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Anup B. Dixit

Publications and source records attributed to Anup B. Dixit.

12 recordsLinked to original sources
arXiv2026

Statistics of the Genus Number of $S_3 \times C_q$ and $D_4$-fields

The genus number of a number field is a fundamental invariant which measures the contribution of ramification to its ideal class group. In this paper, we establish the statistics for the genus number for $S_3\times C_q$-fields for $q\neq 3$ a prime number, $D_4$-fields and pure quartic fields. We also obtain precise results on the average and higher moments of the genus distribution within the family of $S_3\times C_q$-fields. Finally, based on heuristics, we formulate a conjecture identifying families for which one should expect the genus density to be zero, i.e., only a density zero subset of fields in the family attains any fixed genus number.

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arXiv2025

How often are $ \lfloor {n^α} \rfloor $ and $ \lfloor {n^β} \rfloor $ simultaneously primes?

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<α_1 < α_2 < \cdots< α_k < 1$ satisfying a certain condition, we show that there are infinitely many positive integers $n$ for which all of $ \lfloor{n^{α_1}}\rfloor, \lfloor{n^{α_2}}\rfloor,\ldots, \lfloor{n^{α_k}}\rfloor $ are prime numbers. Our approach relies on establishing a simultaneous equidistribution theorem for $ \lfloor{n^{α_i}}\rfloor $ across $k$-many arithmetic progressions.

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arXiv2025

On points of small height in infinite extensions

In this paper, we introduce the notion of asymptotically positive infinite extensions of $\mathbb{Q}$, in the spirit of the Tsfasman-Vlăduţ theory of asymptotically exact families of number fields. For asymptotically positive extensions, we obtain lower bounds on the logarithmic Weil height, establishing the Bogomolov property for a wide range of infinite non-Galois extensions. Our result encompasses the famous theorem of E. Bombieri and U. Zannier on Bogomolov property for totally $p$-adic extensions of type $(e,f)$. Additionally, our theorem can be interpreted as a $p$-adic equidistribution result on conjugates of $α$, resonating with the archimedean equidistribution theorem à la F. Amoroso-M. Mignotte and Y. Bilu. In the parallel setting of elliptic curves, we derive lower bounds on the canonical height for points on an elliptic curve over asymptotically positive extensions, without any restriction on its reduction type. In particular, this extends a result of M. Baker in the context of totally $ν$-adic extensions, where the elliptic curve is assumed to have semistable reduction at $ν$.

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arXiv2025

A p-adic criterion for Lehmer's conjecture

For a non-zero algebraic number $α$ of degree $d$, let $h(α)$ denote its logarithmic Weil height. It is known that when $h(α)$ is small, and $d$ is large, the conjugates of $α$ are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a $p$-adic analogue of this result by obtaining lower bounds for $h(α)$ in terms of the number of its conjugates that lie in a finite extension of $\mathbb{Q}_p$, for some prime $p$. As a consequence, we prove Lehmer's conjecture for all $α$ such that $\gg \sqrt{d\log d}$ many of its conjugates lie in a finite extension of $\mathbb{Q}_p$.

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arXiv2024

On the distribution of $ϕ(σ(n))$

Let $ϕ(n)$ be the Euler totient function and $σ(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $ϕ(σ(n)) > cn$ for any $c>0$. This is a refinement of a result of Alaoglu and Erdős.

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arXiv2023

Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function

In this paper, we establish lower bounds on Weil height of algebraic integers in terms of the low lying zeros of the Dedekind zeta-function. As a result, we prove Lehmer's conjecture for certain infinite non-Galois extensions conditional on GRH. We also introduce and study a condition on prime ideals with small norms for arbitrary infinite extensions, in the spirit of a prime splitting condition for infinite Galois extensions introduced by E. Bombieri and U. Zannier.

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arXiv2023

Linear independence of values of the $q$-exponential and related functions

In this paper, we establish the linear independence of values of the $q$-analogue of the exponential function, $E_q(x)$ and its derivatives at specified algebraic arguments, when $q$ is a Pisot-Vijayraghavan number. We also deduce similar results for cognate functions, such as the Tschakaloff function and certain generalized $q$-series.

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arXiv2020

Large values of $L$-functions on $1$-line

In this paper, we study lower bounds of a general family of $L$-functions on the $1$-line. More precisely, we show that for any $F(s)$ in this family, there exists arbitrary large $t$ such that $F(1+it)\geq e^{γ_F} (\log_2 t + \log_3 t)^m + O(1)$, where $m$ is the order of the pole of $F(s)$ at $s=1$. This is a generalization of the same result of Aistleitner, Munsch and the second author for the Riemann zeta-function. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg $L$-functions of the type $L(s,f\times f)$ on the $1$-line.

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arXiv2019

On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture

As a natural generalization of the Euler-Mascheroni constant $γ$, Y. Ihara introduced the Euler-Kronecker constant $γ_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $γ_K$ in a tower of number fields $\mathcal{K}$ implies the generalized Brauer-Siegel conjecture for $\mathcal{K}$ as formulated by Tsfasman and Vlǎduţ. Moreover, we use known bounds on $γ_K$ for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function $ζ_K(s)$ in the critical strip.

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arXiv2019

A uniqueness property of general Dirichlet series

Let $F(s)=\sum_n a_n/λ_n^s$ be a general Dirichlet series which is absolutely convergent on $\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree $d_F$ and conductor $α_F$. In this paper, we show that there are at most $2d_F$ general Dirichlet series with a given degree $d_F$, conductor $α_F$ and residue $ρ_F$ at $s=1$. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at $s=1$.

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arXiv2017

Value Distribution of L-functions

In 2002, V. Kumar Murty \cite{Km} introduced a class of $L$-functions, namely the Lindelöf class, which has a ring structure attached to it. In this paper, we establish some results on the value distribution of $L$-functions in this class. As a corollary, we also prove a uniqueness theorem in the Selberg class.

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