Can bijection between natural numbers and prime numbers be holomorphically extended to the complex plane?

Question

It is well known that we can map $n\in\mathbb N$ to the $n$-th prime number, can this map be extended holomorphically to the complex plane? If so, are there any good properties of it?

It seems an interesting question but I did not search anything related to it. Thanks!

@AnginaSeng are there? Can you give an example? I am out of complex-analysis business and don’t think this is obvious... — Jonas Linssen, Jun 07 '20 at 09:47
Let $f$ be your function and consider $f(z)+g(z)\sin\pi z$ for any entire $g$. — Angina Seng, Jun 07 '20 at 09:50
For any complex sequence $a_n\to \infty$ and any complex sequence $b_n$ there exists an entire function $f(z)$ satisfying $f(a_n)=b_n$, see Entire function with prescribed values, indeed, infinitely many of them. One can even write an explicit series for it. In your case $a_n=n$ and $b_n=p_n$. But there is no such function that gives us anything interesting, as far as I know. — Conifold, Jun 07 '20 at 09:52

score 1 · Accepted Answer · answered Jun 07 '20 at 15:01

$$e^{x} \sum_{n\ge 1} \frac{p_n}{e^n}\frac{\sin(\pi (x-n))}{\pi (x-n)}$$ It is entire because the summand are entire and the series converges locally uniformly.

The same method works for any sequence, all we have to do is to replace $e^x, 1/e^n$ by $f(x),1/f(n)$ where $f$ is an entire function growing fast enough.

Can bijection between natural numbers and prime numbers be holomorphically extended to the complex plane?

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