Timeline for Studying primes via the gamma function alone: $(x+1)\prod_n \Gamma(\frac{x}{n}+1)^{\mu(n)}$
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Dec 25, 2010 at 21:23 | comment | added | David Feldman | @Daniel Litt The gradual disappearance of all the poles as you multiply out the produce reflects the fundamental theorem of arithmetic. The fact that the infinite product actually converges for every $x$ (indeed the fact that it converges for any $x\not=0$) reflects the prime number theorem. | |
| Dec 25, 2010 at 20:03 | comment | added | David Feldman | Thanks Joe! Yes the idea is that the poles of log gamma line up like the positive integers themselves, so one can get multiplicative identities from combinatorial identities. The multiplication theorems you pointed me to were in my mind, and best understood I think in terms of expressing the sequence of poles of log gamma in two different ways - and here I'm just playing a more complicated version of the game. After a night's sleep, I believe I understand the $e^x$. For $x$ a large integer one plays inclusion-exclusion with the factors of $x!$. In the end one needs $d/dx 1/\zeta(z)$ at $z=1.$ | |
| Dec 25, 2010 at 18:14 | comment | added | David Feldman | @Daniel and Anixx $\mu(n)$ is the Mobius function: -1 to the number of prime factors of a square-free number, 0 for numbers divisible by a square. | |
| Dec 25, 2010 at 15:22 | comment | added | Anixx | What is $\mu (n)$? | |
| Dec 25, 2010 at 14:07 | comment | added | backstoreality | David, the function that you consider is quite interesting. Have you considered that it is based on the product definition of the the $\Gamma(x)$ and the limit form of $e^x$? You also might want to consider the multiplication theorem found at en.wikipedia.org/wiki/Multiplication_theorem Some modification of this theorem might help. If none of this helps then you might want to prove that the function has no roots. Please get back to us on this. Your function is interesting and I haven't seen it in the literature, so I would advise that you definitely keep working on it :) | |
| Dec 25, 2010 at 8:30 | comment | added | Daniel Litt | Wait--fundamental theorem of arithmetic, or prime number theorem? | |
| Dec 25, 2010 at 6:21 | history | asked | David Feldman | CC BY-SA 2.5 |