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I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some mathematical object has some property, I like to know whether I'm working to prove that the object I'm looking has some strange property or if I'm working to prove that it's normal and the numbers aren't just conspiring against me.

For example, when trying to prove whether a number is irrational or not, I know that $100\%$ of numbers are irrational and so I'm trying to show that I didn't happen to pick one of those $0\%$ of numbers.

Sadly, I have no such intuition for analytic continuation. I think that my guess would be that either $100\%$ of $0\%$ of Dirichlet series have analytic continuations, but I could be wrong. To make my question more concrete,

If $\{a_n\}$ is a sequence of complex numbers chosen uniformly randomly in the unit disk, and $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is it's Dirichlet series, what is the probability that $F(s)$ has an analytic continuation past the the line $\Re(s)=1$ (not necessarily to the entire plane).

Answers to variants of this question are also greatly appreciated, like if $a_n$ is chosen uniformly randomly of $[0,1]$ or if we are looking for continuations to the entire complex plane.

EDIT: If it's too complicated to analyse analytic continuation, what about meromorphic ones? Should I expect functions to have meromorphic continuations to $\Re(s)=1$?

I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some mathematical object has some property, I like to know whether I'm working to prove that the object I'm looking has some strange property or if I'm working to prove that it's normal and the numbers aren't just conspiring against me.

For example, when trying to prove whether a number is irrational or not, I know that $100\%$ of numbers are irrational and so I'm trying to show that I didn't happen to pick one of those $0\%$ of numbers.

Sadly, I have no such intuition for analytic continuation. I think that my guess would be that either $100\%$ or $0\%$ of Dirichlet series have analytic continuations, but I could be wrong. To make my question more concrete,

If $\{a_n\}$ is a sequence of complex numbers chosen uniformly randomly in the unit disk, and $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is its Dirichlet series, what is the probability that $F(s)$ has an analytic continuation past the the line $\Re(s)=1$ (not necessarily to the entire plane).

Answers to variants of this question are also greatly appreciated, like if $a_n$ is chosen uniformly randomly of $[0,1]$ or if we are looking for continuations to the entire complex plane.

EDIT: If it's too complicated to analyse analytic continuation, what about meromorphic ones? Should I expect functions to have meromorphic continuations to $\Re(s)=1$?

I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some mathematical object has some property, I like to know whether I'm working to prove that the object I'm looking has some strange property or if I'm working to prove that it's normal and the numbers aren't just conspiring against me.

For example, when trying to prove whether a number is irrational or not, I know that $100\%$ of numbers are irrational and so I'm trying to show that I didn't happen to pick one of those $0\%$ of numbers.

Sadly, I have no such intuition for analytic continuation. I think that my guess would be that either $100\%$ of $0\%$ of Dirichlet series have analytic continuations, but I could be wrong. To make my question more concrete,

If $\{a_n\}$ is a sequence of complex numbers chosen uniformly randomly in the unit disk, and $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is it's Dirichlet series, what is the probability that $F(s)$ has an analytic past the the line $\Re(s)=1$ (not necessarily to the entire plane).

Answers to variants of this question are also greatly appreciated, like if $a_n$ is chosen uniformly randomly of $[0,1]$ or if we are looking for continuations to the entire complex plane.

EDIT: If it's too complicated to analyse analytic continuation, what about meromorphic ones? Should I expect functions to have meromorphic continuations to $\Re(s)=1$?

I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some mathematical object has some property, I like to know whether I'm working to prove that the object I'm looking has some strange property or if I'm working to prove that it's normal and the numbers aren't just conspiring against me.

For example, when trying to prove whether a number is irrational or not, I know that $100\%$ of numbers are irrational and so I'm trying to show that I didn't happen to pick one of those $0\%$ of numbers.

Sadly, I have no such intuition for analytic continuation. I think that my guess would be that either $100\%$ of $0\%$ of Dirichlet series have analytic continuations, but I could be wrong. To make my question more concrete,

If $\{a_n\}$ is a sequence of complex numbers chosen uniformly randomly in the unit disk, and $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is it's Dirichlet series, what is the probability that $F(s)$ has an analytic continuation past the the line $\Re(s)=1$ (not necessarily to the entire plane).

Answers to variants of this question are also greatly appreciated, like if $a_n$ is chosen uniformly randomly of $[0,1]$ or if we are looking for continuations to the entire complex plane.

EDIT: If it's too complicated to analyse analytic continuation, what about meromorphic ones? Should I expect functions to have meromorphic continuations to $\Re(s)=1$?

I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some mathematical object has some property, I like to know whether I'm working to prove that the object I'm looking has some strange property or if I'm working to prove that it's normal and the numbers aren't just conspiring against me.

For example, when trying to prove whether a number is irrational or not, I know that $100\%$ of numbers are irrational and so I'm trying to show that I didn't happen to pick one of those $0\%$ of numbers.

Sadly, I have no such intuition for analytic continuation. I think that my guess would be that either $100\%$ of $0\%$ of Dirichlet series have analytic continuations, but I could be wrong. To make my question more concrete,

If $\{a_n\}$ is a sequence of complex numbers chosen uniformly randomly in the unit disk, and $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is it's Dirichlet series, what is the probability that $F(s)$ has an analytic past the the line $\Re(s)=1$ (not necessarily to the entire plane).

Answers to variants of this question are also greatly appreciated, like if $a_n$ is chosen uniformly randomly of $[0,1]$ or if we are looking for continuations to the entire complex plane.

I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some mathematical object has some property, I like to know whether I'm working to prove that the object I'm looking has some strange property or if I'm working to prove that it's normal and the numbers aren't just conspiring against me.

For example, when trying to prove whether a number is irrational or not, I know that $100\%$ of numbers are irrational and so I'm trying to show that I didn't happen to pick one of those $0\%$ of numbers.

Sadly, I have no such intuition for analytic continuation. I think that my guess would be that either $100\%$ of $0\%$ of Dirichlet series have analytic continuations, but I could be wrong. To make my question more concrete,

If $\{a_n\}$ is a sequence of complex numbers chosen uniformly randomly in the unit disk, and $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ is it's Dirichlet series, what is the probability that $F(s)$ has an analytic past the the line $\Re(s)=1$ (not necessarily to the entire plane).

Answers to variants of this question are also greatly appreciated, like if $a_n$ is chosen uniformly randomly of $[0,1]$ or if we are looking for continuations to the entire complex plane.

EDIT: If it's too complicated to analyse analytic continuation, what about meromorphic ones? Should I expect functions to have meromorphic continuations to $\Re(s)=1$?