In the field of Pure Mathematics, there are so many unsolved problems and billions of dollars are declared as a prize money for solving them. Such a problem is Riemann’s Confusion problem. Here is the historical background and development of the problem.
Let us begin our discussion with a very simple problem:
1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + …. (i)
What is the sum of the series if the series goes up to infinity? It’s ‘Infinity’, i.e. more mathematically it will not ‘converge’. Another example we may discuss in this context is:
1/12 + 1/22+ 1/32 + 1/42 + 1/52 + 1/62 +… (ii)
But interestingly series (ii) converges to a definite value. The sum of this series is π2/6.
Again, taking the sum:
1/13 + 1/23 + 1/33 + 1/43 + 1/53 + 1/63 +… (iii)
We get no interesting result (approx. 1.202) but the likely series:
1/14 + 1/24 + 1/34 + 1/44 + 1/54 + 1/64 +…(iv) gives us the value π4/90.
Now, It became a challenge to mathematicians to find whether for a given real number ‘s’; the sum (Σ 1/ns ) Converges to a ‘interesting number’ or not.
Eventually mathematicians named those sequences as Z-series. Like, (i) is Z (1), (ii) is Z(2), (iii) is Z(3) and so on.
Observations: Z(1) tends to infinity, as well as Z(0)=1+1+1+… also goes to infinity. Similarly If we take Z(-1), the sum will look like 1+2+3+4+.. and blows up. So, clearly, we may assume that, for all ‘s’ ≤ 1, blows up.
The Euler Question:
Z(s)= 1/1s +1/2s +1/3s +1/4s +1/5s+ 1/6s + …. (v)
Are there any s’s for which interesting things happen (besides 2 and 4)?
There was a lot of examples in this contexts, and things were going well, until the day came- when they were patiently waiting for to be summed to produce an ‘interesting’ result, assuming that the operator wouldn’t choose any ‘s’ such that the sum gets blew up.
Prime numbers are culprit (!):
All of a sudden, a gang of ‘Rebel’ numbers started to produce a result like the expression (vi). It was literally a BOOM !! from integers to prime number and they arranged themselves just to produce Z(a):
This made Mathematicians, specially Number Theorists mad, as they found a way to go from the kingdom of integers to the kingdom of prime numbers. Eventually, to the mathematicians, this observation became “as important as the ‘independence of America’ to Americans.”
Despite the fundamental importance of Z(s), Euler’s efforts to tame this function and domesticate it and maybe make it do tricks were hamstrung by the fact that it always seemed to be blowing up at inopportune times1. As a result, he gave up and spent the remainder of his life going blind working on other mathematics in a dark attic.
The function continued to lay prostrate in its useless state for over half of a century until Bernhard Riemann came along. He took the function into his office, watched it self-destructively blow up any time a negative number was mentioned, and decided that it needed help. Although Riemann was not a number theorist by trade, he felt, much like that guy in the movie” Lorenzo’s Oil,” that he could teach himself enough number theory to cure Z of its horrible ailment.
A way from Z-function to Zeta-Function:
From Euler’s Question we had:
Z(s)= 1/1s +1/2s +1/3s +1/4s +1/5s+ 1/6s +…
Now Riemann came with a very simple technique by multiplying both side with 
.
Now the Result became:
Now simply by dividing both side by
We get the result:
And He renamed the expression as ζ(s).
After a lot of works, Riemann came up with the mathematical function that he called ζ (the Greek letter zeta): what he thought was a remedy. And the function in general looks like:
Now in this equation if I put s=2, i.e., ζ (2) the result will be π2/6 and so on.
Riemann’s Result: For any s for which Z(s) doesn’t blow up, ζ(s) = Z(s).
Riemann had found a function that mirrored Z(s). Unlike Z(s), though, ζ(s) didn’t blow up if s was less than 1. And ζ was prepared to handle all kinds of numbers! Fractions, Decimals, Imaginary numbers like √( −1), Combinations of real numbers and imaginary numbers! ζ(s) was like a post spinach-Popeye version of Z(s). Unfortunately, like Achilles, ζ still had one flaw. There was one single value for s that ζ couldn’t handle: s=1. It is actually undefined in case of s=1.
Now, we defined ζ(s). Now we can go to Modified Euler’s Puzzle: Is there anything interesting about ζ(s)?
This question can often be a dog-whistle type question for mathematicians; when we say, “Is it interesting?”, we often mean, “Does it hit zero a lot?” There are actually quite a few places where it is easy to show it hits zero:
Partial Answer: ζ (s) = 0 when s = -2, -4, -6, -8, -10, …
Now mathematicians started to find some non-negative numbers for which this fascinating function hits zero. Luckily, they got some examples which are really amazing. As for example,
ζ(1/2+ 14.134725142i ) = 0;
ζ(1/2+ 21.022039639i ) = 0;
ζ(1/2+ 25.010857580i ) = 0;
ζ(1/2+ 30.424876126i ) = 0;
clearly there is a pattern: ½+some multiple of i. (i= √( −1) )
From such observations a new question emerged out: If ζ (s) = 0, does that mean that s is 1/2 plus a multiple of i ?
But Riemann himself was not so confirm about this.
And finally, this question became so important in the field of mathematics that $10,00,000 is declared a reward to solve the question (!!)
And, the following question became the one of the most important unsolved problem in Mathematics today:
If ζ (s) = 0 and s is not a negative even integer than s = 1/2 + i t (for some real number t.)
OMG!! Gauss Got a question:
After Euler, Riemann; there comes another one Carl Fredrich Gauss. He was then 15 yrs. Old when he came up with the question, “How many prime numbers are there up 100? Or, up to 1000? Or up to 1000000?”
Clearly, there is an obvious method namely, ‘let’s count’. But actually, there are some important memes in Facebook that I must share with friends. So, I need some mathematical formulae so that I can calculate the exact answer without counting. This was the question that a fifteen-year old named Carl Friedrich Gauss
considered in 1792. Unlike me, Gauss did not have an important meme in Facebook to share, so he sat down, looked through the data, and came up with a
pretty good answer:
The number of primes less than x is about x/ ln (x).
This work would have been impressive enough for somebody who wasn’t
yet old enough to drive. However, Gauss wasn’t satisfied and announced,
“No! I can do even better! I shall come up with a function that comes even
closer to the correct number of primes! And the function will be easy to
calculate!”. An election year like 1792, Gauss actually succeeded in his quest and found exactly the function he was looking for. He called this function “Li” because
he, like most 18th century number theorists, was a big fan of kung-Fu legend
Bruce Lee. The “Li” function is given by:
The Importance of Riemann’s hypothesis:
Now we know that, the Li(x) is just an approximated answer to the real answer. If we take P(x) [later it was modified to Euler’s φ function] to be the real answer, then the difference between Li(x) and P(x) is given by Li(x)~P(x) and termed as error term in the prime number theory.
Now, If the Riemann’s Function is correct then the error term must not be greater than √x ln(x).
It can be easily shown that √x ln(x) is pretty small relative to the number things we’re counting, which would mean that Li function does a really good job of approximating P(x).
And here lies the Importance of Zeta-function or Z-function.
So,
Source:
- https://www.britannica.com/science/Riemann-zeta-function
- https://en.wikipedia.org/wiki/Riemann_zeta_function
- A friendly introduction to The Riemann Hypothesis.Thomas Write, Appleton, Wisconsin.Published on Sept. 2010.
By
Subhadip Banerjee
IISER Bhopal