The variance and correlations of the divisor function in $${\mathbb {F}}_q [T]$$, and Hankel matrices

Abstract

Abstract We prove an exact formula for the variance of the divisor function over short intervals in $${\mathcal {A}}:= {\mathbb {F}}_q [T]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where q is a prime power; and for correlations of the form $$d(A) d(A+B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where we average both A and B over certain intervals in $${\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> . We also obtain an exact formula for correlations of the form $$d(KQ+N) d (N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where Q is prime and K and N are averaged over certain intervals with $${{\,\textrm{deg}\,}}N \le {{\,\textrm{deg}\,}}Q -1 \le {{\,\textrm{deg}\,}}K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mtext>deg</mml:mtext> <mml:mspace /> </mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mrow> <mml:mspace /> <mml:mtext>deg</mml:mtext> <mml:mspace /> </mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mrow> <mml:mspace /> <mml:mtext>deg</mml:mtext> <mml:mspace /> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> ; and we demonstrate that $$d(KQ+N)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and d ( N ) are uncorrelated. We generalize our results to $$\sigma _z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>z</mml:mi> </mml:msub> </mml:math> defined by $$\sigma _z (A):= \sum _{E \mid A} |A |^z$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>z</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>∣</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>z</mml:mi> </mml:msup> </mml:mrow> </mml:math> for all monics $$A \in {\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> . Our approach is to use the orthogonality relations of additive characters on $${\mathbb {F}}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> to translate the problems to ones involving the ranks of Hankel matrices over $${\mathbb {F}}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> . We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving $$d_k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> .

Locations

• Research in the Mathematical Sciences - View - PDF