If $P(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\le k, k\le 1,$ Rather, Gulzar and Ahangar [8] proved that for every $\alpha \in \mathbb{C}$ with $|\alpha|\ge k$ and $\gamma >0,$ \[ n(|\alpha|-k)\Bigg\{\int_0^{2\pi}\Big|\frac{P(e^{i\theta})}{D_{\alpha}P(e^{i\theta})}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}&\leq \Bigg\{\int_0^{2\pi}\Big|1+ke^{i\theta}\Big|^\gamma d\theta\Bigg\}^\frac{1}{\gamma}. \] In this paper, we shall obtain a result which generalizes and sharpens the above inequality by obtaining a bound that depends upon the location of all the zeros of $P(z)$ rather than just on the location of the zero of largest modulus.

We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve with an ``additive twist'' and provide asymptotic formulas for the same. As an application of this higher rank sieve, we obtain improvements of results of Heath-Brown and Ho-Tsang on almost prime $k$-tuples.

Let $F(z)$ be a Hecke-Maass form for $SL(m,\mathbb{Z})$ and $A_F(n,1, \dots, 1)$ be the coefficients of $L$-function attached to $F.$ We study the cancellation of $A_F(n,1, \dots, 1)$ for twisted with a nonlinear exponential function at primes, namely the sum \begin{equation*} \sum_{n \leq N} \Lambda (n)A_F(n,1, \dots, 1)e ( \alpha n^\theta ), \end{equation*} where $0<\theta<2/m$. We also strengthen the corresponding previous results for holomorphic cusp forms for $SL(2,\mathbb{Z}),$ and improve the estimates of Ren-Ye on the resonance of exponential sums involving Fourier coefficients of a Maass form for $SL(m,\mathbb{Z})$.

We find bases for the spaces $M_2\Big(\Gamma_0(24),(\frac{d}{\cdot})\Big)$ ($d=1,8,12, 24$) of modular forms. We determine the Fourier coefficients of all $35$ theta products $\varphi[a_1,a_2,a_3,a_4](z)$ in these spaces. We then deduce formulas for the number of representations of a positive integer $n$ by diagonal quaternary quadratic forms with coefficients $1$, $2$, $3$ or $6$ in a uniform manner, of which $14$ are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces $E_2\Big(\Gamma_0(24),(\frac{d}{\cdot})\Big)$ ($d=1,8,12, 24$) and give their Fourier coefficients.

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).

We derive a truncated Voronoi identity for rationally additively twisted sums of Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$, and as an application obtain a pointwise estimate and a second moment estimate for the sums in question.

The aim of this note is to show that any even perfect number, other than $6$, can be written as the sum of at most five positive integral cubes. We also conjecture that any such number can even be written as the sum of at most three positive integral cubes.