Sums of coefficients of general \(L\)-functions over arithmetic progressions and applications (Q6613261)

For \(\mathcal{A}=(a_n)_{n\geq 1}\), a sequence of complex numbers with terms that are not all zero, the author considers the general \(L\)-function \(L(s,\mathcal{A})\) of degree \(d\) as follows\N\[\NL(s,\mathcal{A})=\prod_p\prod_{j=1}^d\left(1-\frac{\alpha_j(p)}{p^s}\right)^{-1}=\sum_{n=1}^\infty\frac{a_n}{n^s},\N\]\Nwhere the Dirichlet series and Euler product are convergent absolutely for \(\Re(s)>1\), and \(L(s,\mathcal{A})\) satisfies several conditions including analytic continuation, functional equation, and growth limitation. The author gives approximation for the sum \(\widehat{\sum}a_n\), where \(\widehat{\sum}\) means that the sum running over \(n\leq x\) and \(n\equiv a\pmod{q}\). For the case \(a_n\geq 0\), his result takes more simpler form. He also gives results related to the more general decomposition of \(L\)-functions. As an application he considers \(\lambda_f(n)\), the Fourier coefficients of holomorphic forms or Maass forms for the full modular group \(\mathrm{SL}(2, \mathbb{Z})\) over arithmetic progressions, and approximates the sums \(\widehat{\sum}\lambda_f(n)^{2j}\) for \(j=2,3,4\) and \(f\in S_r\), where \(S_r\) is the set of arithmetically normalized primitive Maass cusp forms of Laplace eigenvalue \(\lambda=1/4 + r^2\), and the sums \(\widehat{\sum}\lambda_f(n)^{j}\) for \(j\geq 3\) and \(f\in H_k\), where \(H_k\) denote the set of arithmetically normalized primitive holomorphic cusp forms of even integral weight \(k\) for \(\mathrm{SL}(2, \mathbb{Z})\).