Asymptotic estimates for a class of summatory functions. II (Q2769544)

The authors continue their investigations concerning asymptotic formulae for coefficient sums of certain Dirichlet series in [J. Number Theory 51, 147-168 (1995; Zbl 0823.11052), Acta Arith. 75, 39-69 (1996; Zbl 0846.11054) and Part I, J. Number Theory 70, 1-36 (1998; Zbl 0917.11045)]NEWLINENEWLINENEWLINEDefine the sequence \(a(n)\) by NEWLINE\[NEWLINE \sum_{n=1}^\infty {a(n) \over {n^s}} = \zeta(s) \cdot \prod_{i=1}^k \zeta^{\alpha_i} (rs + \beta +\beta_i) \cdot H(s), NEWLINE\]NEWLINE where \(r\) and the \(\alpha_i\) are positive integers, \(-r+1 < \beta <1\), and \(\beta_1 = 0 < \beta_2 < \dots < \beta_k\), and where NEWLINE\[NEWLINE H(s) = \sum_{n=1} {b(n) \over {n^s}} NEWLINE\]NEWLINE has abscissa of absolute convergence \(\sigma_a(H) < {{1-\beta} \over r}\). NEWLINENEWLINENEWLINEThe authors' main result is an asymptotic formula for \(\sum_{n \leq x} a(n)\) with a good remainder term, NEWLINE\[NEWLINE \sum_{n\leq x} a(n) = A_0 \cdot x + \sum_{i=1}^k x^{ {1-\beta - \beta_i} \over r } \cdot \sum_{n=0} A_{n,i} \log^n x + O \left( x^{\mu + \epsilon} \right) NEWLINE\]NEWLINE with an explicitly given constant \(A_0\) and computable constants \(A_{n,i}\). The exponent \(\mu\) is NEWLINE\[NEWLINE \mu = {1\over r} (1-\beta)(1-\delta), NEWLINE\]NEWLINE where \(\delta\) is the solution of NEWLINE\[NEWLINE {1\over r} (1-\beta)(1-\delta) = \delta - \lambda (1-\delta), \text{ if } \lambda \geq 0, \text{ and } = \delta - \lambda \text{ otherwise}; NEWLINE\]NEWLINE the auxiliary parameter \(\lambda\) is given in the paper. Moreover, some assumptions for \(\sigma_a(H) \) in this theorem are supposed. NEWLINENEWLINENEWLINEThe proof uses Perron's formula. NEWLINENEWLINENEWLINEThe authors give applications to NEWLINE\[NEWLINE \sum_{n\leq x} \Bigl( \sigma_b(n)\Bigr)^m \text{ for } |b|<1, NEWLINE\]NEWLINE where \(\sigma_b(n) = \sum_{d |n} d^b\), and \(m\geq 1\) is an integer, and another application to NEWLINE\[NEWLINE \sum_{n\leq x} \Bigl( \sum_{d|n,\;d^{1/r} \in \mathbb{Z}}d^b \Bigr)^2 .NEWLINE\]