I'll start:

Let

let [math]\eta[/math] be a non-trivial zero, [math]\mathbb{C}[/math] doesn't exist because [math]\mathbb{R}[/math] doesn't exist and there are no number above [math]10^{300}[/math] therefore [math]\eta < 10^{300}[/math] and thus...

14 months ago

on the induced topology of the n+1 dimensional semi-spheroids with the Manhattan metric

14 months ago

I actually thought there was something interesting to discover here. Thanks for ruining it for me.

14 months ago

Could someone sum up, how would this function tell us about the number of primes in some region?

And the main question does calculating the number of primes using this function in some region is less demanding than classical (standard) methods?

Also, what are some areas, where knowing how many primes are in some region can be useful?

14 months ago

mfw an infinite number of anons hitting keys at random on their mechanical keyboards will never ever type a proof confirming or refuting the Riemann hypothesis

14 months ago

Prime number theorem (pi(x) ~ (approaches) x/lnx, where pi(x) is the prime counting function, e.g. pi(2)=1, pi(10)=pi(9)=4) is equivalent to the statement [math]\zeta(a+ib) \neq 0 : a=0[/math], or that it has no zeros with real part s = 0. There is no elementary proof of prime number theorem, as of yet, so I'm not going to waste my time explaining exactly how this connection works and why, but basically, let [math]\displaystyle R(x)=1+\sum_{n=1}^\infty \frac {(\ln{x})^n} {nn!\zeta(n+1)} [/math] (that +1 comes in because 1 isn't prime, but some equations act like it is), [math]R(x)-\pi(x)=\sum\limits_{\rho} R(x^\rho)[/math] where [math]\rho[/math] is a nontrivial zero of zeta.

Note that none of this really can be applied to anything, despite what popsci articles tell you. We already know primes larger than computers even bother to use, and lookup tables for factorizations have already been made. Besides, all that the Riemann hypothesis does to solve this is allow approximations of pi(x). For example, if it's true, [math]\pi(x)=Li(x)+O(sqrt(x)\ln(x))[/math] where [math]Li(x)=\int_2^x\frac {dt} {\ln{t}} [/math] (which doesn't have a closed form expression).

14 months ago

Oops, I mean a=1, not a=0 for that top statement. I'm pretty sure the a=0 case is trivial, but I forget now.

14 months ago

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