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Arithmetic Statistics in Function Fields
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The Divisor function The divisor function π π counts the number of divisors of an integer π. π π = π|π 1 = π,π ππ=π 1 Dirichlet divisor problem: Determine the asymptotic behaviour as π₯ββ of the sum π· π₯ = πβ€π₯ π(π) π· π₯ = πβ€π₯ π(π) = πβ€π₯ π|π 1 = π,π ππβ€π₯ 1 This is a count of lattice points under the hyperbola ππ=π₯ Give an example
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Counting method Make use of the symmetry of the region about the line π=π β count twice the number of lattice points under the hyperbola ππ=π₯ and under the line π=π. * On the diagonal line π=π there are π₯ lattice points. * On horizontal lines we have π₯ π βπ lattice points.
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β π« π =π π₯π¨π π + 2πΎβ1 π₯+πΆ( π₯ ) Therefore
πβ€π₯ π(π) =2 πβ€ π₯ ( π₯ π βπ) + π₯ Use π¦ =π¦+π(1) and the Euler summation formula πβ€π₯ 1 π = log π₯ +πΎ+π 1 π₯ β π« π =π π₯π¨π π + 2πΎβ1 π₯+πΆ( π₯ )
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Dirichlet βs divisor problem: Ξ π₯ β πβ€π₯ π π β π₯ log π₯ + 2πΎβ1 π₯ Dirichlet:Ξ π₯ βͺ π₯ 1 2 Voronoi (1903):Ξ π₯ βͺ π₯ 1 3 +π Huxley (2003):Ξ π₯ βͺ π₯ π Problem (Divisor function in short intervals): The limiting distribution of π₯<πβ€π₯+π»(π₯) π(π) when πββ. The trivial range: For π» π = π π and π> , as πββ π₯<πβ€π₯+π»(π₯) π(π) ~π» π log π
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The Divisor function in short intervals
Define Ξ π₯;π» =Ξ π₯+π» βΞ π₯ Ivic (2009): For π π <π»< 1 2 π 1 2 βπ 1 π π 2π Ξ π₯;π» 2 ππ₯~π» π 3 ( log π₯β2 log π» ) with π 3 a certain cubic polynomial.
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The generalized Divisor function
The k-th divisor function π
π π : π π π β#{ π 1 ,β¦, π π : π= π 1 β― π π , π 1 ,β¦, π π β₯1} The classical divisor function π π = π|π 1 being π π = π 2 π . Example: for a prime number π, π π π π =#{ π 1 ,β¦, π π : π= π 1 +β― +π π , π 1 ,β¦, π π β₯0} β π π π π = π+πβ1 πβ1 Generalization of Dirichlet βs divisor problem: Ξ k π₯ β πβ€π₯ π π π βπ₯ π
πβ1 ( log π₯) , where π
πβ1 is a certain polynomial of degree πβ1. I will start by definingβ¦ Delta 2 is the error term in evaluating the sum of the classical divisor function up to x by x times a acertain linear poly in log x
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The generalized Divisor function in short intervals
Lester (2015): π»= π πΏ , 1β 1 πβ1 <πΏ<1β 1 π then (assuming πΏπππππ π π) 1 π π 2π Ξ π π₯;π» 2 ππ₯ ~ π π π π πΏ π»β
log π π 2 β1 Conjecture J.P.Keating, B.Rodgers, ER-G and Z.Rudnick (2015) π»= π πΏ , π<πΏ<1β 1 π , then Where π π πΏ is a piecewise polynomial function in πΏ, of degree π 2 β1
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Function fields π π a finite field (π is a power of an odd prime) π
πͺ π the ring of polynomials with coefficients in π π . π· π β πβ πΉ π π : deg π=π the set of polynomials of degree π π΄ π β πβ πΉ π π : deg π=π , π πππππ be the subset of monic polynomials. The norm of πβπΉ π [π] is defined by π := πͺ πππ π . The k-th divisor function π π π π π π β#{ π 1 ,β¦, π π : π= π 1 β― π π , π 1 ,β¦, π π πππππ} Compare with number fields: πβ π π log πβπ ββ€πβββ π π
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Divisors in function fields
The analogue of Dirichlet divisor problem: π·(π)β πβ π π π π π Over function fields- an easy computation: π π’ = π πππππ π’ deg π = π=0 β π π π’ π = 1 1βππ’ The k-th power of the zeta function is the generating function of πβ π π π π π . π(π’) π = π πππππ π π π π’ πππ π = π=0 β πβ π π π π π π’ π = 1 (1βππ’) π Expand + compare the coefficient of π’ π β πβ π π π π π = π π π+πβ1 πβ1 .
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Short intervals in πΉ π [π]
For π΄β π π and β<π, an interval around π΄ of length β is πΌ π΄;β β π: πβπ΄ β€ π β =π΄+ π β€β Note that Hβ#πΌ π΄;β = π β+1 . The sum over βshort intervalβ π π π π΄;β := πβπΌ π΄;β π π (π) The mean value is π π π β;β = 1 π π π΄β π π π π π π΄;β Define Ξ π π΄;β β π π π π΄;β β π π π β;β Our goal: to study the variance of π π π (in the limit of a large field size) var π π π β;β = 1 π π π΄β π π | Ξ π π΄;β β 2
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Short intervals as arithmetic progressions
There is a bijection between Intervals and arithmetic progressions: πΌ π΄;β β πβ‘ π π π΄ 1 π πππ π πββ When π΄= π β+1 π΅ , deg π΅=πβββ1 This covers all intervals since π π = βͺ π΅β π πβββ1 πΌ( π β+1 π΅;β)
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Variance in short intervals
Theorem (J.P.Keating, B.Rodgers, E.R-G and Z.Rudnick 2015) Let nβ₯5, and ββ€minβ‘( 1β 1 π πβ1 , πβ5) , then as πββ 1 π π π΄β π π Ξ π π΄;β 2 ~ π β+1 I k (n;nβhβ2) Where I π π;π β π π π 1 +β―+ π π =π 0β€ π 1 ,β¦, π π β€π π π π 1 π β―π π π π π 2 ππ and π π π π are the secular coefficients: det πΌ+π₯π = π=0 π π π π π π₯ π Corollary: If nβ₯8 and β< π 2 β1 , then as πββ 1 π π π΄β π π Ξ 2 π΄;β 2 ~ π β+1 (πβ2β+5)(πβ2β+6)(πβ2β+7) 6
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Comparison between number field and function field results (short intervals)
π π π log π» β+1 π» π β+1 For π π <π»< 1 2 π 1/2 1 π π 2π Ξ 2 π₯;β 2 ππ₯~ π» π 3 ( log π₯β2 log π» ) If β<π/2 then 1 π π π΄β π π Ξ 2 π΄;β 2 ~ π β+1 Poly 3 (nβ2h)
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Ingredients of the proof
Orthogonality relation for Dirichlet characters mod Q: 1 Ξ¦(π) π πππ π π π΄ π π = π=π΄ πππ π ππ‘βπππ€ππ π 1 Ξ¦(π) π΄ πππ π π 1 π΄ π 2 π΄ = π 1 = π ππ‘βπππ€ππ π Even characters π ππ =π(π) for all πβ πΉ π Γ πΏ π’,π := π,π =1 1βπ π π deg π β1 = π=1 deg π β1 (1β πΌ π π π’) Riemann Hypothesis (proved by Weil) : |πΌ π π |= π 1 2 Spectral interpretation (for primitive even characters): πΏ π’,π =(1βπ’) det (πΌβπ’ π Ξ π ) , Ξ π =ππππ π π π 1 ,β¦, π π π deg π β1
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The main ingredient Theorem (Nick Katz 2013)
The unitarized Frobenii Ξ π when π is an even primitive character mod π π+1 become equidistributed in PU(m-1) as πβ β. 1 π ππ£ β π πββ π πππ π πββ πβ π 0 π ππ ππ£ππ πππ ππππππ‘ππ£π πΉ( Ξ π ) ~ ππ(πβββ2) πΉ π ππ
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Sketch of the proof Write the L-functions in terms of unitary matrices
πβ π π π π π β evaluated in terms of π π’ πβπΌ π΄;β π π (π) β restrict to an interval using orthogonality relations for Dirichlet character β evaluate in terms of the associated Dirichlet L- functions Write the L-functions in terms of unitary matrices πΏ π’,π = 1βπ’ det πΌβπ’ π Ξ π The variance ~ 1 π ππ£ β π πββ π πππ π πββ πβ π 0 π ππ ππ£ππ πππ ππππππ‘ππ£π π 1 +β―+ π π =n 0β€ π 1 ,β¦, π π β€nβhβ2 π π π 1 Ξ π β―π π π π Ξ π 2
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Sketch of the proof Apply Katz equidistribution result
1 π ππ£ β π πββ π πππ π πββ πβ π 0 π ππ ππ£ππ πππ ππππππ‘ππ£π π 1 +β―+ π π =n 0β€ π 1 ,β¦, π π β€nβhβ2 π π π 1 Ξ π β―π π π π Ξ π 2 ~ π nβhβ π 1 +β―+ π π =n 0β€ π 1 ,β¦, π π β€nβhβ2 π π π 1 π β―π π π π π 2 ππ
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Matrix Integral What do we know about I π π;π β π π π 1 +β―+ π π =π 0β€ π 1 ,β¦, π π β€π π π π 1 π β―π π π π π 2 ππ ? For π>ππ , I π π;π =0. I π π;π = π+ π 2 β1 π 2 β1 , π<π Functional equation I π π;π = I π ππβπ;π . β I π π;π = ππβπ+ π 2 β1 π 2 β1 , πβ1 π<π<ππ
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Theorem (J. P. Keating, B. Rodgers, ER-G and Z
Theorem (J.P.Keating, B.Rodgers, ER-G and Z.Rudnick) I π π;π is equal to the count of lattice points π₯=( π₯ π π )β (β€) π 2 satisfying each of the relations 0β€ π₯ π π β€π for all 1β€π,πβ€π. π₯ 1 π + π₯ 2 πβ1 +β―+ π₯ π 1 =ππβπ. π₯β π΄ π , where π΄ π is the collection of πΓπ matrices whose entries satisfy the following system of equalities,
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Theorem (J.P.Keating, B.Rodgers, E.RβG and Z.Rudnick 2015)
Let πβπ/π. Then for πβ 0,π , I π π;π = πΎ π π π π 2 β1 +π( π π 2 β2 ) With πΎ π π = 1 π!πΊ 1+π ,1 π πΏ π π€ 1 +β―+ π€ π π<π π€ π β π€ π d k w Here πΊ is the Barnes G-function, so that for positive integers k, πΊ 1+π =1!β2!β3!ββ― πβ1 ! πΎ π π is a piecewise polynomial which changes when ever π reaches an integer.
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