1. Arithmetic Statistics in Function Fields

2. The Divisor function
The divisor function 𝑑 𝑛 counts the number of divisors of an integer 𝑛. 𝑑 𝑛 = 𝑚|𝑛 1 = 𝑚,𝑞 𝑞𝑚=𝑛 1 Dirichlet divisor problem: Determine the asymptotic behaviour as 𝑥→∞ of the sum 𝐷 𝑥 = 𝑛≤𝑥 𝑑(𝑛) 𝐷 𝑥 = 𝑛≤𝑥 𝑑(𝑛) = 𝑛≤𝑥 𝑚|𝑛 1 = 𝑚,𝑞 𝑞𝑚≤𝑥 1 This is a count of lattice points under the hyperbola 𝑞𝑚=𝑥
Give an example

3. Counting method
Make use of the symmetry of the region about the line 𝑞=𝑚
⇒ count twice the number of lattice points under the hyperbola 𝑞𝑚=𝑥 and under the line 𝑞=𝑚.
* On the diagonal line 𝑞=𝑚 there are 𝑥 lattice points.
* On horizontal lines we have 𝑥 𝑚 −𝑚 lattice points.

4. ⇒ 𝑫 𝒙 =𝒙 𝐥𝐨𝐠 𝒙 + 2𝛾−1 𝑥+𝑶( 𝑥 ) Therefore
𝑛≤𝑥 𝑑(𝑛) =2 𝑚≤ 𝑥 ( 𝑥 𝑚 −𝑚) + 𝑥
Use 𝑦 =𝑦+𝑂(1) and the Euler summation formula
𝑚≤𝑥 1 𝑚 = log 𝑥 +𝛾+𝑂 1 𝑥
⇒ 𝑫 𝒙 =𝒙 𝐥𝐨𝐠 𝒙 + 2𝛾−1 𝑥+𝑶( 𝑥 )

5. Dirichlet ‘s divisor problem: Δ 𝑥 ≔ 𝑛≤𝑥 𝑑 𝑛 − 𝑥 log 𝑥 + 2𝛾−1 𝑥 Dirichlet:Δ 𝑥 ≪ 𝑥 1 2 Voronoi (1903):Δ 𝑥 ≪ 𝑥 1 3 +𝜖 Huxley (2003):Δ 𝑥 ≪ 𝑥 𝜖 Problem (Divisor function in short intervals): The limiting distribution of 𝑥<𝑛≤𝑥+𝐻(𝑥) 𝑑(𝑛) when 𝑋→∞. The trivial range: For 𝐻 𝑋 = 𝑋 𝜃 and 𝜃> , as 𝑋→∞ 𝑥<𝑛≤𝑥+𝐻(𝑥) 𝑑(𝑛) ~𝐻 𝑋 log 𝑋

6. The Divisor function in short intervals
Define Δ 𝑥;𝐻 =Δ 𝑥+𝐻 −Δ 𝑥 Ivic (2009): For 𝑋 𝜀 <𝐻< 1 2 𝑋 1 2 −𝜀 1 𝑋 𝑋 2𝑋 Δ 𝑥;𝐻 2 𝑑𝑥~𝐻 𝑃 3 ( log 𝑥−2 log 𝐻 ) with 𝑃 3 a certain cubic polynomial.

7. The generalized Divisor function
The k-th divisor function 𝒅 𝒌 𝒏 : 𝑑 𝑘 𝑛 ≔#{ 𝑎 1 ,…, 𝑎 𝑘 : 𝑛= 𝑎 1 ⋯ 𝑎 𝑘 , 𝑎 1 ,…, 𝑎 𝑘 ≥1} The classical divisor function 𝑑 𝑛 = 𝑚|𝑛 1 being 𝑑 𝑛 = 𝑑 2 𝑛 . Example: for a prime number 𝑝, 𝑑 𝑘 𝑝 𝑚 =#{ 𝑏 1 ,…, 𝑏 𝑘 : 𝑚= 𝑏 1 +⋯ +𝑏 𝑘 , 𝑏 1 ,…, 𝑏 𝑘 ≥0} ⇒ 𝑑 𝑘 𝑝 𝑚 = 𝑚+𝑘−1 𝑘−1 Generalization of Dirichlet ‘s divisor problem: Δ k 𝑥 ≔ 𝑛≤𝑥 𝑑 𝑘 𝑛 −𝑥 𝑅 𝑘−1 ( log 𝑥) , where 𝑅 𝑘−1 is a certain polynomial of degree 𝑘−1.
I will start by defining…
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

8. The generalized Divisor function in short intervals
Lester (2015): 𝐻= 𝑋 𝛿 , 1− 1 𝑘−1 <𝛿<1− 1 𝑘 then (assuming 𝐿𝑖𝑛𝑑𝑒𝑙 𝑜 𝑓) 1 𝑋 𝑋 2𝑋 Δ 𝑘 𝑥;𝐻 2 𝑑𝑥 ~ 𝑎 𝑘 𝑃 𝑘 𝛿 𝐻⋅ log 𝑋 𝑘 2 −1 Conjecture J.P.Keating, B.Rodgers, ER-G and Z.Rudnick (2015) 𝐻= 𝑋 𝛿 , 𝜖<𝛿<1− 1 𝑘 , then Where 𝑃 𝑘 𝛿 is a piecewise polynomial function in 𝛿, of degree 𝑘 2 −1

9. Function fields
𝑭 𝒒 a finite field (𝑞 is a power of an odd prime) 𝐅 𝐪 𝐗 the ring of polynomials with coefficients in 𝑭 𝒒 . 𝑷 𝒏 ≔ 𝑓∈ 𝐹 𝑞 𝑋 : deg 𝑓=𝑛 the set of polynomials of degree 𝑛 𝑴 𝒏 ≔ 𝑓∈ 𝐹 𝑞 𝑋 : deg 𝑓=𝑛 , 𝑓 𝑚𝑜𝑛𝑖𝑐 be the subset of monic polynomials. The norm of 𝑓∈𝐹 𝑞 [𝑋] is defined by 𝐟 := 𝐪 𝐝𝐞𝐠 𝐟 . The k-th divisor function 𝑑 𝑘 𝑓 𝑑 𝑘 𝑓 ≔#{ 𝑓 1 ,…, 𝑓 𝑘 : 𝑓= 𝑓 1 ⋯ 𝑓 𝑘 , 𝑓 1 ,…, 𝑓 𝑘 𝑚𝑜𝑛𝑖𝑐} Compare with number fields: 𝑋↔ 𝑞 𝑛 log 𝑋↔𝑛 ℎ≤𝑋↔ℎ∈ 𝑀 𝑛

10. Divisors in function fields
The analogue of Dirichlet divisor problem:
𝐷(𝑛)≔ 𝑓∈ 𝑀 𝑛 𝑑 𝑘 𝑓
Over function fields- an easy computation:
𝑍 𝑢 = 𝑓 𝑚𝑜𝑛𝑖𝑐 𝑢 deg 𝑓 = 𝑛=0 ∞ 𝑞 𝑛 𝑢 𝑛 = 1 1−𝑞𝑢
The k-th power of the zeta function is the generating function of 𝑓∈ 𝑀 𝑛 𝑑 𝑘 𝑓 .
𝑍(𝑢) 𝑘 = 𝑓 𝑚𝑜𝑛𝑖𝑐 𝑑 𝑘 𝑓 𝑢 𝑑𝑒𝑔 𝑓 = 𝑛=0 ∞ 𝑓∈ 𝑀 𝑛 𝑑 𝑘 𝑓 𝑢 𝑛 = 1 (1−𝑞𝑢) 𝑘
Expand + compare the coefficient of 𝑢 𝑛
⇒ 𝑓∈ 𝑀 𝑛 𝑑 𝑘 𝑓 = 𝑞 𝑛 𝑛+𝑘−1 𝑘−1 .

11. Short intervals in 𝐹 𝑞 [𝑋]
For 𝐴∈ 𝑀 𝑛 and ℎ<𝑛, an interval around 𝐴 of length ℎ is
𝐼 𝐴;ℎ ≔ 𝑓: 𝑓−𝐴 ≤ 𝑞 ℎ =𝐴+ 𝑃 ≤ℎ
Note that H≔#𝐼 𝐴;ℎ = 𝑞 ℎ+1 .
The sum over “short interval” 𝑁 𝑑 𝑘 𝐴;ℎ := 𝑓∈𝐼 𝐴;ℎ 𝑑 𝑘 (𝑓)
The mean value is
𝑁 𝑑 𝑘 ∎;ℎ = 1 𝑞 𝑛 𝐴∈ 𝑀 𝑛 𝑁 𝑑 𝑘 𝐴;ℎ
Define Δ 𝑘 𝐴;ℎ ≔ 𝑁 𝑑 𝑘 𝐴;ℎ − 𝑁 𝑑 𝑘 ∎;ℎ
Our goal: to study the variance of 𝑁 𝑑 𝑘 (in the limit of a large field size) var 𝑁 𝑑 𝑘 ∎;ℎ = 1 𝑞 𝑛 𝐴∈ 𝑀 𝑛 | Δ 𝑘 𝐴;ℎ ​ 2

12. Short intervals as arithmetic progressions
There is a bijection between Intervals and arithmetic progressions:
𝐼 𝐴;ℎ ↔ 𝑔≡ 𝑋 𝑛 𝐴 1 𝑋 𝑚𝑜𝑑 𝑋 𝑛−ℎ
When 𝐴= 𝑋 ℎ+1 𝐵 , deg 𝐵=𝑛−ℎ−1
This covers all intervals since
𝑀 𝑛 = ∪ 𝐵∈ 𝑀 𝑛−ℎ−1 𝐼( 𝑋 ℎ+1 𝐵;ℎ)

13. Variance in short intervals
Theorem (J.P.Keating, B.Rodgers, E.R-G and Z.Rudnick 2015) Let n≥5, and ℎ≤min⁡( 1− 1 𝑘 𝑛−1 , 𝑛−5) , then as 𝑞→∞ 1 𝑞 𝑛 𝐴∈ 𝑀 𝑛 Δ 𝑘 𝐴;ℎ 2 ~ 𝑞 ℎ+1 I k (n;n−h−2) Where I 𝑘 𝑚;𝑁 ≔ 𝑈 𝑁 𝑗 1 +⋯+ 𝑗 𝑘 =𝑚 0≤ 𝑗 1 ,…, 𝑗 𝑘 ≤𝑁 𝑆 𝑐 𝑗 1 𝑈 ⋯𝑆 𝑐 𝑗 𝑘 𝑈 2 𝑑𝑈 and 𝑆 𝑐 𝑗 𝑈 are the secular coefficients: det 𝐼+𝑥𝑈 = 𝑗=0 𝑁 𝑆 𝑐 𝑗 𝑈 𝑥 𝑗 Corollary: If n≥8 and ℎ< 𝑛 2 −1 , then as 𝑞→∞ 1 𝑞 𝑛 𝐴∈ 𝑀 𝑛 Δ 2 𝐴;ℎ 2 ~ 𝑞 ℎ+1 (𝑛−2ℎ+5)(𝑛−2ℎ+6)(𝑛−2ℎ+7) 6

14. Comparison between number field and function field results (short intervals)𝑋
𝑞 𝑛
log 𝐻
ℎ+1 𝐻
𝑞 ℎ+1
For 𝑋 𝜀 <𝐻< 1 2 𝑋 1/2
1 𝑋 𝑋 2𝑋 Δ 2 𝑥;ℎ 2 𝑑𝑥~
𝐻 𝑃 3 ( log 𝑥−2 log 𝐻 )
If ℎ<𝑛/2 then
1 𝑞 𝑛 𝐴∈ 𝑀 𝑛 Δ 2 𝐴;ℎ 2 ~
𝑞 ℎ+1 Poly 3 (n−2h)

15. Ingredients of the proof
Orthogonality relation for Dirichlet characters mod Q:
1 Φ(𝑄) 𝜒 𝑚𝑜𝑑 𝑄 𝜒 𝐴 𝜒 𝑁 = 𝑁=𝐴 𝑚𝑜𝑑 𝑄 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
1 Φ(𝑄) 𝐴 𝑚𝑜𝑑 𝑄 𝜒 1 𝐴 𝜒 2 𝐴 = 𝜒 1 = 𝜒 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Even characters 𝜒 𝑐𝑓 =𝜒(𝑓) for all 𝑐∈ 𝐹 𝑞 ×
𝐿 𝑢,𝜒 := 𝑃,𝑄 =1 1−𝜒 𝑃 𝑈 deg 𝑃 −1 = 𝑗=1 deg 𝑄 −1 (1− 𝛼 𝑗 𝜒 𝑢)
Riemann Hypothesis (proved by Weil) : |𝛼 𝑗 𝜒 |= 𝑞 1 2
Spectral interpretation (for primitive even characters):
𝐿 𝑢,𝜒 =(1−𝑢) det (𝐼−𝑢 𝑞 Θ 𝜒 ) , Θ 𝜒 =𝑑𝑖𝑎𝑔 𝑒 𝑖 𝜃 1 ,…, 𝑒 𝑖 𝜃 deg 𝑄 −1

16. The main ingredient Theorem (Nick Katz 2013)
The unitarized Frobenii Θ 𝜒 when 𝜒 is an even primitive
character mod 𝑇 𝑚+1 become equidistributed in PU(m-1) as
𝑞→ ∞.
1 𝜙 𝑒𝑣 ∗ 𝑇 𝑛−ℎ 𝜒 𝑚𝑜𝑑 𝑇 𝑛−ℎ 𝜒≠ 𝜒 0 𝜒 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝐹( Θ 𝜒 ) ~ 𝑃𝑈(𝑛−ℎ−2) 𝐹 𝑈 𝑑𝑈

17. Sketch of the proof Write the L-functions in terms of unitary matrices
𝑓∈ 𝑀 𝑛 𝑑 𝑘 𝑓 → evaluated in terms of 𝑍 𝑢
𝑓∈𝐼 𝐴;ℎ 𝑑 𝑘 (𝑓) → restrict to an interval using orthogonality relations for Dirichlet character
→ evaluate in terms of the associated Dirichlet L- functions
Write the L-functions in terms of unitary matrices
𝐿 𝑢,𝜒 = 1−𝑢 det 𝐼−𝑢 𝑞 Θ 𝜒
The variance ~
1 𝜙 𝑒𝑣 ∗ 𝑇 𝑛−ℎ 𝜒 𝑚𝑜𝑑 𝑇 𝑛−ℎ 𝜒≠ 𝜒 0 𝜒 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝑗 1 +⋯+ 𝑗 𝑘 =n 0≤ 𝑗 1 ,…, 𝑗 𝑘 ≤n−h−2 𝑆 𝑐 𝑗 1 Θ 𝜒 ⋯𝑆 𝑐 𝑗 𝑘 Θ 𝜒 2

18. Sketch of the proof Apply Katz equidistribution result
1 𝜙 𝑒𝑣 ∗ 𝑇 𝑛−ℎ 𝜒 𝑚𝑜𝑑 𝑇 𝑛−ℎ 𝜒≠ 𝜒 0 𝜒 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑝𝑟𝑖𝑚𝑖𝑡𝑖𝑣𝑒 𝑗 1 +⋯+ 𝑗 𝑘 =n 0≤ 𝑗 1 ,…, 𝑗 𝑘 ≤n−h−2 𝑆 𝑐 𝑗 1 Θ 𝜒 ⋯𝑆 𝑐 𝑗 𝑘 Θ 𝜒 2
~ 𝑈 n−h− 𝑗 1 +⋯+ 𝑗 𝑘 =n 0≤ 𝑗 1 ,…, 𝑗 𝑘 ≤n−h−2 𝑆 𝑐 𝑗 1 𝑈 ⋯𝑆 𝑐 𝑗 𝑘 𝑈 2 𝑑𝑈

19. Matrix Integral
What do we know about I 𝑘 𝑚;𝑁 ≔ 𝑈 𝑁 𝑗 1 +⋯+ 𝑗 𝑘 =𝑚 0≤ 𝑗 1 ,…, 𝑗 𝑘 ≤𝑁 𝑆 𝑐 𝑗 1 𝑈 ⋯𝑆 𝑐 𝑗 𝑘 𝑈 2 𝑑𝑈 ?
For 𝑚>𝑘𝑁 , I 𝑘 𝑚;𝑁 =0.
I 𝑘 𝑚;𝑁 = 𝑚+ 𝑘 2 −1 𝑘 2 −1 , 𝑚<𝑁
Functional equation I 𝑘 𝑚;𝑁 = I 𝑘 𝑘𝑁−𝑚;𝑁 .
⇒ I 𝑘 𝑚;𝑁 = 𝑘𝑁−𝑚+ 𝑘 2 −1 𝑘 2 −1 , 𝑘−1 𝑁<𝑚<𝑘𝑁

20. Theorem (J. P. Keating, B. Rodgers, ER-G and Z
Theorem (J.P.Keating, B.Rodgers, ER-G and Z.Rudnick) I 𝑘 𝑚;𝑁 is equal to the count of lattice points 𝑥=( 𝑥 𝑖 𝑗 )∈ (ℤ) 𝑘 2 satisfying each of the relations
0≤ 𝑥 𝑖 𝑗 ≤𝑁 for all 1≤𝑖,𝑗≤𝑘.
𝑥 1 𝑘 + 𝑥 2 𝑘−1 +⋯+ 𝑥 𝑘 1 =𝑘𝑁−𝑚.
𝑥∈ 𝐴 𝑘 , where 𝐴 𝑘 is the collection of 𝑘×𝑘 matrices whose entries satisfy the following system of equalities,

21. Theorem (J.P.Keating, B.Rodgers, E.R−G and Z.Rudnick 2015)
Let 𝑟≔𝑚/𝑁. Then for 𝑟∈ 0,𝑘 ,
I 𝑘 𝑚;𝑁 = 𝛾 𝑘 𝑟 𝑁 𝑘 2 −1 +𝑂( 𝑁 𝑘 2 −2 )
With
𝛾 𝑘 𝑟 = 1 𝑘!𝐺 1+𝑘 ,1 𝑘 𝛿 𝑟 𝑤 1 +⋯+ 𝑤 𝑘 𝑖<𝑗 𝑤 𝑖 − 𝑤 𝑗 d k w
Here 𝐺 is the Barnes G-function, so that for positive integers k,
𝐺 1+𝑘 =1!∙2!∙3!∙⋯ 𝑘−1 !
𝛾 𝑘 𝑟 is a piecewise polynomial which changes when ever 𝑟 reaches an integer.

22. Thank you!