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The variance of the number of sums of two squares in $\mathbb{F}_q[T]$ in short intervals

Ofir Gorodetsky, Brad Rodgers

Abstract

Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in the large $q$ limit, finding a connection to the $z$-measures first investigated in the context of harmonic analysis on the infinite symmetric group. A similar connection to $z$-measures is established for sums over short intervals of the divisor functions $d_z(n)$. We use these results to make conjectures in the setting of the integers which match very well with numerically produced data. Our proofs depend on equidistribution results of N. Katz and W. Sawin.

The variance of the number of sums of two squares in $\mathbb{F}_q[T]$ in short intervals

Abstract

Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in the large limit, finding a connection to the -measures first investigated in the context of harmonic analysis on the infinite symmetric group. A similar connection to -measures is established for sums over short intervals of the divisor functions . We use these results to make conjectures in the setting of the integers which match very well with numerically produced data. Our proofs depend on equidistribution results of N. Katz and W. Sawin.

Paper Structure

This paper contains 29 sections, 21 theorems, 164 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Classical theory
  3. The function field analogy
  4. The divisor functions dz(n)
  5. Acknowledgements
  6. Dirichlet characters
  7. The strategy of the proof
  8. From short interval variance to character sums
  9. A computation related to the Dirichlet series of b
  10. The passage to zeros of L-functions
  11. Equidistribution and random matrix integrals
  12. z-measures on partitions
  13. Definitions
  14. The evaluation of random matrix integrals
  15. Arithmetic functions and z-measures
  16. ...and 14 more sections

Key Result

Theorem 1.1

For a fixed odd prime $p$, and fixed $n \geq 6$, take $0 \leq h \leq n-7$ and let $N:= n-h-1$ and $q = p^k$. Define For $N(N-1) \geq n$, as $q \rightarrow \infty$ (that is $k\rightarrow \infty$).

Figures (5)

  • Figure 1: Numerically produced data compared to the $z$-measure induced prediction given in Conjecture \ref{['b_variance_conj']} for variance in short intervals. Let $V_b(X;H)$ be the variance of counts of $S$ in random short intervals $[x,x+H]$ for $x \leq X$. For $X = 10^8$ and $H\leq X$, set $\delta = \delta_H = \log(H)/\log(X)$. For a selection of $H$, we plot the points $(\delta, V_b(X,H)/(H/\sqrt{\log X}))$ under the label data, and the curve $(\delta, K\, G(1-\delta))$ under prediction. See Section \ref{['sec:making_integer_conjectures']} for further discussion.
  • Figure 2: Numerically produced data compared to the prediction of Conjecture \ref{['b_variance_conj_q']} for variance in arithmetic progressions. For $X = 9\cdot 10^8$ and a selection of primes $p$, set $\delta \log(X/p)/\log(X)$; we plot the points $(\delta, \mathbb{V}_b(X,p)/((X/p)/\sqrt{\log X}))$ under the label data, and the curve $(\delta, K\, G(1-\delta))$ under prediction.
  • Figure 3: (a) A finite approximation of $F_z(s)$ for $z=0.25$, $0.5$, $0.75$. (b) A finite approximation for $F_z(s)$ for $z=1.5$, $2.0$, $2.5$, $3.0$, $3.5$. In both graphs, the finite approximations are the measures $\mathbb{P}_z^{(n)}(\lambda_1 \leq sn)$ for $n = 60$ with data points taken at $sn$ an integer. (c) A finite approximation of $F^\prime_z(s)$, obtained from the discrete derivative of the graphs in (b).
  • Figure 4: (a) A finite approximation of $G(s)$, obtained from $T(n;N)/T(n;n)$ for $n=50$. (b) A finite approximation of $G'(s)$, obtained from the discrete derivative of the graph in (a).
  • Figure 5: Numerically produced data compared to the Connors-Keating induced prediction: Consider the count of elements of $S$ less than $X = 9 \cdot 10^8$ congruent to a random $a$ modulo $p$, and let $\mathbb{V}_b(X,p)$ be the variance of these counts as $a$ varies. Let $\delta = \delta_p = \log(X/p)/\log(X)$. For a selection of primes $p$ in between $X/2$ and $X$ -- for which $\delta \in (0,\log 2 / \log (X))$ -- we plot the points $(\delta, \mathbb{V}_b(X,p)/((X/p)/\sqrt{\log X}))$ under the label data, while the prediction is a plot of the curve $(\delta, K + (-K^2 X^{\delta} + 1-X^{-\delta})/\sqrt{\log X})$.

Theorems & Definitions (54)

  • Conjecture 1
  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • ...and 44 more