The coefficient of Selberg Class L-function satisfy:


$a\_n <M\_{\epsilon} n^{\epsilon}$ (for any $\epsilon >0$) and the $a\_n$ are multiplicative.


So I would like to know if it can be shown that we also have the following partial sum bounded by a constant (maybe using also other properties of $a\_n$ coefficients):


$\sum\_{n=1}^{2N} a\_n - a\_2 a\_n < M$ 


(As $a\_n$ are multiplicative, we see that some terms disappear from the sum as $a\_{2k}=a\_2 a\_k$ for k odd, but is there a chance to have this sum bounded by a constant?)


More generally what can you advise to read on the characterisation of coefficent of Selberg Class L-function?



On properties of coefficients of Selberg Class L-function
I think this fails for the Dirichlet coefficients of $L(s):=\zeta(s+i\gamma)\zeta(s-i\gamma)$ whenever $\gamma>0$ satisfies $2^{i\gamma}+2^{-i\gamma}\neq 1$. Indeed, for such $\gamma$, the inequality would imply that $S(x):=\sum\_{n\leq x}a\_n$ is bounded by a constant either from below or from above. However, $L(s)$ has simple poles at $1\pm i\gamma$ and it is holomorphic elsewhere, hence a quantitative form of Landau's oscillation theorem yields $S(x)=\Omega\_{\pm}(x)$. See Theorem 11.8 in Bateman-Diamond: Analytic number theory - An introductory course.


