An \(\Omega\)-result for the difference of the coefficients of two \(L\)-functions (Q2910116)

Let \(a_F(n)\), \(n\in\mathbb{N}\), denote the coefficients of a Dirichlet series defining, for \(\Re s =:\sigma>1\), a function \(F(s)\) from the extended Selberg class \(\mathcal{S}^\sharp\). Moreover, \(d_F\) denotes the degree of \(F(s)\), \(\sigma_a(F)\) denotes the abscissa of absolute convergence and NEWLINE\[NEWLINE A_F(x) = \sum_{n\leq x}a_F(n) = \text{res}_{s=1}F(s)\frac{x^s}{s}+R_F(x). NEWLINE\]NEWLINE It is known that \(R_F(x)=\Omega(x^{1/2-1/2d_F})\) for any \(F\in\mathcal{S}^\sharp\).NEWLINENEWLINEIn the reviewed paper the authors deal with \(\Omega\)-results for NEWLINE\[NEWLINE \delta(F,G) = \limsup_{x\to 0^+}\frac{\log(1+\sum_{n=1}^\infty|a_F(n)-a_G(n)|e^{-nx})}{\log(1/x)}\qquad (F,G\in\mathcal{S}^\sharp). NEWLINE\]NEWLINE As the main result of the paper they prove that NEWLINE\[NEWLINE \delta(F,G)\geq \frac{1}{2}+\frac{1}{2}\min\left(\frac{1}{d_F},\frac{1}{d_G}\right), NEWLINE\]NEWLINE if \(F,G\in\mathcal{S}^\sharp\) with \(d_F,d_G>0\).NEWLINENEWLINETherefore, on can deduce that NEWLINE\[NEWLINE\sum_{n\leq x}|a_F(n)-a_G(n)| = \Omega\left(\left(\frac{x}{\log x}\right)^{\frac{1}{2}+\frac{1}{2}\min\left(\frac{1}{d_F},\frac{1}{d_G}\right)}\right). NEWLINE\]NEWLINE What is more, it turns out that the ``min'' in the lower bound of \(\delta(F,G)\) can be replaced by ``max'', provided \(F,G\) are functions from the Selberg class \(\mathcal{S}\). Under this restriction, the authors prove as well that NEWLINE\[NEWLINE\sum_{n\leq x}|a_F(n)-a_G(n)| = \Omega\left(x^{1-\varepsilon}\right)NEWLINE\]NEWLINE for every \(\varepsilon>0\) and all \(F,G\in\mathcal{S}\) with \(\max(d_F,d_G)>0\) if and only if \(\sigma_a(F)=1\) for all \(F\in\mathcal{S}\), \(F\neq 1\).