Sign changes in Fourier coefficients of the symmetric power $L$-functions on sums of two squares
Amrinder Kaur
TL;DR
The paper studies sign changes in the $m$th Fourier coefficients $\lambda_{\text{sym}^j f}(m)$ of the $j$th symmetric power $L$-function attached to a normalized Hecke eigen cusp form $f$, restricted to indices $m$ that are sums of two squares, i.e., $m=c^2+d^2$. Using subconvexity bounds for twisted and untwisted symmetric power $L$-functions and a Dirichlet-series decomposition $F(s)=G(s)H(s)$ for $F(s)=\sum_{m\ge1} \frac{\lambda_{\text{sym}^j f}^2(m) r(m)}{m^s}$, they obtain asymptotics and error terms for sums $\sum_{m\le x} \lambda_{\text{sym}^j f}(m) r_2(m)$ and $\sum_{m\le x} \lambda_{\text{sym}^j f}^2(m) r_2(m)$. A weighted Meher–Murty sign-change framework then yields a quantitative lower bound on sign changes: for $f\in H_k$ and $j\ge 2$, there are at least $x^{1-\delta_j}$ sign changes in the subsequence $\{ \lambda_{\text{sym}^j f}(m) \mid m\le x, olinebreak[4] m=c^2+d^2 \}$ with $\delta_j$ in $\left(\frac{21j^2+42j+19}{21j^2+42j+40},1\right)$. The results extend sign-change phenomena to higher-degree automorphic $L$-functions on sparse subsequences under standard automorphy assumptions (Newton–Thorne) and Ramanujan-type conjectures, highlighting robustness of sign changes in symmetric power families.
Abstract
Let $f$ be a normalized primitive Hecke eigen cusp form of even integral weight $k$ for the full modular group $SL(2,\mathbb{Z})$. For integers $j \geq 2$, let $λ_{sym^j f}(m)$ denote the $m$th Fourier coefficient of the $j$th symmetric power $L$-function associated with $f$. We give a quantitative result on the number of sign changes of $λ_{sym^j f}(m)$ for the indices $m$ that are the sum of two squares in the interval $[1,x]$ for sufficiently large $x$.