## JSH: Prime counting and the PDE

Three simple steps show how you go from a prime counting function to a PDE--partial differential equation:
1. Prime counting function sieve form:

With natural numbers where p_j is the j_th prime:

P(x,n) = x - 1 - sum for j=1 to n of {P([x/p_j],j-1) - (j-1)}

where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.

The [] is the floor function and is actually redundant as you're to use natural numbers.

The sqrt() is the integer square root, for instance sqrt(10) = 3, as that's the closest integer, and it's automatically positive because natural numbers are only the positive integers greater than 0.

2. Prime counting function difference equation form:

With natural numbers

P(x,y) = x - 1 - sum for j=1 to y of {(P([x/j],j-1) - P(j-1, sqrt(j-1)))(P(j, sqrt(j)) - P(j-1, sqrt(j-1)))}

where if y>sqrt(x), then P(x,y) = P(x,sqrt(x)).

And that is an interesting advance as notice it cannot be done with any other known prime counting function! And also notice there is no longer need to tell the equation of any numbers that are prime! If you program it, you'll notice it plucks out primes on its own as it counts, as only when j is prime does:

P(j, sqrt(j)) - P(j-1, sqrt(j-1)) = 0

And it was a difference equation form (slightly different from the above) which I found first! And later I introduced the sieve form.

3. Continuous function:

In the complex plane

P'_y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))

and a differential equation, which is of course a continuous function, is found easily enough from the difference equation from above.
The partial differential equation form is rather succinct like the sieve form, and I just gave the PDE versus talking about summation as here there is no actual prime count involved. Integrating the PDE is of interest.

Now mathematicians have looked for why prime counting and continuous functions like x/ln x are connected for some time.

If you didn't know they were connected, just notice 10/ln 10 = 4.3 to one significant digit, while of course there are 4 primes up to 10 and those are 2, 3, 5 and 7.

While 100/ln 100 = 21.7 to one significant digit, with the count of primes to 100 being 25.

And 1000/ln 1000 = 144.7 to one sig (no rounding), with the count of primes to 1000 being 168.

The famous Riemann Hypothesis was about musings by Riemann in search for a reason for the connection as it seems to indicate some deep connection between prime numbers and continuous functions like ln x.

His complex hypothesis though loses its luster when you properly interpret the progression of equations above, as now it appears that the connection is easily explained by a P(x,y) function, versus the pi(x) function that mathematicians traditionally use.

Now I found the difference equation form back in 2002 and have bugged mathematicians about it off and on ever since, but to no avail and one tempting answer as to why IS that the answer is just not "sexy" enough for people who have pushed solving the Riemann Hypothesis as a "Holy Grail" of their field!!!

BUT if you DO know of the equations above, it seems hard to imagine you'd have much joy remaining for the Riemann Hypothesis so one check to see if mathematicians are knowingly avoiding the answer is to see if there has been a drop off in research into that area!!!

I suggest to you there has.

It has been 7 years since I found the answer to the prime distribution's connection to continuous functions.

More than enough time for "top" mathematicians in the field to move away from a dead area.

If you are a physicist who did not know better who is wasting your time with RH, then you can thank your mathematical colleagues for letting you waste years out of your life, for nothing.