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Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a pole at $1$?

This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above is true, then we get the result.

Related question: What if we assume $a_n$ to be only a bounded sequence of complex numbers?

Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a pole at $1$?

This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above is true, then we get the result.

Related question: What if we assume $a_n$ to be only a bounded sequence of complex numbers?

Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero?

This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above is true, then we get the result.

Related question: What if we assume $a_n$ to be only a bounded sequence of complex numbers?

Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a pole at $1$?

This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above is true, then we get the result.

Related question: What if we assume $a_n$ to be only a bounded sequence of complex numbers?

Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a pole at $1$?

This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above is true, then we get the result.

Related question: What if we assume $a_n$ to be only a bounded sequence of complex numbers?

Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero?

This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above is true, then we get the result.

Related question: What if we assume $a_n$ to be only a bounded sequence of complex numbers?