@article{Liu2024, 
author = {Huafeng Liu and Rui Liu},
title = {The sum of a hybrid arithmetic function over a sparse sequence},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {2},
pages = {4830-4843},
keywords = {Fourier coefficients, cusp forms, automorphic L-function},
url = {https://www.sciopen.com/article/10.3934/math.2024234},
doi = {10.3934/math.2024234},
abstract = {Let        λ          f        (  n  ) be the    n-th normalized Fourier coefficient of    f, which is a primitive holomorphic cusp form of even integral weight    k  ≥  2 for the full modular group    S      L    2    (      Z    ). Let also    σ  (  n  ) and    ϕ  (  n  ) be the sum-of-divisors function and the Euler totient function, respectively. In this paper, we are able to establish the asymptotic formula of the sum of the hybrid arithmetic function        λ          f              l        (  n  )      σ          c        (  n  )      ϕ          d        (  n  ) over the sparse sequence    {  n  :  n  =      a    2    +      b    2    }, namely,        ∑          n      ≤      x            λ          f              l        (  n  )      σ          c        (  n  )      ϕ          d        (  n  )      r    2    (  n  ) for    1  ≤  l  ≤  8, where    x is a sufficiently large real number, the function        r    2    (  n  ) denotes the number of representations of    n as    n  =      a    2    +      b    2  ,    a  ,  b  ,  l  ∈      Z   and    c  ,  d  ∈      R  .}
}