MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am wondering what real and complex analysis say about the primes and twin primes.

According to Wikipedia

analytic variety is defined locally as the set of common zeros of finitely many analytic functions


$$ P(x) = \sin^2(\pi x) + \sin^2( \frac{\pi \cdot (\Gamma(x)+1)}{ x})$$

The real roots of $P(x)$ greater than one are exactly the primes. Since it is a sum of squares, both terms must vanish. $\sin(\pi x)$ vanishes at integers and the second term is Wilson's primality test.

Let $P_2(x) = P(x)^2 + P(x+2)^2$.

The real roots of $P_2(x)$ greater than one are exactly the smaller twin of twin primes.

I believe $P(x)$ is of no practical interest since $\Gamma(x)$ grows prohibitively fast.

Q1 Are there computationally better real functions with zeros exactly the primes?

Prime (resp. twin prime) counting is counting the zeros of $P(x)$ (resp. $P_2(x)$).

Q2 Does analysis give bounds for the number of zeros in an interval?

Q3 Are some properties of the complex roots of $P(x)$ known?

The complex XRays look messy to me.

The primes appear to be zeros of multiplicity $2$.


share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.