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Martin Sleziak
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A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. herehere for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.

A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. here for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.

A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. here for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.

broken link fixed, cf. https://math.meta.stackexchange.com/a/34713/228959
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Glorfindel
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A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. herehere for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez MarcoE. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.

A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. here for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.

A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. here for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.

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dke
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A formula that surely belongs here linking $\pi$ and the primes is $$2.3.5.7...=4\pi^2.$$ This is obtained via a zeta regularization in a similar way to the more well-known $\infty!=\sqrt{2\pi}$ (see e.g. here for a short discussion of this). However, to find the product of the primes, one uses the prime zeta function $$\sum_{p\; prime} \frac{1}{p^s}$$ which has the unfortunate property of having infinitely many singularities between 0 and 1 which breaks the standard regularization procedure. E. Muñoz García and R. Pérez Marco circumvent this problem (literally) by adding in an extra variable and taking the limit from a different direction.