On high power sums of a hybrid arithmetic function

Abstract Let λ f ( n ), σ ( n ) and ϕ ( n ) denote n th Fourier coefficient of the primitive holomorphic cusp form f , the sum-of-divisors function, and the Euler totient function, respectively. In this paper, we first refine related previous results on the power sum <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable columnalign="center" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:munder> <m:mo movablelimits="false">∑</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>x</m:mi> </m:mrow> </m:munder> <m:msubsup> <m:mi>λ</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>f</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>j</m:mi> </m:mrow> </m:msubsup> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>σ</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>b</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>ϕ</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>c</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle \sum\limits_{n\leq x} \lambda_{f}^{j}(n)\sigma^{b}(n)\phi^{c}(n) \end{array}$ and then investigate the average behavior of the power sum <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable columnalign="center" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:munder> <m:mo movablelimits="false">∑</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mtable rowspacing="0.1em" columnspacing="0em 0em 0em 0em"> <m:mtr> <m:mtd> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>c</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>d</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>≤</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>∈</m:mo> <m:msup> <m:mrow class="MJX-TeXAtom-ORD"> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> </m:mrow> <m:mn>4</m:mn> </m:msup> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:munder> <m:msubsup> <m:mi>λ</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>f</m:mi> </m:mrow> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>j</m:mi> </m:mrow> </m:msubsup> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>σ</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>b</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>ϕ</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>c</m:mi> </m:mrow> </m:msup> <m:mo stretc