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Let $R$ be a graded ring and $n\ge 1$ an integer. We introduce and study $n$-strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that $n$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be $m$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever $n>m$. Many properties of the $n$-strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate...

We give a simple proof that critical values of any Artin $L$-function attached to a representation $\rho$ with character ${\chi }_{\rho }$ are stable under twisting by a totally even character $\chi$, up to the $dim\rho$-th power of the Gauss sum related to $\chi$ and an element in the field generated by the values of ${\chi }_{\rho }$ and $\chi$ over $\mathbb{Q}$. This extends a result of Coates and Lichtenbaum as well as the previous work of Ward.