arXiv subjects

Michael Yiasemides

Publications and source records attributed to Michael Yiasemides.

4 recordsLinked to original sources

arXiv2023

Lattice Point Variance in Thin Elliptic Annuli over $\mathbb{F}_q [T]$

For fixed coprime polynomials $U,V \in \mathbb{F}_q [T]$ with degrees of different parities, and a general polynomial $A \in \mathbb{F}_q [T]$, define the arithmetic function $S_{U,V} (A)$ to be the number of representations of $A$ of the form $UE^2 + VF^2$ with $E,F \in \mathbb{F}_q [T]$. We study the mean and variance of $S_{U,V}$ over short intervals in $\mathbb{F}_q [T]$, and this can be interpreted as the function field analogue of the mean and variance of lattice points in thin elliptic annuli, where the scaling factor of the ellipses is rational. Our main result is an asymptotic formula for the variance even when the length of the interval remains constant relative to the absolute value of the centre of the interval. In terms of lattice points, this means we obtain the variance in the so-called ``local'' or ``microscopic'' regime, where the area of the annulus remains constant relative to the inner radius. We also obtain asymptotic or exact formulas for almost all other lengths of the interval, and we see some interesting behaviour at the boundary between short and long intervals. Our approach is that of additive characters and Hankel matrices that we employed for the divisor function and a restricted sum-of-squares function in previous work, and we develop further results on Hankel matrices in this paper.

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arXiv2022

The Variance of the Sum of Two Squares over Intervals in $\mathbb{F}_q [T]$: I

For $B \in \mathbb{F}_q [T]$ of degree $2n \geq 2$, consider the number of ways of writing $B=E^2 + γF^2$, where $γ\in \mathbb{F}_q^*$ is fixed, and $E,F \in \mathbb{F}_q [T]$ with $\mathrm{deg} \hspace{0.25em} E = n$ and $\mathrm{deg} \hspace{0.25em} F = m < n$. We denote this by $S_{γ; m} (B)$. We obtain an exact formula for the variance of $S_{γ; m} (B)$ over intervals in $\mathbb{F}_q [T]$. We use the method of additive characters and Hankel matrices that the author previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach; and we briefly discuss the possible extension of our result to the number of ways of writing $B=E^2 + T F^2$.

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arXiv2021

The Variance and Correlations of the Divisor Function in $\mathbb{F}_q [T]$, and Hankel Matrices

We prove an exact formula for the variance of the divisor function over short intervals in $\mathcal{A} := \mathbb{F}_q [T]$, where $q$ is a prime power. A slight adaption of the proof allows us to obtain an exact formula for correlations of the form $d(A) d(A+B)$, where we average both $A$ and $B$ over certain intervals in $\mathcal{A}$. We also consider correlations of the form $d(KQ+N) d (N)$, where $Q$ is prime and $K$ and $N$ are averaged over certain intervals. If $\mathrm{deg } K < \mathrm{deg } Q -1$, then these correlations appear in the off-diagonal terms for the fourth moment of Dirichlet $L$-functions. We consider the case $\mathrm{deg } K \geq \mathrm{deg }Q -1$ and obtain an exact formula for the correlations. Further, we demonstrate that $d(KQ+N)$ and $d (N)$ are uncorrelated for the given ranges of $K$ and $N$. Our approach to these problems is to use the orthogonality relations of additive characters on $\mathbb{F}_q$ to translate the problems to ones involving the ranks of Hankel matrices over $\mathbb{F}_q$. Most of the paper is dedicated to proving several results regarding the rank and kernel structure of these matrices, and thus demonstrating their number-theoretic properties. We briefly discuss extending our method to moments higher than the second (the variance) over intervals; to the $k$-th divisor function; and to correlations of the divisor function with applications to moments of Dirichlet $L$-functions in function fields.

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arXiv2021

The Hybrid Euler-Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q [T]$

For Dirichlet $L$-functions in $\mathbb{F}_q [T]$ we obtain a hybrid Euler-Hadamard product formula. We make a splitting conjecture, namely that the $2k$-th moment of the Dirichlet $L$-functions at $\frac{1}{2}$, averaged over primitive characters of modulus $R$, is asymptotic to (as $\mathrm{deg} R \longrightarrow \infty$) the $2k$-th moment of the Euler product multiplied by the $2k$-th moment of the Hadamard product. We explicitly obtain the main term of the $2k$-th moment of the Euler product, and we conjecture via random matrix theory the main term of the $2k$-th moment of the Hadamard product. With the splitting conjecture, this directly leads to a conjecture for the $2k$-th moment of Dirichlet $L$-functions. Finally, we lend support for the splitting conjecture by proving the cases $k=1,2$. This work is the function field analogue of the work of Bui and Keating. A notable difference in the function field setting is that the Euler-Hadamard product formula is exact, in that there is no error term.

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