On the distribution of the sequence of integers $d(n^2)$
Venkatasubbareddy Kampamolla, Sankaranarayanan Ayyadurai
TL;DR
This paper studies the distribution of the divisor-type function $d(n^2)$ under the strong Riemann hypothesis and derives a refined asymptotic for $\sum_{n\le x} d(n^2)$. The authors employ the Dirichlet series $F(s)=\frac{\zeta^3(s)}{\zeta(2s)}$ and Perron’s formula, extracting main terms from the poles at $s=1$ and $s=0$ and from the zeros of $\zeta(2s)$ on $\Re(s)=\tfrac14$, which yields oscillatory terms of the form $x^{1/4+i\gamma}$ accompanied by a leading polynomial in $\log x$. Under the strong RH, they prove an explicit refinement: $\sum_{n\le x} d(n^2) = \mathcal{A}_1 x(\log x)^2 + \mathcal{A}_2 x\log x + \mathcal{A}_3 x + \sum_{\rho} \mathcal{A}_{\gamma} x^{1/4+i\gamma} + O(x^{1/3+\varepsilon})$, with $|\gamma|<x^{2/3}$. The method balances main-term extraction and error control via contour optimization $T\sim x^{2/3}$, extending prior conditional results and clarifying the oscillatory structure dictated by zeros of $\zeta(2s)$.
Abstract
In this paper, we study the distribution of the sequence of integers $d(n^2)$ under the assumption of the strong Riemann hypothesis. Under this assumption, we provide a refined asymptotic formula for the sum $\displaystyle\sum_{n\leq x}d(n^2)$ with an improved error term by extracting some more main terms.