arXiv subjects

Bittu Chahal

Publications and source records attributed to Bittu Chahal.

6 recordsLinked to original sources

arXiv2026

Distribution of Farey fractions with $k$-free denominators

We investigate the distributional properties of the sequence of Farey fractions with $k$-free denominators in residue classes, defined as \[\mathscr{F}_{Q,k}^{(m)}:=\left\{\frac{a}{q}\ |\ 1\leq a\leq q\leq Q,\ \gcd(a,q)=1,\ q\ \text{is}\ k\text{-free}\ \&\ q\equiv b\pmod{m} \right\}.\] We show that $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$ is equidistributed modulo one, and prove analogues of the classical results of Franel, Landau, and Niederreiter for $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$, particularly, deriving an equivalent form of the generalized Riemann hypothesis (GRH) for Dirichlet $L$-functions in terms of the distribution of $\left(\mathscr{F}_{Q,k}^{(m)}\right)_{Q\ge 1}$. Beyond examining the global distribution, we also study the local statistics of these sequences. We establish formulas for all levels ($ν\ge 2$) of correlation measure. Specifically, we show the existence of the limiting pair ($ν=2$) correlation function and provide an explicit expression for it. Our results are based upon the estimation of weighted Weyl sums and weighted lattice point counting in restricted domains.

math.NT↗

arXiv2025

On the distribution of polynomial Farey points and Chebyshev's bias phenomenon

We study two types of problems for polynomial Farey fractions. For a positive integer $Q$, and polynomial $P(x)\in\mathbb{Z}[X]$ with $P(0)=0$, we define polynomial Farey fractions as \[\mathcal{F}_{Q,P}:=\left\{\frac{a}{q}: 1\leq a\leq q\leq Q,\ \gcd (P(a),q)=1\right\}.\] The classical Farey fractions are obtained by considering $P(x)=x$. In this article, we determine the global and local distribution of the sequence of polynomial Farey fractions via discrepancy and pair correlation measure, respectively. In particular, we establish that the sequence of polynomial Farey fractions is uniformly distributed modulo one and show that the limit superior of the pair correlation measure of $(\mathcal{F}_{Q,P})_{Q\ge1}$ is bounded. For the specific polynomial $P(x)=x(x+1)$, we show the existence of the limiting pair correlation measure of $(\mathcal{F}_{Q,P})_{Q\ge1}$ and also provide an explicit formula for the pair correlation function which is non-Poissonian. Further, restricting the polynomial Farey denominators to certain subsets of primes, we explicitly find the pair correlation measure and show it to be Poissonian. Finally, we study Chebyshev's bias type of questions for the classical and polynomial Farey denominators along arithmetic progressions and obtain an $Ω$-result for the error term of its counting function.

math.NT↗

arXiv2025

Chebyshev's bias for irrational factor function

In this article, we study the distribution of the irrational factor function of order $k$, introduced first by Atanassov for $k=2$ and later it was generalized by Dong et al. for all $k\geq 2$. We introduce the irrational factor function in both number field and function field settings, derive asymptotic formulas for their average value, and further establish omega results for the error term in the asymptotic formulas. Moreover, we study the Chebyshev's bias phenomenon for number field and function field analogues of sum of the irrational factor function.

math.NT↗

arXiv2025

On the Second Hardy-Littlewood Conjecture

The second Hardy-Littlewood conjecture asserts that the prime counting function $π(x)$ satisfies the subadditive inequality \begin{align*} π(x+y)\leqslant π(x)+π(y) \end{align*} for all integers $x,y\geqslant 2$. By linking the subadditivity of $π(x)$ to the error term in the Prime Number Theorem, we obtain unconditional improvements on the range of $y$ for which $π(x)$ is known to be subadditive. Moreover, assuming the Riemann Hypothesis, we show that for all $ε>0$, there exists $x_ε \geqslant 2$ such that for all $x\geqslant x_ε$ and $y$ in the range \begin{align*} \frac{(2+ε)\sqrt{x}\log^2x}{8π}\leqslant y\leqslant x, \end{align*} the inequality $π(x+y)\leqslant π(x) + π(y)$ holds.

math.NT↗

arXiv2024

Pair correlation of Farey fractions with square-free denominators

In this article, we study the pair correlation of Farey fractions by proving that the limiting pair correlation function of the sequence of Farey fractions with square-free denominators exists and provide an explicit formula for the limiting pair correlation function.

math.NT↗

arXiv2023

Distribution of values of general Euler totient function

Let $Φ_k(n)=|\{ (x_1, x_2, \cdots, x_k)\in \left(\mathbb{Z}/n\mathbb{Z}\right)^k; \ \gcd(x_1^2+x_2^2+ \cdots+ x_k^2, n)=1\}|$ be a general totient function introduced first by Caldéron et. al. Motivated by the classical works of Schoenberg, Erdős, Bateman and Diamond on the distribution of $Φ_1(n)$, we prove results on the joint distribution of $Φ_k(n)$ for any $k\ge 1$. Additionally, we also exhibit the extremal order of $Φ_k(n)$.

math.NT↗