Let $\lambda_{sym^{r}f}(n)$ be the $n-$th coefficient in the Dirichlet series representing the symmetric power $L-$function attached to a primitive form $f$ of weight $k$ and level $N$. It is known that $\lambda_{sym^{r}f}(n)$ is a multiplicative function. My question is the following: Can we write $\lambda_{{\rm sym}^rf}(n)$ as a convolution product of two arithmetic functions? And what are explicitly those functions ?
$\begingroup$ $\endgroup$
4 
- 2$\begingroup$ If you mean Dirichlet convolution then I believe the answer to your question is yes. $\endgroup$Steven Clark– Steven Clark2024-12-26 19:24:00 +00:00Commented Dec 26, 2024 at 19:24
- $\begingroup$ Yes it is and what are these functions? $\endgroup$Khadija Mbarki– Khadija Mbarki2024-12-26 19:41:32 +00:00Commented Dec 26, 2024 at 19:41
- 4$\begingroup$ Every arithmetic function $f$ can be decomposed into a convolution. For example, $f=1\ast g$, where $g:=\mu\ast f$. So you need to be more specific about what you want. $\endgroup$GH from MO– GH from MO2024-12-27 08:49:20 +00:00Commented Dec 27, 2024 at 8:49
- 1$\begingroup$ If $b(1)\ne0$ and $b^{-1}(n)$ is the Dirichlet inverse of $b(n)$, then $a(n)$ can be evaluated as the Dirichlet convolution $a(n)=c(n) * b(n)$ where $c(n)$ is the Dirichlet convolution $c(n)=a(n) * b^{-1}(n)$ . There is no requirement for $a(n)$ to be multiplicative. $\endgroup$Steven Clark– Steven Clark2024-12-27 14:48:46 +00:00Commented Dec 27, 2024 at 14:48
Add a comment |