Fomenko [8] proved that n≤x λ sym 2 f (n)x 1/2 (log x) 2 .Since then, this sum has been studied by many scholars, including Sankaranarayanan, Ichihara, Tang, Lü and Jiang (see [31,12,16,33,28]). The higher degree cases were considered by Lau and Lü [25] and Tang and Wu [35]. In particular, under the assumption that L(sym j f, s) is automorphic for j ≥ 1, Tang and Wu showed thatwhere δ j is given by [35, (1.14)].On the other hand, Fomenko [9] studied the sum of λ 2 sym 2 f (n). Later, this result was improved by Tang [34]. X. G. He [11] where δ * 2 = 13 17 , δ * 3 = 248 269 and P 2 (t) is a polynomial in t of degree 2. Lately, Sankaranarayanan, Singh and Srinivas [32] showed that n≤x λ 2 sym 3 f (n) = c 1 x + O(x 15 17 +ε ), n≤x λ 2 sym 4 f (n) = c 2 x + O(x 12 13 +ε ).
Let f and g be two distinct holomorphic cusp forms for S L 2 ℤ , and we write λ f n and λ g n for their corresponding Hecke eigenvalues. Firstly, we study the behavior of the signs of the sequences λ f p λ f p j for any even positive integer j . Moreover, we obtain the analytic density for the set of primes where the product λ f p i λ f p j is strictly less than λ g p i λ g p j . Finally, we investigate the distribution of linear combinations of λ f p j and λ g p j in a given interval. These results generalize previous ones.
The generalized rook monoid is introduced.The order of the generalized rook monoid is calculated and description of the generalized rook monoid is obtatined by matrices.
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