Analytic continuation of Dirichlet L series via nonstandard limits

Let $a_1,a_2,a_3,\ldots$ be a sequence of complex numbers, and let $\{b_n\}$ be its partial sum sequence, assumed to be bounded. Choose a hyperreal number system and define $\lim b_n$ as in Definition 1.5: when it exists, it is the common standard part of $c_N$ for every infinite integer $N$ divisible by any finite integer, where $\{c_n\}$ is the arithmetic average sequence of $\{b_n\}$. Analytic-continuation conjecture. If $\lim b_n$ exists under this definition, and both its existence and its value are independent of the choice of hyperreal number system, then the Dirichlet $L$ series

$$\sum_{n=1}^{\infty}\frac{a_n}{n^s}$$

can be analytically continued to $s=0$, with the single value given by $\lim b_n$. The conjecture proposes a nonstandard-analysis criterion for assigning the value of a potentially divergent Dirichlet series at $s=0$, extending the agreement with analytic continuation known in the special cases discussed earlier in the paper. The general assertion is presented as a technical conjecture, and no resolution is supplied here.