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Q1 Is there real or complex analytic function $f(x)$ such that its positive real zeros are the primes and it is given in closed form of compositions of already named functions and elementary operations?

If we drop the analytic constraint, from the linked question, $\sin(\pi x)$ vanishes at the integers and via sums of squares we get the function $P(x)=\sin^2{(\pi x)} + \sin^2{( \pi(\Gamma(x)+1) / x)}$ whose positive real zeros are the primes.

Q2 Can we use counting of zeros to count primes in an interval?

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    $\begingroup$ An entire function whose zeros are the primes should be $\prod_{n\ge1}\big(1-\frac z{p_n}\big)\exp\big(\frac z{p_n}\big)$, but I don't think it is related to any elementary function. $\endgroup$
    – Pietro Majer
    Commented Oct 18, 2022 at 12:15
  • $\begingroup$ @PietroMajer Thanks. If we drop the analytic constraint, how do you explain the elementary form of $P(x)$? $\endgroup$
    – joro
    Commented Oct 18, 2022 at 12:19
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    $\begingroup$ Can you clarify the question? Certainly for positive real $x$ this $P(x)$ vanishes only if $x$ is an integer and $x$ divides $(x-1)!+1$, which by Wilson's theorem is true exactly for primes $x$ (as said in the linked question) $\endgroup$
    – Pietro Majer
    Commented Oct 18, 2022 at 12:31
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    $\begingroup$ @PietroMajer Yes, I meant this. I asked why such closed form exists, if we don't have other constraints on $f$. $\endgroup$
    – joro
    Commented Oct 18, 2022 at 12:36
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    $\begingroup$ @joro That infinite product expansion follows from Hadamard's factorization theorem, which allows you to construct entire functions with prescribed zeros. $\endgroup$
    – TravorLZH
    Commented Oct 20, 2022 at 16:25

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H. Laurent introduced the function

$$f(z)=\sum_{n=1}^\infty(\sin\pi z)^2\left(\frac{1}{n^2\sin\frac{\pi z}{n}}-\frac{1}{\pi n (n-z)}\right)^2$$

whose real roots are the positive primes and have no negative zeros. (Interméd. Math. 5 (1898) p. 78; 15 (1908) p. 265 [Question 1263]. With a rectification by P. Fatou (Interméd. Math. 16 (1909) p. 248)

I read this on a French translation of a German Encyclopedia that I have not at this moment to give his complete reference.

There is an amusing history of this Encyclopedia here at the University of Sevilla. The hole Encyclopedia and his French translation was acquired in the first years after the Civil War in Spain by the then only professor of mathematics in Seville, Mr. Patricio Peñalver. His successor was my thesis director, D Antonio de Castro Brzezicki. The Encyclopedia was in a cabinet with doors. One day when D. Antonio wanted to consult it, he noticed that it had diminished considerably. The maid was earning a bonus by selling the encyclopedia by weight as paper. Curiously, the paper was very light, the kind that yellows with time.

we should not be too critical of the cleaner, hunger is very bad, and the post-war years were terrible in Spain.

One of the volumes that was saved was the translation of P. Bachman's original by J. Hadamard and E. Maillet on "Propositions transcendentes de la théorie des nombres". Of which I have a photocopy at home.

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    $\begingroup$ Thanks. I asked about closed form, not series, but your answer is interesting. $\endgroup$
    – joro
    Commented Oct 19, 2022 at 13:19

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