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Let f and g be two distinct newforms which are normalized Hecke eigenforms of weights k1, k2 ≥ 2 and levels N1, N2 ≥ 1 respectively. Also let a f (n) and ag(n) be the n-thFourier-coefficients of f and g respectively. In this article, we investigate the first sign change ofwhere p is a prime number. We further study the nonvanishing of the sequence {a f (n)ag(n)} n∈N and derive bounds for first non-vanishing term in this sequence. 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 ∈ S, the sequence {a f (p n )ag(p m )} n,m∈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.2010 Mathematics Subject Classification. 11F30, 11F11.

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Let F be a Siegel cusp form of weight k and degree n>1 with Fourier‐Jacobi coefficients {ϕm}m∈N. In this article, we investigate the Ramanujan–Petersson conjecture (formulated by Kohnen) for the Petersson norm of ϕm. In particular, we show that this conjecture is true when F is a Hecke eigenform and a Duke–Imamoğlu–Ikeda lift. This generalizes a result of Kohnen and Sengupta. Further, we investigate an omega result and a lower bound for the Petersson norms of ϕm as m→∞. Interestingly, these results are different depending on whether F is a Saito–Kurokawa lift or a Duke–Imamoğlu–Ikeda lift of degree n⩾4.