arXiv subjects
Publications and source records attributed to Mayank Pandey.
We obtain an asymptotic formula for the L^1 norm of the exponential sum $M(α) = \sum_{n\le X}τ(n)e(nα)$ where $τ(n) = \sum_{d|n} 1$ is the divisor function. In particular, we show that it is $\sim C\sqrt{X}\log X$ with $C = \frac{18}{π^3} - \frac{12\log 2}{π^3} - \frac{1}{2π}\approx 0.153\dots$.
We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}τ_k(n)e(nα)\bigg|dα$ where $τ_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for the exponential sum with $τ_k$, and asymptotics for high moments of the Dirichlet kernel.
We use the circle method to obtain tight bounds on the $L^p$ norm of an exponential sum involving the divisor function for $p > 2$.
In this paper, we prove that there are infinitely many primes of the form $\ell^2 - \ell m + m^2$ such that $2\ell - m$ is also prime. To prove this, we follow along the lines of the work of Fouvry and Iwaniec (1997) who showed that there are infinitely many primes of the form $\ell^2 + m^2$ for $\ell$ prime, but use $\mathbb{Z}[ω]$ instead of $\mathbb{Z}[i]$ to work with the bilinear forms that arise in both.
Using sharper Weyl sum estimates, we show that $H(10)\le 105$, improving upon the previous bound of $107$.