Wikipedia:Reference desk/Archives/Mathematics/2024 January 28

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

January 28

Riemann Hypothesis

Consider the non-trivial zeros of the Riemann zeta function, denoted by ρ=β+iγ, where β and γ are real and imaginary parts, respectively. The Riemann Hypothesis posits that all non-trivial zeros lie on the critical line β=21​.

Now, the question is: Assuming the Riemann Hypothesis is true, prove that there are infinitely many prime numbers.Harvici (talk) 13:24, 28 January 2024 (UTC)

I think you mean β=1/2, not β=21. The infinity of primes doesn't depend on the Riemann Hypothesis. See Euclid's theorem. --Amble (talk) 17:03, 28 January 2024 (UTC)

Yeah its β=1/2 , and infinty of primes doesn't depend on it, but many papers and researchers show that the existence of infinitely many zeros on the critical line implies the existence of infinitely many prime numbers. Harvici (talk) 05:39, 29 January 2024 (UTC)

OK, but it has been proven that there are infinitely many zeros on the critical line, independent of the Riemann hypothesis being true. See Riemann_hypothesis#Zeros_on_the_critical_line. There are interesting connections between the primes and the zeta function whether or not the hypothesis is true. --Amble (talk) 17:06, 29 January 2024 (UTC)

Euler's proof in that article may be what you're thinking of though it doesn't require any of the extra mechanism in the Riemann Hypothesis. A particularly nice bit there is that one can then show the divergence of the sum of the reciprocals of the primes. NadVolum (talk) 17:18, 28 January 2024 (UTC)

Right, and there's also a section "Stronger results" that uses the prime number theorem in one case. So you can do it that way if you want, but it's a lot of extra work if all you want is the infinity of primes. --Amble (talk) 17:32, 28 January 2024 (UTC)

Inverse problem about scalar multiplication on koblitz elliptic curves (or more exactly the secp256k1)

My problem is given Q=nP to find point P given 257 bits integer n and point Q. It’s something possible on other curves but Koblitz curves have extra characteristic and can’t be converted to Montgomery or Edwards curves.
P is a fixed point while n and thus Q can vary leading to several examples using the same unknown P.

So is such operation possible on the secp256k1 curve given a 257 bits integer n ? If yes, how to actually compute it in full ? 2A01:E0A:401:A7C0:D470:71EF:EDBF:3354 (talk) 13:47, 28 January 2024 (UTC)

The problem you're describing is commonly referred to as the "elliptic curve discrete logarithm problem" (ECDLP), where the goal is to find the point P on the elliptic curve. This problem is assumed to be hard, and the security of many elliptic curve-based cryptographic systems relies on the difficulty of solving the ECDLP.(Here is some work done by the University of Auckland.) Harvici (talk) 13:50, 31 January 2024 (UTC)

Wrong, ECDLP is finding $n$ given $P$ and $Q=nP$. Here the problem is to find $P$ given $n$ and $Q=nP$. This problem is not as hard as ECDLP. If you know the order $l$ of the curve, then you can find $m$ such that $mn=1({\text{mod }}l)$ and then compute $mQ=mnP=P$. --2A0D:6FC2:4C20:B000:C847:374F:66AE:340A (talk) 16:55, 2 February 2024 (UTC)