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In this paper we consider the mean value of hybrid arithmetic function of the form λ6f (n)σb(n)Øc(n), where λf (n) is the n-th Fourier coefficients of the holomorphic cusp form f, σ(n) is the sum-of-divisors function, Ø(n) is the Euler's totient function. In detail, we prove that for any e > 0, ∑ n⩽x λ6f (n)σb(n)Øc(n) = x(b+c+1)P4(log x) + O(xb+c+631/652+ε); where b; c ∊ ℝ. [ABSTRACT FROM AUTHOR]

Let f( z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λ( n). Let d( n) be the Dirichlet divisor function. In this paper, we establish that where $${0 < \delta_{j} < 1}$$ , $${0 < \delta_{i,j}\leqq 1}$$ ( $${\delta_{i,j} = 1}$$ if and only if i = j), and . [ABSTRACT FROM AUTHOR]