In cosmology, we struggle to understand the true geometric shape of our immense Universe.  What type of curvature does it have? Or is the Universe flat where parallel lines never meet and the angles of a triangle always add up to 180 degrees as in Euclidian geometry?  If the Universe is not flat then straight lines will eventually intersect if the curvature is positive or inversely diverge to infinity in a negatively curved Universe.  We think of lines in mathematics as being infinite straight so that parallel lines never meet but can that thinking be valid if our physical Universe converges as on a sphere [positive curvature] or diverges as on a riding saddle for horseback [negative curvature] ?  Zany, crazy, unlikely but what if we graphed Riemann’s famous zeta function and his non-trivial zeros on straight lines that mimic what could be the real geometry of the Universe? Is that innovative or just foolishly outrageous?

Mathematics has functioned magnanimously in describing and quantifying the physical mechanics of our Universe.  What if beyond Reimann, does a curved universe challenge our conceptual ideation of how we abstractly envision the perfection of mathematics in illustrating our physical world?  Infinity and an infinite series goes on for a long time…for what does forever look like in a non-flat Universe? What does an infinite lineation of Reimann’s nontrivial zeros look like in a possibly curved Universe?  Do they as I suspect continue to fall neatly on the critical line ad infinitum plotted on the north-south line defined by those complex numbers whose real part is one-half?  But what occurs when that critical line which bisects the critical strip lineation goes askew in curved geometry that perhaps is ultimately embedded in our reality?  The zeta zeros are a mirror of the primes themselves and what happens if the straight lines of Reimann’s vertical imaginary axis and the horizontal real axis as envisioned in our mathematical concepts can’t match the true reality of such physical lines in a possibly  curved Universe?  Does it matter at all?

 

Riemann Hypothesis-Please Click Here for My Article
I

Excerpt from Famous Scientist: Mathematicians-The Art Of Genius.   http://www.famousscientists.org/bernhard-riemann/ :

A prime number is a natural number greater than 1 with no positive divisors apart from itself and 1.
The prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,… have been a source of fascination, concern, excitement, amusement, frustration, and pleasure for mathematicians since our ancestors first started thinking mathematically.
Over 2,000 years ago, the Greek mathematician and geographer Eratosthenes devised his famous sieve as a means of finding primes.
One reason for the love affair is that primes are the natural numbers from which all other natural numbers can be constructed – it’s often said that the primes are as fundamental to numbers as the periodic table is to chemistry.

Prime Numbers are Random.

They Decline, but Never Die
Between 1 – 100 you will find 25 prime numbers.
Between 901 – 1000 you will find 14 prime numbers.
Between 9,901 – 10,000 you will find 8 prime numbers.
These observations represent a relentless trend. As we count to ever higher numbers, prime numbers become rarer. They never disappear though. Euclid furnished us with a rather simple and beautiful proof that there is an infinite number of primes.
Another crucial property of primes is that they seem to be scattered randomly among the natural numbers. There is no formula to predict when the next prime will show up, and they continue materializing apparently at random for as long as you care to continue counting.

Organization of Prime Numbers

Is there a way to predict how many primes are present within any section of the number line, such as within the hundred numbers between 999,999,999,901 and 1,000,000,000,000? (There are 4, if you are interested.)
Gauss had been fascinated by this question. He spent many hours in his youth in intense calculations identifying the prime numbers between 2 – 1,000,000. This was a truly phenomenal piece of work – it is hard to express how overwhelming the sheer scale of calculations required must have been. It was only achievable because the raw calculating power of Gauss’s brain seems to have been close to superhuman.
Gauss’s work revealed mathematics in a light people do not often think about. When he counted the prime numbers, Gauss was acting like an experimental scientist, gathering data from which he could draw conclusions.
As a result of his calculations, Gauss discovered the prime number theorem. This theorem, expressed informally, says the number of prime numbers up to any number x is given by:

In this case, log x is the natural logarithm of x. The calculation becomes ever more accurate for ever higher values of x.

The Riemann Hypothesis

Much of Riemann’s work concerned complex analysis – the realm of complex numbers. These are numbers which have a real part, corresponding to the normal number line, and an ‘imaginary’ part which contains √-1, identified by the letter i.
Any complex number can be graphed on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. For example, the complex number plotted below is 3 + 2i, which is the same as 3 + 2√-1.

Riemann attacked the distribution of primes using a mathematical equation known as the Riemann zeta function, which was actually first worked on by Leonhard Euler.
Riemann was interested in the zeta function because he noticed it emerged when he was deriving an alternative formula to Gauss’s prime number theorem.
The zeta function is an infinite series:

Riemann asked himself what values of s would allow the zeta function to equal zero. The answer required complex numbers.
His work allowed him to form a hypothesis. His hypothesis was evidence-based, as all of his calculations supported it. All subsequent calculations, using computers to explore ever higher numbers, also support it.
So, at last, we can state the Riemman hypothesis, which is that the zeta function is equal to zero in only two circumstances:
For s values that are real, even, and negative – i.e. when s is equal to -2, -4, -6, -8,… (A mathematically trivial result, although not obvious.)
For s values that are complex numbers whose real part is equal to ½. (Not mathematically trivial.)

Where there is discord, may we bring harmony
Riemann showed that the distribution of prime numbers seems to be profoundly linked to the non-trivial zeros of the zeta function, where the complex number’s real part equals ½. The mathematical magician had pulled a rabbit out of a hat: although everyone had believed prime numbers were distributed randomly, there in fact seems to be some sort of pattern determined by the zeta function’s zeros.
But Riemann couldn’t provide a mathematical proof of his hypothesis.
This meant he could not prove the zeta function’s zeros and the distribution of prime numbers are linked in all circumstances.
“Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study…”
——————————————————————Bernhard Riemann,  1859