arXiv subjects
Publications and source records attributed to Kamalakshya Mahatab.
We obtain improved omega bounds for the error term in the Piltz divisor problem over number fields. Our proof combines Lamzouri's resonance argument with a counting theorem for nonnegative multiplicative functions. The counting theorem gives the estimates needed for the divisor coefficients over a number field and allows us to remove the factor involving the third iterated logarithm from the earlier bounds.
Let $G$ be a finite group with $t$ conjugacy classes, and let $S_N$ be the symmetric group. Let $Z_t(N)$ be the number of zeros in the character table of the wreath product $G\wr S_N$. We prove \begin{equation*} Z_t(N)\ge \frac{2p_t(N)^{2}}{\log \frac{N}{t}}\left(1+O\left(\frac{\log \log \frac{N}{t}}{\log \frac{N}{t}}\right)\right), \end{equation*} where $p_t(N)$ is the number of $t$-multipartitions of $N$.
Let $K_1,\cdots,K_r$ be fixed number fields, and let $L$ be the compositum of their Galois closures. Assuming GRH for $ζ_L$, 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.
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.
Let $p_t(N)$ denote the number of $t$-multipartitions of a positive integer $N$. In this article, we obtain an asymptotic formula for $p_t(N)$, when $t \ll N^{1 - ε}$, for any $ε> 0$.
For any two partitions $λ$ and $μ$ of a positive integer $N$, let $χ_λ(μ)$ be the value of the irreducible character of the symmetric group $S_{N}$ associated with $λ$, evaluated at the conjugacy class of elements whose cycle type is determined by $μ$. Let $Z(N)$ be the number of zeros in the character table of $S_N$, and $Z_{t}(N)$ be defined as $$ Z_{t}(N):= \#\{(λ,μ): χ_λ(μ) = 0 \; \text{with $λ$ a $t$-core}\}. $$ We prove $$ Z(N) \ge \frac{2\, p(N)^{2}}{\log N} \left(1+O\left(\frac{1}{\sqrt{\log N}}\right)\right), $$ where $p(N)$ denotes the number of partitions of $N$. We also give explicit lower bounds for $Z_t(N)$ in various ranges of $t$.
A partition is $t$-regular if none of its parts is divisible by $t$. Let $p(N,t)$ be the number of $(t+1)$-regular partitions of a positive integer $N$. In 1971, Hagis proved an asymptotic formula for $p(N,t)$ using the circle method, when $t$ fixed. In this article, we use the saddle point method and extend the result of Hagis in different ranges of $t$, obtaining explicit bounds. We also discuss an application of our result to estimate zeros in the character table of the symmetric group.
We apply the resonance method to obtain large values of general exponential sums with positive coefficients. As applications, we show improved $Ω$-bounds for Dirichlet and Piltz divisor problems, Gauss circle Problem, and error term for the mean square of the Riemann zeta function and the Dirichlet $L$-function.
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 $Ω-$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.
Let $S(t)$ denote the argument of the Riemann zeta-function, defined as $$ S(t)=\dfrac{1}π\,\Im\logζ(1/2+it). $$ Assuming the Riemann hypothesis, we prove that $$ S(t)=Ω_{\pm}\bigg(\dfrac{\log t\log\log\log t}{\log\log t}\bigg). $$ This improves the classical omega results of Montgomery and matches with the $Ω$-result obtained by Bondarenko and Seip.
Let $S(σ,t)=\frac{1}π\argζ(σ+it)$ be the argument of the Riemann zeta-function at the point $σ+it$ in the critical strip. For $n\geq 1$ and $t>0$, we define \begin{equation*} S_{n}(σ,t) = \int_0^t S_{n-1}(σ,τ) \,dτ+ δ_{n,σ\,}, \end{equation*} where $δ_{n,σ}$ is a specific constant depending on $σ$ and $n$. Let $0\leq β<1$ be a fixed real number. Assuming the Riemann hypothesis, we establish lower bounds for the maximum of $S_n(σ,t+h)-S_n(σ,t)$ near the critical line, on the interval $T^β\leq t \leq T$ and in a small range of $h$. This improves some results of the first author and generalizes a result of the authors on $S(t)$. We also give new omega results for $S_n(t)$, improving a result by Selberg.
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.
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.
Let $Z(t):=ζ\left(\frac{1}{2}+it\right)χ^{-\frac{1}{2}}\left(\frac{1}{2}+it\right)$ be Hardy's function, where the Riemann zeta function $ζ(s)$ has the functional equation $ζ(s)=χ(s)ζ(1-s)$. We prove that for any $ε>0$, \begin{align*} &\quad\max_{T^{3/4}\leq t\leq T} Z(t) \gg \exp\left(\left(\frac{1}{2}-ε\right)\sqrt{\frac{\log T\log\log\log T}{\log\log T}}\right)\\ \text{ and }& \max_{T^{3/4}\leq t\leq T}- Z(t) \gg \exp\left(\left(\frac{1}{2}-ε\right)\sqrt{\frac{\log T\log\log\log T}{\log\log T}}\right). \end{align*}
In recent years a variant of the resonance method was developed which allowed to obtain improved $Ω$-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper we show how this method can be adapted to prove the existence of large values of $|L(σ, χ)|$ in the range $σ\in (1/2,1]$, and to estimate the proportion of characters for which $|L(σ, χ)|$ is of such a large order. More precisely, for every fixed $σ\in (1/2,1)$ we show that for all sufficiently large $q$ there is a non-principal character $χ$ (mod $q$) such that $\log |L(σ,χ)| \geq C(σ) (\log q)^{1-σ} (\log \log q)^{-σ}$. In the case $σ=1$ we show that there is a non-principal character $χ$ (mod $q$) for which $|L(1,χ)| \geq e^γ\left(\log_2 q + \log_3 q - C \right)$. In both cases, our results essentially match the prediction for the actual order of such extreme values, based on probabilistic models.
We prove that for the Steinhaus Random Variable $z(n)$ \[\mathbb{E}\left(\left|\sum_{n\in E_{N, m}}z(n)\right|^6\right)\asymp |E_{N, m}|^3 \text{ for } m\ll(\log\log N)^{\frac{1}{3}},\] where \[E_{N, m}:=\{1\leq n:Ω(n)=m\}\] and $Ω(n)$ denotes the number of prime factors of $N$.
We estimate the number of composite elements in the $n$-th grade of the group semiring of finite boolean groups. In view of this result we may conjecture that the composites in the semiring of finite groups are thinly dispersed.
We consider fluctuations of error terms $Δ(x)$ appearing in the asymptotic formula for a summatory function of coefficients of the Dirichlet series. These are quantified via $Ω$ and $Ω_{\pm}$ estimates. We obtain $Ω$ bounds for Lebesgue measure of the sets $$\{T\leq x \leq 2T: Δ(x)>λx^α\} \text{ and } \{T\leq x \leq 2T: Δ(x)< -λx^α\}$$ for some $α, λ>0$. Primary aim of this article is to develop a general framework to approach these problems. We rediscover several classical results in general setting with weak assumptions. Moreover, several applications of these methods have been discussed and new results have been obtained for some Dirichlet series.