Soumyarup Banerjee
In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï's formula concerning the error term in the Dirichlet divisor problem.
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We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this function for many cases. Our proof uses the estimate of exponential sums. We also show uniform results for ideal counting function and relatively $r$-prime lattice points as one of applications.
The Piltz divisor Problem in Number Fields Using The Resonance Method
本文在数域上研究Piltz除数问题，通过推广Soundararajan的Voronoi型公式并应用第二作者的近期结果，成功获得了误差项改进的$\Omega-$界。
Omega bound for the Piltz divisor problem
本文通过结合共振论证与计数定理，在数域上改进了Piltz除数问题误差项的omega界，并去除了先前界中的三次迭代对数因子。
Ergodic averages with the Hecke eigenvalue square weights and the Piltz divisor function weights
本文证明Hecke特征值平方与Piltz除数函数可作为逐点遍历定理的良好权重，部分解决Cuny和Weber的问题，并推广至其他算术函数。
On the Lindelöf Hypothesis for the Riemann Zeta function and Piltz divisor problem
本文通过给出临界带内ζᵏ(s)的傅里叶展开推导林德勒夫假设的充要条件，并针对皮尔茨除数问题的误差项Δₖ(x)，得到了几乎所有x≥1时其与傅里叶系数及拉盖尔多项式相关的极限表达式。
On the sum of $Δ_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$
本文研究Piltz除数问题中误差项$\Delta_{k}(n)$的Dirichlet级数解析性质，利用Perron公式估计$k=3,4$时误差项的部分和。
Omega Results for The Divisor and Circle Problems Using The Resonance Method
该研究将共振法用于获取带正系数的一般指数和的大值，改进了数论中多个经典问题的Ω界，包括除数问题、圆问题及zeta函数和L函数的均方误差问题。
Sign changes of the error term in the Piltz divisor problem
本文研究Piltz除数问题误差项Δ_k(x)的符号变化，在Lindelöf假设或黎曼假设下证明其存在大量子区间使误差项保持较大，推广了k=2和k=3情形的已有结果。
In this paper we study the problem of the first moment of the Dedekind zeta function of a number field $K$ and improve the error term. As a ready generalization of our proof, we improve the error term in the Piltz divisor problem.
In this note, we establish a Voronoï--Oppenheim summation formula for divisor functions over an arbitrary number field.
We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …
It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{X-H}^{X+H}Δ^4_k(x)\d x \ll_εX^ε\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $Δ_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{\"ı}--type formula for $Δ_k(x)$, and the bound of Robert--Sargos for the number of integers when the difference of four $k$--th roots is small. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $Δ_2(x)$.
Sharp omega results for the divisor and circle problems
本文通过建立猜想中sharp的omega结果，并确定所获得的大值的符号，改进了Soundararajan以及Sourmelidis和Mahatab后续的独立改进，首次提升了Hafner在1981年对除数问题的Ω_+结果和圆问题的Ω_-结果。主要的新成分是一种共振方法，它直接与Voronoï求和公式中出现的相位工作，通过将通常的正核替换为单侧扇形核，即Gamma分布的密度，其傅里叶变换位于复平面上合适的扇形内。
In this article, we study a divisor function in an arbitrary number field akin to Koshliakov's work on Vorono\"{\dotlessi} summation formula. More precisely, we generalize Koshliakov's kernel and Koshliakov's transform over any number field to obtain identities for the Lambert series associated to the divisor function in an arbitrary number field.
A weighted divisor problem and exponential sum
本文针对 $ζ'(s)^2$ 的 Dirichlet 级数系数 $D_{(1)}(n)$ 的加权指数和，建立截断 Voronoï 型公式，并由此得到误差项的均方估计，同时研究了相应 Riesz 和及其误差项的均方估计。
We focus on three pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -- each is vitiated by divergent series. We concentrate on only one of the two incorrect "identities", which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of …
The mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip
本文改进了Matsumoto和Ishikawa的工作，通过获得Atkinson型误差项显式公式，给出了临界带内黎曼zeta函数与Dirichlet多项式乘积均方值的渐近公式，与广义Dirichlet除数问题紧密相关。
The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $σ_{\mathbb{K},α}(n)$. The Dirichlet series associated to this function is $ζ_{\mathbb{K}}(s)ζ_{\mathbb{K}}(s-α)$. We give an expression for the Riesz sum associated to $σ_{\mathbb{K},α}(n),$ and also extend the validity of this formula by using convergence theorems. As a special case, when $\mathbb{K}=\mathbb{Q}$, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for $α=0$, gives the Vorono\"ı summation formula associated to the d…
The shifted convolution problem in function fields
本文研究函数域中除数函数的移位卷积问题，证明了特定次数范围内的渐近公式，还研究了Dirichlet特征卷积的相关性，并首次得到函数域上的Voronoi求和公式。
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