arXiv subjects

Akshaa Vatwani

Publications and source records attributed to Akshaa Vatwani.

8 recordsLinked to original sources

arXiv2026

Equivalence between the Functional Equation and Vorono\"ı-type summation identities for a class of $L$-Functions

To date, the best methods for estimating the growth of mean values of arithmetic functions rely on the Vorono\"ı summation formula. By noticing a general pattern in the proof of his summation formula, Vorono\"ı postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ``nice" test functions $f(n)$, provided $a(n)$ is an ``arithmetic function". These arithmetic functions $a(n)$ are called so because they are expected to appear as coefficients of some $L$-functions satisfying certain properties. It has been well-known that the functional equation for a general $L$-function can be used to derive a Vorono\"ı-type summation identity for that $L$-function. In this article, we show that such a Vorono\"ı-type summation identity in fact endows the $L$-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

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arXiv2025

Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$

For a fixed $z\in\mathbb{C}$ and a fixed $k\in\mathbb{N}$, let $σ_{z}^{(k)}(n)$ denote the sum of $z$-th powers of those divisors $d$ of $n$ whose $k$-th powers also divide $n$. This arithmetic function is a simultaneous generalization of the well-known divisor function $σ_z(n)$ as well as the divisor function $d^{(k)}(n)$ first studied by Wigert. The Dirichlet series of $σ_{z}^{(k)}(n)$ does not fall under the purview of Chandrasekharan and Narasimhan's fundamental work on Hecke's functional equation with multiple gamma factors. Nevertheless, as we show here, an explicit and elegant Vorono\"{\dotlessi} summation formula exists for this function. As its corollaries, some transformations of Wigert are generalized. The kernel $H_{z}^{(k)}(x)$ of the associated integral transform is a new generalization of the Bessel kernel. Several properties of this kernel such as its differential equation, asymptotic behavior and its special values are derived. A crucial relation between $H_{z}^{(k)}(x)$ and an associated integral $K_{z}^{(k)}(x)$ is obtained, the proof of which is deep, and employs the theory of linear differential equations and the properties of Stirling numbers of the second kind and elementary symmetric polynomials.

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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.

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arXiv2022

Riesz type criteria for $L$-functions in the Selberg class

We formulate a generalization of Riesz-type criteria in the setting of $L$-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for $L$-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for $L$-functions in this subclass. Identities of Ramanujan-Hardy-Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer $G$-function of the type $G^{n \ 0}_{0 \ n}$.

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arXiv2022

A modular relation involving non-trivial zeros of the Dedekind zeta function, and the Generalized Riemann Hypothesis

We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for $ζ_K(s)$. New elegant transformations are obtained when $K$ is a quadratic extension, one of which involves the modified Bessel function of the second kind.

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arXiv2020

Joint Extreme values of $L$-functions

We consider $L$-functions $L_1,\ldots,L_k$ from the Selberg class which have polynomial Euler product and satisfy Selberg's orthonormality condition. We show that on every vertical line $s=σ+it$ with $σ\in(1/2,1)$, these $L$-functions simultaneously take large values of size $\exp\left(c\frac{(\log t)^{1-σ}}{\log\log t}\right)$ inside a small neighborhood.

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arXiv2019

Zeros of Dirichlet polynomials

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these functions. We prove that the known non-trivial zero-free half plane for Dirichlet polynomials associated to this class of multiplicative functions is optimal. We also introduce a characterization of elements in this class based on a new parameter depending on the Dirichlet series $F(s) = \sum_{n=1}^\infty f(n) n^{-s}$. In this context, we obtain non-trivial regions in which the associated Dirichlet polynomials do have zeros.

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arXiv2018

Zeros of partial sums of $L$-functions

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of $f$. More precisely, we prove estimates for the sum $\sum_{n=1}^x f(n)/n$ in terms of the size of $|F(1+1/\log x)|$ and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums $F_N(s)$. In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field $K$. More precisely, we give some improved results for the number of zeros up to height $T$ as well as new zero density results for the number of zeros up to height $T$, lying to the right of $\Re(s) =σ$, where $σ> 1/2$.

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