LAUSR

Let $f$ and $g$ be two distinct newforms with weights $k_1, k_2 \ge 2$ and square-free levels $N_1, N_2 \ge 1$ respectively and which are normalized Hecke eigenforms. Further, for $n\ge 1$, let $ a_f(n)$ and $a_g(n)$ be the $n$-th Fourier-coefficients of $f$ and $g$ respectively. In this article, we investigate the first sign change as well as non-vanishing of the sequence $\{a_f(n)a_g(n) \}_{n \in \N}$. Using Rankin-Selberg method and an idea of Iwaniec, Kohnen and Sengupta, we show that for any $\epsilon>0$, there exists a natural number $$ n \ll_{\epsilon} \left[ \left(1+\frac{k_2-k_1}{2}\right) \left(\frac{k_1+k_2}{2}\right) N\right]^{1+\epsilon} $$ such that $a_f(n)a_g(n) < 0$. Here $N = {\rm{lcm}}[N_1, N_2]$. We then use sub-convexity bound of Kowalski, Michel and Vanderkam to get a stronger result in level aspect. We further study the non-vanishing of the sequence $\{a_f(n)a_g(n) \}_{n \in \N}$ and derive bounds for first non-vanishing term in this sequence. These bounds are much stronger as compared to the bounds for the first sign change. We also show, using ideas of Kowalski-Robert-Wu and Murty-Murty, that there exists a set of primes $S$ of natural density one such that for any prime $p \in S$, the sequence $\{a_f(p^n)a_g(p^m) \}_{n,m \in \N}$ has no zero elements. This improves a recent work of Kumari and Ram Murty. Finally, using $\B$-free numbers, we investigate simultaneous non-vanishing of coefficients of $m$-th symmetric power $L$-functions of non-CM forms in short intervals.