# User:Chrisdecorte

Chris De Corte is a freelance consultant living in Aalst (Belgium) and is among others also a mathematical hobbyist.

Chris is interested in unsolved problems

Chris found that the equality between the Riemann zèta function and the Euler product does not seem to hold (for s<=1). This is explained here.

Chris independently developed his own sieve: the "Sine Sieve" which has some resemblance with the Sieve of Eratosthenes and which is explained here.

Chris independently derived 2 new formulas to determine if a given number n is prime or not:

${\displaystyle P(n)=\prod _{i=2}^{n-1}\sin({\frac {\pi n}{i}})<>0\Rightarrow n={\textrm {Prime}}}$

and:

${\displaystyle {\textrm {n=Prime}}(=p_{i+1})\iff {\textrm {P(n)=}}\prod _{p_{i}=2}^{\forall p_{i}0}$

Using these formulas, he can prove twice Goldbach's Conjecture & Twin prime Conjecture.

This is done by replacing n with 2n=p+q (p and q being 2 primes) and working out the sinus terms.

One of the 2 primes primes that compose the Goldbach requirement needs to be a solution to the following equation:

${\displaystyle G(x)=\prod _{i=2}^{x-1}\sin({\frac {\pi x}{i}})\cdot \prod _{j=2}^{(2n-x)-1}\sin({\frac {\pi (2n-x)}{j}})<>0\Rightarrow x={\textrm {q\quad where\quad 2n=p+q}}}$

Other Prime testing formula's he developed are (for those who can't see the beauty of the sine function):

${\displaystyle P(n)=\prod _{i=1or2}^{n-1}GCD(n,i)=1\Rightarrow n={\textrm {Prime}}}$

and:

${\displaystyle P(n)=\sum _{i=1}^{n-1}(-1)^{i}.GCD(n,i)=0\Rightarrow n{\textrm {\quad could\quad be\quad Prime}}}$

This last formula is true for primes but is also true for some non-primes as 4, 9, 15, ...

Chris also developed multiple prime counting formula:

His probabilistic prime counting formula is his final one and can be represented as follows:

${\displaystyle \pi (x=p_{i})=\alpha .\int _{2}^{x}\prod _{i=2}^{x=p_{i}}(1-1/p_{i}).dx\ with\ \alpha \approx 1.7810292}$

where ${\displaystyle \alpha }$ can be very closely approximated as:

${\displaystyle \alpha =e^{\gamma }\ where\ \gamma \approx 0.57721\ is\ the\ Euler-Mascheroniconstant}$

The origin of this formula can be found here and it will take a long time before someone will improve the accurateness of this formula. A video about this formula can be found here.

His previous formula had also very good accuracy:

${\displaystyle \pi (x)={x \over 2}\cdot {(1-{\sqrt {1-{4 \over \ln(x)}}})}-7}$

Others are:

${\displaystyle \pi (x)=\alpha .x^{\beta }\quad {\textrm {with:}}\quad \alpha =0.2083666\quad and\quad \beta =0.9294465}$

This formula seems to be better than the pure version of the Logarithmic Integral x/lnx up to approximately 1E+10 (except for a short range between 2 and 9000).

Therefore, he would like to propose the following improved formula:

${\displaystyle \pi (x)={\alpha .x^{\beta }.10^{(\gamma -x)}+x/\ln x \over 1+10^{(\gamma -x)}}\quad {\textrm {with:}}\quad \alpha =0.2083666\quad \beta =0.9294465\quad \gamma =1E+10}$

Other works:

Chris found a very close approximation to the angle trisection problem

He also found an approximation to the squaring of the circle and made some interesting but unanswered comments.

I calculated the approximation of some famous constants as a fractal.