I'm A Shameless Exponent of MatLab (LOL!)

I'd be a happy fellow, now that I have the following :

.... after wrestling the MATLAB Variable Precision Arithmetic into line. This time using PI/12 as the angle of interest - it's tangent is, analytically, the sqrt((1-sqrt(3)/2)/(1+sqrt(3)/2)). Now the above three numbers are all equal and accurate to the book value @ 100 decimal places for (a) the intrinsic MATLAB tan^-1 function atan, (b) the roll-my-own high precision version arctan and (c) the intrinsic PI constant that MATLAB produces by whatever algorithm. The key MATLAB language point is to use the vpa type declarations on the innermost elements of an expression, the language interpreter/parser causes the rest of the constructs to follow that precision. A stunning feature actually. If you want more precision then change max_digits and wait for a longer time ( tic starts a timer, toc and reads it out ).

It's all there, for your cosmic measuring needs .... LOL :-)

Cheers, Mike.

Another rabbit hole : prime numbers.

Fortunately MATLAB has a primes(n) command which returns ( as a vector ) the prime numbers less than or equal to the given number number n. On my machine I can get all the primes up to 10 billion before a resource limit is reached. What now ? Well I've always been fascinated by the fact that once you're past 10 then all primes have their last digit equal to 1 or 3 or 7 or 9. But what about having 4 primes in a given decade like 11, 13, 17, 19 or 5651, 5653, 5657, 5659 say. A group of four such as these are called a prime quadruplet. One  conjecture states that there is no highest quadruplet ie. it is an infinite set. Conjecture means it hasn't been proven one way or the other. An open problem.

So now here's some code that selects all the prime quadruplets up to 10 billion :

.... and saves the first prime from each quadruplet in a file to be later worked on. I've chosen, based on a Wikipedia page, to use these primes as additive factors in a particular ( analytically infinite ) series, like so :

.... but first doing a loglog plot of the distribution of these quadruplets. This reveals a straight line for sufficiently large numbers ( upper right side on the graphic below ). This is a clue that there may be a power law operating, and by eye it looks to have an exponent of about 4/3 :

... this hinting that there may be indeed no greatest prime quadruplet if said power law holds for all magnitudes. BTW if you take the total sum of the series, multiply it by ten and then add two : the result is very close to the natural logarithm base e. :-)

But all this is proof of nothing, just a muck-about. All I've shown is how to spend a pleasant afternoon on MATLAB. Mind you if one does prove that the prime quadruplets form an infinite set then you also prove, as a lemma on the side, the Twin Prime conjecture .... that pairs of primes differing by two also form an infinite set. Which is where other distributed projects like PrimeGrid come in.

Cheers, Mike.

The Riemann zeta function. It is defined as a series for that half of the complex number plane where Re(z) > 1 thus :

zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + .....

where s = x + iy is a complex variable. So one picks a value of s and then uses the formula to get the zeta value for that s. However the function doesn't use that series formula directly for Re(z) < 1 as it doesn't converge. But by a technique/trick of complex mathematics called analytic continuation there is effectively a 'mirroring' of the function values. That is, there is a formula connecting the regions either side of that line :

zeta(s) = 2^s * PI^s-1 * sin(PI*s/2) * gamma(1 - s) * zeta(1 - s)

 ...... and so one can evaluate the series when it converges and use the result to assign function values where the series directly couldn't otherwise apply. For example to evaluate for s = -3 using the above connecting formula one needs to calculate the series when s = 1-(-3) = 4, etc. The function gamma() is an extended definition of the factorial function for non-integer arguments. Also note that PI, again, turns up in the formula, almost without invitation as it were. I'm beginning to think of PI as some sort of universal constant in mathematics which, as an aside, also has something to do with circles ! :-0

Now zeta(s) has so-called trivial zeros when s = - 2, -4, -6 ...... because that makes sin(k*PI) = 0 for some integer k.

Now for the money shot. There are other zeros of the zeta function, not of the trivial type, and to date of all such non-trivial zeros discovered lie on the line Re(z) = 1/2. The Riemann Hypothesis is that all non-trivial zeros of the zeta function lie on the line Re(z) = 1/2. There have been discovered billions & billions of zeros - calculated on computers - which satisfy Re(s) = 1/2 and no exception to this rule has been found. If you can be the first person to prove that this must be so then you have yourself a million dollar prize. This has been offered by the Clay Mathematics Institute as one of it's millennium problems. Needless to say this question requires some acute understanding of complex analysis in order to even begin to appreciate what is being asked. I am working on that .... ;-)

Why the big deal then ? So what if the zeta function has some set of zeros all at Re(s) = 1/2 ? Sounds pretty arcane. Well there are hundreds of other important results in number theory that depend on the Riemann conjecture being true. In the several centuries since it was first mentioned many results in pure mathematics have assumed it to be true ie. mathematicians have ventured forth without the firm knowledge that this Riemann hypothesis is correct. So if one cracks this problem then a blizzard of other stuff becomes valid also. For example the error in the predicting the distribution of prime numbers is one of them. There has yet to be a general formula for prime numbers, you have to just go from one number to the next and test for it's multiplicative factors. 

Cheers, Mike.

( edit ) One warning about visualising complex functions : the zeta function, say, has a complex number as an input and a complex number as it's output. Because a complex number is defined as x + iy where x and y are real numbers then zeta is a two-input, two-output function. You can't graph it like an ordinary one-input/one-output function. You'd need four dimensions to get it right ....  and if you are familiar with the ordinary derivatives of real functions then even sensing what complex differentiation is, and is not, is difficult. It's probably not ideal to think of slopes, tangents and such-like for complex functions. Perhaps it is best to consider the first derivative as being the best linear approximation of the complex function near some point in it's domain. Differentiation of a complex function is weird. If you can take the first derivative - an adapted form of the limiting process in the real numbers - then you can take the derivative as many times as you like. One corollary is that a complex function can't be only differentiable a finite number of times, it either can't be done at all or if so then infinitely many times. Also integration is weird too, but that's another story ...

( edit ) You could legitimately ask : what is 1/2^s when s is complex ? Well it follows the usual arithmetic of using powers with some base where s = x + i y ( both x and y real numbers ) :

1/2^s = (1/2)^x+iy = (1/2)^x * (1/2)^iy

with (1/2)^x being an ordinary real number. However (1/2)^iy encodes a phase, or the angle in polar form with the positive real axis. If you vary x then the distance to the origin changes, if you vary y then that phase changes ie. it rotates around the origin. You may be more familiar with Euler's use of e^iy, that is the natural logarithm base being used for the polar form of complex numbers. You can use other bases if you like - but it effectively changes the scaling of the y input. For example :

(1/3)^iy = 3^-iy = e^-ln(3)*iy

.... where ln(3) is the natural logarithm of 3. Likewise for 1/k^s generally. Indeed this is the core of the calculation with the zeta function ( and complex power series ) : it is the summation of lot's of vectors in the two dimensional ( Argand ) plane that are placed head to tail to achieve some resultant displacement from the origin. The series converges if that resultant vector settles down to some finite length and fixed direction. For a zero of the zeta function the chain of vectors goes around some path with the final vectors as close as possible to the centre of the Argand plane ie. 0 + i0. Hence the phrase 'zero of the function'. For the argument s = 1 = 1 + i0 it doesn't converge, as the magnitude will grow to be as big as one pleases ( see discussion here ), a slow crawl to infinity but unbounded nonetheless. There isn't anything one can do about that for the zeta function.