arXiv subjects

Mithun Kumar Das

Publications and source records attributed to Mithun Kumar Das.

11 recordsLinked to original sources

arXiv2025

Littlewood's estimates for $L$-functions in the hyperelliptic ensemble

We investigate the analogues of certain classical estimates of Littlewood for the Riemann zeta-function in the context of quadratic Dirichlet $L$-functions over function fields. In some situations, we are actually able to establish finer results in the function field setup than what is currently known in the original number field setup, and this leads us to an educated guess on what could happen for the Riemann zeta-function in such situations. Fourier analysis techniques play an important role in our approach.

math.NT↗

arXiv2025

Fourier optimization and pair correlation problems

We introduce a generic framework to provide bounds related to the pair correlation of sequences belonging to a wide class. We consider analogues of Montgomery's form factor for zeros of the Riemann zeta function in the case of arbitrary sequences satisfying some basic assumptions, and connect their estimation to two extremal problems in Fourier analysis, which are promptly studied. As applications, we provide average bounds of form factors related to some sequences of number theoretic interest, such as the zeros of primitive elements of the Selberg class, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, our results bear an implication to a conjecture of Gonek and Ki (2018), showing it cannot hold in some situations.

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arXiv2024

Effective equidistribution of Galois orbits for mildly regular test functions

In this paper we provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus, identifying the quantitative dependence of the convergence in terms of the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005), and by D'Andrea, Narv\'aez-Clauss and Sombra (2017).

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arXiv2024

Distribution of the zeros of polynomials near the unit circle

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1} in 2008. Furthermore, by combining this result with Euclidean geometry, we derive an upper bound on the number of zeros of such a polynomial within a region resembling a gear wheel. Additionally, we obtain a sharp upper bound on the annular discrepancy of such zeros near the unit circle. Our approach builds upon a modified version of the method described in \cite{BE1}, combined with the refined version of the best-known upper bound for angular discrepancy of zeros of polynomials.

math.CV↗

arXiv2023

Poissonian pair correlation for higher dimensional real sequences

In this article, we examine the Poissonian pair correlation (PPC) statistic for higher-dimensional real sequences. Specifically, we demonstrate that for $d\geq 3$, almost all $(\alpha_1,\ldots,\alpha_d) \in \mathbb{R}^d$, the sequence $\big(\{x_n\alpha_1\},\dots,\{x_n\alpha_d\}\big)$ in $[0,1)^d$ has PPC conditionally on the additive energy bound of $(x_n).$ This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [1]. More generally, we derive the PPC for $\big(\{x_n^{(1)}\alpha_1\},\dots,\{x_n^{(d)}\alpha_d\}\big) \in [0,1)^d$ for almost all $(\alpha_1,\ldots,\alpha_d) \in \mathbb{R}^d.$ As a consequence we establish the metric PPC for $(n^{\theta_1},\ldots,n^{\theta_d})$ provided that all of the $\theta_i$'s are greater than two.

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arXiv2023

On higher dimensional Poissonian pair correlation

In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any $d\geq 2$, strictly increasing sequences $(a_n^{(1)}),\ldots, (a_n^{(d)})$ of natural numbers have metric Poissonian pair correlation with respect to sup-norm if their joint additive energy is $O(N^{3-δ})$ for any $δ>0$. Further, in two dimension, we establish an analogous result with respect to $2$-norm. As a consequence, it follows that $(\{nα\}, \{n^2β\})$ and $(\{nα\}, \{[n\log^An]β\})$ ($A \in [1,2]$) have Poissonian pair correlation for almost all $(α,β)\in \mathbb{R}^2$ with respect to sup-norm and $2$-norm. This gives a negative answer to the question raised by Hofer and Kaltenböck [15]. The proof uses estimates for 'Generalized' GCD-sums.

math.NT↗

arXiv2022

The variance of a general class of multiplicative functions in short intervals

We study a general class of multiplicative functions by establishing a connection between their ``short averages" and ``long average". More precisely, we employ Fourier analysis and the counting of rational points on specific binary forms to provide asymptotic estimates for the variance of this class within short intervals. Our results apply to notable multiplicative functions such as $\mu_k(n)$, $\frac{\phi(n)}{n}$, $\sigma_{\alpha}(n)$, among others, yielding several new results and improvements in the realm of short interval analysis. Remarkably, our results disprove a conjecture of van Overbeeke concerning the variance of $\frac{\phi(n)}{n}$.

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arXiv2021

Hilbert transforms and the equidistribution of zeros of polynomials

We improve the current bounds for an inequality of Erdős and Turán from 1950 related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of Soundararajan, we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan (2019), refinements of the discrepancy inequality of Erdős and Turán had been obtained by Ganelius (1954) and Mignotte (1992).

math.CA↗

arXiv2021

Zeros of higher derivatives of Riemann zeta function

In this article, we extend the result of Conrey [5, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of $\zeta^{(k)}$. Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's $\eta_k$-function cluster near the critical line.

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arXiv2020

Sparse subsets of the natural numbers and Euler's totient function

In this article, we investigate sparse subsets of the natural numbers and study the sparseness of some sets associated with the Euler's totient function $ϕ$ via the property of `Banach Density'. These sets related to the totient function are defined as follows: $V:=ϕ(\mathbb{N})$ and $N_i:=\{N_i(m)\colon m\in V \}$ for $i = 1, 2, 3,$ where $N_1(m)=\max\{x\in \mathbb{N}\colon ϕ(x)\leq m\}$, $N_2(m)=\max(ϕ^{-1}(m))$ and $N_3(m)=\min(ϕ^{-1}(m))$ for $ m\in V$. Masser and Shiu call the elements of $N_1$ as `sparsely totient numbers' and construct an infinite family of these numbers. Here we construct several infinite families of numbers in $N_2\setminus N_1$ and an infinite family of composite numbers in $N_3$. We also study (i) the ratio $\frac{N_2(m)}{N_3(m)}$, which is linked to the Carmichael's conjecture, namely, $|ϕ^{-1}(m)|\geq 2 ~\forall ~ m\in V$, and (ii) arithmetic and geometric progressions in $N_2$ and $N_3$. Finally, using the above sets associated to the totient function, we generate an infinite class of subsets of $\mathbb{N}$, each with asymptotic density zero and containing arbitrarily long arithmetic progressions.

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arXiv2019

Combinatorial properties of sparsely totient numbers

Let $N_1(m)=\max\{n \colon ϕ(n) \leq m\}$ and $N_1 = \{N_1(m) \colon m \in ϕ(\mathbb{N})\}$ where $ϕ(n)$ denotes the Euler's totient function. Masser and Shiu \cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. Next, we construct explicit infinite families of sparsely totient numbers and describe their relationship with the distribution of consecutive primes. We also study the sparseness of $N_1$ and prove that it is multiplicatively piecewise syndetic but not additively piecewise syndetic. Finally, we investigate arithmetic/geometric progressions and other additive and multiplicative patterns like $\{x, y, x+y\}, \{x, y, xy\}, \{x+y, xy\}$ and their generalizations in the sparsely totient numbers.

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