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Kamalakshya Mahatab

Publications and source records attributed to Kamalakshya Mahatab.

22 records · Page 2Linked to original sources
arXiv2018

Omega Theorems for The Twisted Divisor Function

For a fixed $θ\neq 0$, we define the twisted divisor function $$ τ(n, θ):=\sum_{d\mid n}d^{iθ}\ .$$ In this article we consider the error term $Δ(x)$ in the following asymptotic formula $$ \sum_{n\leq x}^*|τ(n, θ)|^2=ω_1(θ)x\log x + ω_2(θ)x\cos(θ\log x) +ω_3(θ)x + Δ(x),$$ where $ω_i(θ)$ for $i=1, 2, 3$ are constants depending only on $θ$. We obtain $$Δ(T)=Ω\left(T^{α(T)}\right) \text{ where } α(T) =\frac{3}{8}-\frac{c}{(\log T)^{1/8}} \text{ and } c>0,$$ along with an $Ω$-bound for the Lebesgue measure of the set of points where the above estimate holds.

math.NT↗
arXiv2017

Extreme values of the Riemann zeta function on the 1-line

We prove that there are arbitrarily large values of $t$ such that $|ζ(1+it)| \geq e^γ (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.

math.NT↗
arXiv2015

Chinese Remainder Theorem for Cyclotomic Polynomials in $\mathbf{Z}[X]$

By the Chinese remainder theorem, the canonical map \[Ψ_n: R[X]/(X^n-1)\to \oplus_{d|n} R[X]/Φ_d(X)\] is an isomorphism when $R$ is a field whose characteristic does not divide $n$ and $Φ_d$ is the $d$th cyclotomic polynomial. When $R$ is the ring $\mathbf{Z}$ of rational integers, this map is injective but not surjective. In this paper, we give an explicit formula for the elementary divisors of the cokernel of $Ψ_n$(when $R=\mathbb{Z}$) using the prime factorisation of $n$. We also give a pictorial algorithm using Young Tableaux that takes $O(n^{3+ε})$ bit operations for any $ε> 0$ to determine a basis of Smith vectors (see Definition 3.1) for the codomain of $Ψ_n$. In general when $R$ is an integral domain, we prove that the determinant of $Ψ: R[X]/(\prod_j f_j) \to \bigoplus_j R[X]/(f_j)$ written with respect to the standard basis is $\prod_{1 \leqslant i < j \leqslant n} \mathcal{R}(f_j, f_i)$, where $f_i$'s are pairwise relatively prime monic polynomials and $\mathcal{R}(f_j, f_i)$ is the resultant of $f_j$ and $f_i$.

math.NT↗