombieri and Hejhal [], although they too require a strong zero density result near the critical line We should also point out that by just focussing on the central limit theorem, we have not obtained asymptotic formulae for the moments of log ζ + it which Selberg established In Selberg s approach, it was easier to handle Im log ζ +it, and the case of log ζ +it entailed additional technicalities In contrast, our method works well for log ζ + it but requires substantial modifications to handle Im log ζ + it he reason is that Proposition 4 guarantees that typically ζσ 0 + it Mσ 0 + it, but it could be that Im log ζ + it and Im log Mσ 0 + it are not typically close but differ by a substantial integer multiple of π In this respect our argument shares some similarities with Laurinčikas s proof of Selberg s central limit theorem [6], which relies on bounding small moments of ζ + it using Heath-Brown s work on fractional moments [4] In particular, Laurinčikas s argument also breaks down for the imaginary part of log ζ + it We can quantify the argument given here, providing a rate of convergence to the limiting distribution With more effort in particular taking higher moments in Lemma to obtain better bounds on the exceptional set there we can recover previous results in this direction, but have not been able to obtain anything stronger We also remark that the argument also gives the joint distribution of log ζ + it and log ζ + it + iα for any fixed non-zero α R and shows that these are distributed like independent Gaussians One can allow for more than one shift, and also keep track of the uniformity in α

SELBERG S CENRAL LIMI HEOREM FOR log ζ + it Our proof of Proposition 3 in Section 4 involved splitting the mollifier Ms into two factors, or equivalently of the prime sum Ps into two pieces We would have liked to get away with just one prime sum, but this barely fails In order to use Proposition successfully, we are forced to take W = o log log o mollify successfully on the + W log line see Proposition 4 we need to work with primes going up to roughly W If W = o log log then this length is A/ log log for a large parameter A, and if we try to expand expps into a series as in Section 4 we will be forced to take more than log log terms in the exponential series his leads to Dirichlet polynomials that are just a little too long We resolve this see Section 4 by splitting P into two terms, exploiting the fact that the longer sum P has a significantly smaller variance Propositions,, and 3 in our argument are quite general and analogs may be established for higher degree L-functions in the t-aspect An analog of Proposition 4 however can at present only be established for L-functions of degree relying here upon information on the shifted convolution problem, and unknown for degrees 3 or higher However, some hybrid results are possible For example, by adapting the techniques in [, 3] we can establish an analog of Proposition 4 for twists of a fixed GL3 L-function by primitive Dirichlet characters with conductor below Q In this way one can show that as χ ranges over all primitive Dirichlet characters with conductor below Q, and t ranges between and, the distribution of log L + it, f χ is approximately normal with mean 0 and variance log log Q; here f is a fixed eigenform on GL3 Keating and Snaith [5] have conjectured that central values of L-functions in families have a log normal distribution with an appropriate mean and variance depending on the family For example, we may consider the family of quadratic Dirichlet L-functions L, χ d where d ranges over fundamental discriminants of size X In this setting, we may carry out the arguments of Propositions, 3 and 4 and conclude that log Lσ 0, χ d has a normal distribution with mean log log X and variance log log X, provided that σ 0 = + W log X where W is any function with W as X and with log W = olog log X However in this situation we do not have an analog of Proposition allowing us to pass from this to the central value; indeed, our knowledge at present does not exclude the possibility that L, χ d = 0 for a positive proportion of discriminants d Finally we remark that the proof presented here was suggested by earlier work of the authors [7], where general one sided central limit theorems towards the Keating-Snaith conjectures are established References [] E Bombieri and D A Hejhal On the distribution of zeros of linear combinations of Euler products Duke Math J, 803:8 86, 995 [] J B Conrey, H Iwaniec, and K Soundararajan he sixth power moment of Dirichlet L-functions Geom Funct Anal, 5:57 88, 0 [3] J Brian Conrey, Henryk Iwaniec, and Kannan Soundararajan Critical zeros of Dirichlet L-functions J Reine Angew Math, 68:75 98, 03

MAKSYM RADZIWI L L AND K SOUNDARARAJAN [4] D R Heath-Brown Fractional moments of the Riemann zeta function J London Math Soc, 4:65 78, 98 [5] J P Keating and N C Snaith Random matrix theory and L-functions at s = / Comm Math Phys, 4:9 0, 000 [6] A Laurinchikas A limit theorem for the Riemann zeta-function on the critical line II Litovsk Mat Sb, 73:489 500, 987 [7] Maksym Radziwi l l and K Soundararajan Moments and distribution of central L-values of quadratic twists of elliptic curves Invent Math, pages 40, 05 [8] Atle Selberg Contributions to the theory of the Riemann zeta-function Arch Math Naturvid, 485:89 55, 946 [9] Atle Selberg Old and new conjectures and results about a class of Dirichlet series In Proceedings of the Amalfi Conference on Analytic Number heory Maiori, 989, pages 367 385 Univ Salerno, Salerno, 99 Department of Mathematics, Rutgers University, 0 Frelinghuysen Rd, Piscataway, NJ 08854-809 E-mail address: maksymradziwill@gmailcom Department of Mathematics, Stanford University, 450 Serra Mall, Bldg 380, Stanford, CA 94305-5 E-mail address: ksound@mathstanfordedu