arXiv subjects

Shivani Goel

Publications and source records attributed to Shivani Goel.

At least 19 recordsLinked to original sources

arXiv2025

The second moment of cubic Dirichlet L-functions over function fields

In this article, we study the second moment of cubic Dirichlet L-functions at the central point $s=1/2$ over the rational function field $\mathbb{F}_q(T)$, where $q$ is a power of an odd prime satisfying $q \equiv 2 \pmod{3}$. Our result extends prior work of David, Florea and Lalin, who obtained an asymptotic formula for the first moment. Our approach relies on analytic techniques (Perron's formula, approximate functional equation, etc), adapted to the function field context. A key step in the construction is to relate second moment to certain averages of Gauss sums, which are estimated in loc. cit. using results of Kubota and Hoffstein.

math.NT↗

arXiv2025

Diophantine approximation and the subspace theorem

Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results to higher dimensions, with profound implications to Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth's theorem and Schlickewei's refinement of the subspace theorem, with an emphasis on proofs. The arguments presented are classical and approachable for readers with a background in algebraic number theory, serving as a streamlined, yet condensed reference for these fundamental results.

math.NT↗

arXiv2024

Explicit Deuring-Heilbronn phenomenon for Dirichlet $L$-functions

Assuming the existence of a Landau-Siegel zero, we establish an explicit Deuring-Heilbronn zero repulsion phenomenon for Dirichlet $L$-functions modulo $q$. Our estimate is uniform in the entire critical strip, and improves over the previous best known explicit estimate due to Thorner and Zaman.

math.NT↗

arXiv2024

On the moments of averages of Ramanujan sums

Chan and Kumchev studied averages of the first and second moments of Ramanujan sums. In this article, we extend this investigation by estimating the higher moments of averages of Ramanujan sums using the Br\`eteche Tauberian theorem. We also give a result for the moments of averages of Cohen-Ramanujan sums.

math.NT↗

arXiv2024

Moments of Averages of Ramanujan Sums over Number Fields

Assuming the generalized Lindel\"of hypothesis, we provide asymptotic formulas for the mean values of the first and second moments of Ramanujan sums over any number field. Additionally, unconditionally, we estimate the second moment of Ramanujan sums over cyclotomic number fields.

math.NT↗

arXiv2023

Pair correlation of real-valued vector sequences

In this article, we investigate the fine-scale statistics of real-valued arithmetic sequences. In particular, we focus on real-valued vector sequences and show the Poissonian behavior of the pair correlation function for certain classes of such sequences, thereby extending previous works of Boca et al. and the first author on local statistics of integer-valued and rational-valued vector sequences.

math.NT↗

arXiv2023

Distance matrix of enhanced power graphs of finite groups

The enhanced power graph of a group $G$ is the graph $\mathcal{G}_E(G)$ with vertex set $G$ and edge set $ \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}$. In this paper, we compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order $pq$, dihedral groups, dicyclic groups, elementary abelian groups $\mathrm{El}(p^n)$ and the non-cyclic abelian groups $\mathrm{El}(p^n)\times\mathrm{El}(q^m)$ and $\mathrm{El}(p^n)\times \mathbb{Z}_m$, where $p$ and $q$ are distinct primes. For the non-cyclic abelian group $\mathrm{El}(p^n)\times \mathrm{El}(q^m)$, we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph.

math.CO↗

arXiv2022

On the distribution of index of Farey Sequences

In this article, we study the distribution of index of Farey fractions which was first introduced and studied by Hall and Shiu. We provide asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with $\mathcal{B}$-free denominators. Additionally, we reconsider the squarefree case earlier done in [ALVZ08], and obtain new results for moments of indices with square-free denominators. We also study higher level correlations of the index function generalizing earlier known results on two level correlations.

math.NT↗

arXiv2021

On resistance matrices of weighted balanced digraphs

Let $G$ be a connected graph with $V(G)=\{1,\dotsc,n\}$. Then the resistance distance between any two vertices $i$ and $j$ is given by $r_{ij}:=l_{ii}^† + l_{jj}^†-2 l_{ij}^†$, where $l_{ij}^†$ is the $(i,j)^{\rm th}$ entry of the Moore-Penrose inverse of the Laplacian matrix of $G$. For the resistance matrix $R:=[r_{ij}]$, there is an elegant formula to compute the inverse of $R$. This says that \[R^{-1}=-\frac{1}{2}L + \frac{1}{τ' R τ} ττ', \] where \[τ:=(τ_1,\dotsc,τ_n)'~~\mbox{and}~~ τ_{i}:=2- \sum_{\{j \in V(G):(i,j) \in E(G)\}} r_{ij}~~~i=1,\dotsc,n. \] A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.

math.CO↗

arXiv2021

On the distribution of Ramanujan Sums over number fields

For a number field $\mathbb{K}$, and integral ideals $\mathcal{I}$ and $\mathcal{J}$ in its number ring $\mathcal{O}_{\mathbb{K}}$, Nowak studied the asymptotic behaviour of the average of Ramanujan sums $C_{\mathcal{J}}({\mathcal{I}})$ over both ideals $\mathcal{I}$ and $\mathcal{J}$. In this article, we extend this investigation by establishing asymptotic formulas for the second moment of averages of Ramanujan sums over quadratic and cubic number fields, thereby generalizing previous works of Chen, Kumchev, Robles, and Roy on moments of averages of Ramanujan sums over rationals. Additionally, using a special property of certain integral domains, we obtain second moment results for Ramanujan sums over some other number fields.

math.NT↗

arXiv2021

Steiner distance matrix of caterpillar graphs

For a connected graph $G:=(V,E)$, the Steiner distance $d_G(X)$ among a set of vertices $X$ is the minimum size among all the connected subgraphs of $G$ whose vertex set contains $X$. The $k-$Steiner distance matrix $D_k(G)$ of $G$ is a matrix whose rows and columns are indexed by $k-$subsets of $V$. For $k$-subsets $X_1$ and $X_2$, the $(X_1,X_2)-$entry of $D_k(G)$ is $d_G(X_1 \cup X_2)$. In this paper, we show that the rank of $2-$Steiner distance matrix of a caterpillar graph on $N$ vertices and with $p$ pendant veritices is $2N-p-1$.

math.CO↗

arXiv2021

Generalized Euclidean distance matrices

Euclidean distance matrices (EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices (GDMs) that include EDMs. Each GDM is an entry-wise nonnegative matrix. A GDM is not symmetric unless it is an EDM. By some new techniques, we show that many significant results on Euclidean distance matrices can be extended to generalized Euclidean distance matrices. These contain results about eigenvalues, inverse, determinant, spectral radius, Moore-Penrose inverse and some majorization inequalities. We finally give an application by constructing infinitely divisible matrices using generalized Euclidean distance matrices.

math.FA↗

arXiv2020

On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices

Let $n \geq 4$. The helm graph $H_n$ on $2n-1$ vertices is obtained from the wheel graph $W_n$ by adjoining a pendant edge to each vertex of the outer cycle of $W_n$. Suppose $n$ is even. Let $D := [d_{ij}]$ be the distance matrix of $H_n$. In this paper, we first show that $\det(D) = 3(n-1)2^{n-1}.$ Next, we find a matrix $Ł$ and a vector $u$ such that \[D^{-1} = -\frac{1}{2}Ł+\frac{4}{3(n-1)}uu'.\] We also prove an interlacing property between the eigenvalues of $Ł$ and $D$.

math.CO↗

arXiv2020

Mean value estimates of gcd and lcm-sums

We study the distribution of the generalized gcd and lcm functions on average. The generalized gcd function, denoted by $(m,n)_b$, is the largest $b$-th power divisor common to $m$ and $n$. Likewise, the generalized lcm function, denoted by $[m,n]_b$, is the smallest $b$-th power multiple common to $m$ and $n$. We derive asymptotic formulas for the average order of the arithmetic, geometric, and harmonic means of $(m,n)_b$. Additionally, we also deduce asymptotic formulas with error terms for the means of $(n_1,n_2,\cdots, n_k)_b$, and $[n_1,n_2,\cdots, n_k]_b$ over a set of lattice points, thereby generalizing some of the previous work on gcd and lcm-sum estimates.

math.NT↗

arXiv2020

On distance matrices of wheel graphs with odd number of vertices

Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \[d_{ij}:= \begin{cases} 0 & \mbox{if}~i=j \\ 1 & \mbox{if}~i~ \mbox{and} ~j~ \mbox{are adjacent} \\ 2 & \mbox{else}. \end{cases}\] Let $D$ be the $n \times n$ matrix with $(i,j)^{\rm th}$ entry equal to $d_{ij}$. The matrix $D$ is called the distance matrix of $W_n$. Suppose $n \geq 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of $D$. More precisely, we obtain an $n\times n$ matrix $\widetilde{L}$ and a rank one matrix $ww'$ such that \[D^\dagger = -\frac{1}{2} \widetilde{L}+\frac{4}{n-1}ww'.\] Here, $\widetilde{L}$ is positive semidefinite, ${\rm rank}(\widetilde{L})=n-2$ and all row sums are equal to zero.

math.CO↗

arXiv2020

An inverse formula for the distance matrix of a wheel graph with even number of vertices

Let $n \geq 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n \times n$ symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to $d_{ij}$. In this paper, we find a positive semidefinite matrix $\widetilde{L}$ such that ${\rm rank}(\widetilde{L})=n-1$, all row sums of $\widetilde{L}$ equal to zero and a rank one matrix $ww^T$ such that \[D^{-1}=-\frac{1}{2}\widetilde{L} + \frac{4}{n-1}ww^T. \] An interlacing property between the eigenvalues of $D$ and $\widetilde{L}$ is also proved.

math.CO↗

arXiv2019

Resistance distance in directed cactus graphs

Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set $V=\{1,\dotsc,n\}$. The classical distance $d_{ij}$ between any two vertices $i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$ and $j$. The resistance distance (or, simply the resistance) between any two vertices $i$ and $j$ in $V$ is defined by $r_{ij}:=l_{ii}^†+l_{jj}^†-2l_{ij}^†$, where $l_{pq}^{\dagger}$ is the $(p,q)^{\rm th}$ entry of the Moore-Penrose inverse of $L$ which is the Laplacian matrix of $G$. In practice, the resistance $r_{ij}$ is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between $i$ and $j$ is always less than or equal to the classical distance, i.e. $r_{ij} \leq d_{ij}$. However, no proof for this inequality is known. In this paper, we show that this inequality holds for all directed cactus graphs.

math.CO↗

arXiv2019

Distance matrices perturbed by a Laplacian

Let $T$ be a tree with $n$ vertices. To each edge of $T$, we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$. Define $D:=[D_{ij}]$, where $D_{ii}$ is the $s \times s$ null matrix and for $i \neq j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order $s$. If $i$ and $j$ are adjacent, then define $L_{ij}:=-W_{ij}^{-1}$, where $W_{ij}$ is the weight of the edge $(i,j)$. Define $L_{ii}:=\sum_{i \neq j,j=1}^{n}W_{ij}^{-1}$. The Laplacian of $G$ is now the $ns \times ns$ block matrix $L:=[L_{ij}]$. In this paper, we first note that $D^{-1}-L$ is always non-singular and then we prove that $D$ and its perturbation $(D^{-1}-L)^{-1}$ have many interesting properties in common.

math.CO↗