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Nilmoni Karak

Publications and source records attributed to Nilmoni Karak.

5 recordsLinked to original sources
arXiv2026

Sharp lower bounds for shifted moments of Dedekind zeta functions

Let $K_1,\cdots,K_r$ be fixed number fields, and let $L$ be the compositum of their Galois closures. Assuming GRH for $\zeta_L$, we prove a sharp lower bound for products of shifted Dedekind zeta functions on the critical line, for arbitrary fixed positive real exponents and uniformly for shifts of size at most $T/2$. The correlation factor is expressed as a product of Dedekind zeta functions of the fixed fields of double-coset stabilisers in $\textrm{Gal}(L/\mathbb{Q})$. Combined with the corresponding upper bound by the authors, determines the order of magnitude of these shifted moments for both Galois and non-Galois fields.

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arXiv2026

Sharp Upper Bounds for Moments of Dedekind Zeta Functions

Assuming the Generalised Riemann Hypothesis, we establish conjecturally sharp upper bounds for shifted moments of products of Dedekind zeta functions of arbitrary number fields. This improves results of Milinovich and Turnage-Butterbaugh and extends a recent result of Hagen. As applications, we obtain mean-square bounds for short-interval sums of the coefficients of Dedekind zeta functions and upper bounds for the large deviations of Dedekind zeta functions. Our results apply to both Galois and non-Galois extensions.

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arXiv2025

The Piltz divisor Problem in Number Fields Using The Resonance Method

The Piltz divisor problem is a natural generalization of the classical Dirichlet divisor problem. In this paper, we study this problem over number fields and obtain improved $\Omega-$bounds for its error terms. Our approach involves generalizing a Voronoi-type formula due to Soundararajan in the number field setting, and applying a recent result due to the second author.

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arXiv2023

A Dirichlet character analogue of Ramanujan's formula for odd zeta values

In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, $$ \sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1}, $$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$. Recently, Dixit and the last author pointed out that this series has already been present in the Lost Notebook of Ramanujan with a more general form. Although, Ramanujan did not provide any transformation identity for it. In the same paper, Dixit and the last author found an elegant generalization of Ramanujan's celebrated identity for $ζ(2m+1)$ while extending the results of Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored another extended version of the aforementioned series, namely, $$\sum_{r=1}^{q}\sum_{n=1}^{\infty} \frac{χ(r)n^{N-2h}{\exp\left(-\frac{r}{q}n^N x\right)}}{1-\exp({-n^N x})},$$ where $χ$ denotes a Dirichlet character modulo $q$, $N\in 2\mathbb{N}$ and with some restriction on the variable $h$. In the current paper, we investigate the above series for {\it any} $N \in \mathbb{N}$ and $h \in \mathbb{Z}$. We obtain a Dirichlet character analogue of Dixit and the last author's identity and there by derive a two variable generalization of Ramanujan's identity for $ζ(2m+1)$. Moreover, we establish a new identity for $L(1/3, χ)$ analogous to Ramanujan's famous identity for $ζ(1/2)$.

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