Statistics of the Zeta zeros: Mesoscopic and macroscopic phenomena

Transcription

1 Statistics of the Zeta zeros: Mesoscopic and macroscopic phenomena Department of Mathematics UCLA Fall 2012

2 he Riemann Zeta function Non-trivial zeros: those with real part in (0, 1). First few: i14.13, i21.02, i We assume the Riemann Hypothesis in what follows: all nontrivial zeros have the form 1 + iγ, for γ R. 2 Around height, zeros have density roughly log /. More precisely: heorem (Riemann - von Mangoldt) Figure : ζ(s). Hue is argument, brightness is modulus. Made with Jan Homann s ComplexGraph Mathematica code. N( ) = #{γ (0, ), ζ( 1 + iγ) = 0} 2 = log( ) + O(log )

3 1 and 2-level density 1-level density: For large random s [, 2 ] and dx small, ( P one γ [ ] ) dx s, s + dx log Since s is random, for fixed x, we can translate by x and have the log same statement P ( one γ s + log [x, x + dx]) dx 2-level density (pair correlation): Does the presence of one zero in a location affect the likelihood of other zeros being nearby? Conjecture: P ( one γ s + [x, x + dx], one log γ s+ [y, y + dy]) log ( 1 ( sin π(x y) π(x y) ) 2 ) dx dy sin π(x y) 1 ( ) 2 0 when x y, so very low likelihood of two zeros π(x y) being much nearer than average. Compare probability dx dy for poisson process.

4 A histogram of the pair correlation conjecture Figure : A histogram of log (γ γ ) for the first zeros, in intervals of size.05, compared to the appropriately scaled prediction 1 ( sin πx ) 2. πx

5 k-level density k-level density: Conjecture: P ( one γ 1 s + [x1, x1 + dx1], one γ2 s + [x2, x2 + dx2], log log..., one γ k s + [x log k, x k + dx k ] ) 1 S(x 1 x 2) S(x 1 x k ) S(x 2 x 1) 1 S(x 2 x k ) det dx1 dx2 dx k S(x k x 1) S(x k x 2) 1 where S(x) = sin πx πx. his is the same probability as P ( log log (γ1 s) [x1, x1 + dx1],..., (γ k s) [x k, x k + dx k ] )

6 A more formal statement and a comparison with the unitary group More formally: Conjecture (GUE) For fixed k and fixed η (Schwartz, say) 1 2 η ( log (γ1 s),..., log γ 1,...,γ k distinct (γ k s) ) ds R k η(x) det k k ( S(xi x j ) ) d k x his is known to be the case for unitary matrices. Let U(N) be the Haar-probability space of N N random unitary matrices g, and label g s eigenvalues {e iθ 1,..., e iθ N } with θ j [ 1/2, 1/2) for all j. heorem (Dyson-Weyl) For fixed k and η, U(N) i 1,...,i k distinct η(nθ i1,..., Nθ ik ) dg R k η(x) det k k ( S(xi x j ) ) d k x

7 More on GUE he GUE conjecture says that for a fixed interval J, the random variables ( ) # J { log (γ s)} s [, 2 ] and ) # J ({Nθ j } g U(N) tend in distribution as, N to the same random variable. For certain band-limited test functions, the GUE conjecture is known (on RH) to be true. heorem (Mongtomery, Hejhal, Rudnick-Sarnak) For fixed k and η with supp ˆη {y : y y k < 2} 1 2 η ( log (γ1 s),..., log (γ k s) ) ds η(x) det R k k k γ 1,...,γ k distinct ( S(xi x j ) ) d k x

8 he statistics we have been talking about, of zeros at height separated by O(1/ log ), are microscopic statistics. If we limit our knowledge to what I have so far talked about, we suffer two restrictions: (1) We can t say anything rigorous about the distribution of zeros when we count with test functions that are too oscillatory (too narrowly concentrated, that is, by the uncertainty principle) at the microscopic level. (2) We can t say anything about the distribution of zeros when counted by test functions that are not essentially supported at the microscopic level. We can t say anything, for instance, about the effect the position of a zero will have on the statistics of a zero a distance of 1 away. Philosophy: (1) is a serious obstruction to our knowledge of zeta statistics, (2) is not. Any question that can be asked about zeta zeros, provided answering it does not require counting with functions that are too oscillatory in the microscopic regime, can be rigorous answered.

9 Mesoscopic collections of zeros heorem (Fujii) Let n( ) be a function as but so that n( ) = o(log ), and let s be random and uniformly distributed on [, 2 ]. Let J = [ n( )/2, n( )/2], and define = # J ({ log (γ s)}) = N(s + n( ) log 2 ) N(s log n( ) 2 ) we have E = n( ) + o(1) Var := E( E ) 2 1 log n( ) π2 and in distribution E Var N(0, 1) as. hat n( ) = o(log ) is important! Collections of zeros in this range are known as mesoscopic.

10 Mesoscopic collections of eigenvalues heorem (Costin-Lebowitz) Let n(m) be a function as M, but so that n(m) = o(m). Let I M = [ n(m)/2, n(m)/2]. Consider the counting function M = # IM ({Mθ i }). hen and in distribution E U(M) M = n(m) Var U(M) M 1 log n(m) π2 M E M Var M N(0, 1) Here n(m) = o(m) is a natural boundary. Heuristic conjecture of Berry (1989): he zeros look like eigenvalues not only microscopically, but also mesoscopically.

11 Macroscopic collections of zeros heorem (Backlund) N( ) = 1 π arg Γ( 1 + i ) log π S( ) 4 2 S( ) := 1 π arg ζ( i ) S( ) is small and oscillatory, and may be thought of as an error term. heorem (Fujii) Let = S(s + log n( ) n( ) ) S(s ) 2 log 2 and n( ) we have E = o(1) { 1 log n( ) if n( ) = o(log ) Var π 2 1 log log if log n( ) = o( ). π 2 We still have /Var N(0, 1). his phase change does not correspond to phenomena in random matrix theory. What causes it?

12 Macroscopic pair correlation 1 Figure : A histogram of γ γ for the first 5000 zeros, in intervals of size.1.

13 Macroscopic pair correlation 2 Figure : A histogram of γ γ for the first 7500 zeros, in intervals of size.1.

14 Macroscopic pair correlation 3 Figure : A histogram of γ γ for the first zeros, in intervals of size.1.

15 he Bogomolny - Keating prediction Figure : A histogram of γ γ for the first zeros, in intervals of size.1, compared with the prediction of Bogomolny and Keating.

16 Montgomery s strong pair correlation heorem (Montgomery) For fixed ɛ > 0 and w(u) = 4/(4 + u 2 ), 1 log 0 γ,γ R ( ) e α log (γ γ ) w(γ γ ) = 1 (1 α ) + + o(1) + (1 + o(1)) 2α log = (1 + o(1)) e(αx)w ( ) [ ( x sin πx δ(x) + 1 log πx uniformly for α 1 ɛ. For fixed M, this is conjectured to be true uniformly for α M. Corollary For a fixed interval J, #{γ, γ (0, ) : γ γ J} ( log ) 2 J ) 2 ] dx

17 Idea of proof: Use the Fourier pair g(ν) = 1 2 ( ) sin πν/2 2, πν/2 ĝ(α) = (1 2α )+ Integrate in α with respect to e( α log r)ĝ(α): 1 ) log 0 γ,γ = (1 + o(1)) R uniformly, for any r. ( g log (γ γ r) w(γ γ ) g(x log r)w( x log ) [ ( sin πx ) 2 ] δ(x) + 1 dx πx log B log g(x u) du = log A B A g(x log r) dr = 1 [ log log A, } {{ B](x) } width log + ɛ(x) }{{} width O(1) Implies 1 log 0 γ,γ 1 [A,B] (x)(γ γ )w(γ γ ) = (1 + o(1)) log 1 [A,B] w(x) dx R

18 Macroscopic pair correlation: An exact formulation heorem (R.) For fixed ɛ > 0 and fixed ω with a smooth and compactly supported Fourier transform, 1 0<γ γ ( 1 ) = O δ δ + ω(γ γ )e ( α log (γ γ ) ) R ω(u)e ( α log u)[ 1 0 for any δ < ɛ/2, uniformly for α < 1 ɛ. where ( log(t/) ) 2 + Qt(u) dt] du Q t (u) := 1 ( ( ζ ) (1 ( ζ ) (1 + iu) B(iu) + iu) B( iu) 4π 2 ζ ζ + ( t ) ) iuζ(1 ( t ) iuζ(1 iu)ζ(1 + iu)a(iu) + + iu)ζ(1 iu)a( iu), defined by continuity at u = 0, and A(s) := p (1 1 p 1+s )(1 p p 1+s ) ( 1 1 ) 2 = (1 (1 p s ) 2 ) p p (p 1) 2 = 1 + O(s 2 ), and B(s) := p log 2 p (p 1+s 1) 2.

19 Macroscopic pair correlation: A reformulation for α < 1 heorem For fixed ɛ > 0 and fixed ω with a smooth and compactly supported Fourier transform, 1 0<γ γ ( 1 ) = O δ δ + ω(γ γ )e ( α log (γ γ ) ) R ω(u)e ( α log u)[ 1 0 for any δ < ɛ/2, uniformly for α < 1 ɛ. where Q t(u) := 1 ( Λ 2 (n) 4π 2 n 1+iu defined by continuity at u = 0. + Λ 2 (n) e( + n1 iu ( log(t/) ) 2 + Q t(u) dt] du log(t/) u) + e( log(t/) u 2 u) ),

20 Ideas in proof: Explicit formulas Ex: Λ(n) 1 γ n 1/2+iγ + lower order Must use an explicit formula which is exact, or extremely close to being exact Must use an explicit formula which takes into account the functional equation Recall N( ) = 1 arg Γ( 1 + i ) log π S( ). π 4 2 : dn(ξ) = γ δ γ(ξ)dξ = Ω(ξ) + ds(ξ) where Ω(ξ)/ is regular and log(ξ)/ Pair correlation Knowing about dn(ξ 1 + t)dn(ξ 2 + t) on average Knowing about ds(ξ 1 + t)ds(ξ 2 + t) on average

21 Ideas in proof: Explicit formulas heorem (Riemann-Guinand-Weil) For nice g ( ξ ) ĝ ds(ξ) = R Here ψ(x) = n x Λ(n). [g(x) + g( x)]e x/2 d ( e x ψ(e x ) ) his is a Fourier duality between the error term of the prime counting function, and the error term of the zero counting function.

22 Ideas in proof: Smooth averages Replace 1 2 ds = R 1 [1,2] (s/ ) ds with R σ(s/ ) for ˆσ compactly supported, and σ of mass 1 (so ˆσ(0) = 1). ds We want to know about: σ(s/ ) A = e ( α log R R 2 (ξ 1 s) α log (ξ 2 s) ) ( ) ( ) r ξ1 s r ξ2 s ds(ξ 1 ) ds(ξ 2 ) ds = ) ˆσ( (ε 1x 1 + ε 2 x 2 ) ˆr(ε 1 x 1 α log )ˆr(ε 2 x 2 + α log ) ε { 1,1} 2 e (x 1+x 2 )/2 d ( e x 1 ψ(e x 1 ) ) d ( e x 2 ψ(e x 2 ) ) his is really four integrals, over different measures: d ( e x 1 ψ(e x 1 ) ) d ( e x 2 ψ(e x 2 ) ) = d(e x 1 )d(e x 2 ) d(e x 1 )dψ(e x 2 ) dψ(e x 1 )d(e x 2 ) + dψ(e x 1 )dψ(e x 2 )

23 Ideas in proof ( he term ˆσ ( 1 ) A = O + 1 α ( 1 ) = O + 1 α n (ε1x1 + ε2x2) forces ε1x1 + ε2x2 = O(1/ ): ε { 1,1} 2 ( ˆσ )ˆr(ε (ε1x1 + ε2x2) 1x 1 α log ) ˆr(ε 2x 2 + α log )e (x 1+x 2 )/2 dψ(e x 1 )dψ(e x 2 ) Λ 2 (n) [ˆr( log n α log )ˆr(log n α log ) n + ˆr(log n α log )ˆr( log n α log ) ] his can be untangled with some complex analysis to give the form we re after. Some additional work is needed to untangle ds(ξ 1 + t)ds(ξ 2 + t).

24 Another application of this philosophy: An analogue of Szegő s theorem heorem (R., Bourgade-Kuan) Let n( ), but n( ) = o(log ). For a fixed η define η, = γ η ( log n( ) (γ s)), For all η with compact support and bounded variation when x ˆη(x) 2 dx diverges, and nearly all such η when the integral converges, we have E η, = n( ) η(ξ)dξ + o(1), R n( ) Var η, x ˆη(x) 2 dx n( ) and in distribution η, E η, Var η, N(0, 1) as.

25 A more ambitious application: Moments 1 ζ( it) 2k dt = U(n) U(n) det(1 e iθ g) 2k dg dθ det(1 g) 2k dg Macroscopic information in k-point correlation functions, with microscopic band-limitations: Fourier support in {y : y y k 2} Using only knowledge of the k-point correlation functions η(e iθ j 1,..., e iθ jk ) dg j 1,...,j k distinct for η : k R, supp ˆη {r Z k : r r k 2n}

26 A more ambitious application: Moments But with this information, we can deduce n det(1 g) 2k dg = (2 e iθ j e iθ j ) k dg j=1 for k = 1, 2 but no higher. Classical knowledge about the zeta function, having nothing to do with random matrix theory, let s us deduce the asymptotics of for k = 1, 2, but no higher. 1 0 ζ( it) 2k dt Question: Is there a way to understand these computations in terms of macroscopic k-point correlation functions? What about the conjectured asymptotics of higher moments? (Keating-Snaith conjecture)

27 hanks: hanks!