Riemann hypothesis
Want to earn a million dollar? Try Riemann hypothesis ! It is one of the most famous problem in mathematics and simple enough to be understood by a 12th grade teen.
For understanding the problem we need to define what Zeta or sometimes called Riemann zeta function.
Famous mathematician Leonhard Euler showed that this function can expressed in another form, the form over the product of prime numbers.
Now, this formula is very important in mathematics as it can be seen that the information about prime numbers is embedded in this formula.
Now, the series can be convergent or divergent depending on the values of s.
For s > 1, the series will be convergent as we can see that the value of successive term is always less than the previous one and hence we can assign a finite value to this.
For example, say s=2 the ζ(s) = 1 + 1/ 4 + 1/9 + 1/16.... and there is well defined proof that this value is equivalent to π^2/6. (See the proof here)
Further we can extend the values of s to the negative values and we will observe divergent nature of the series.
For example, say s = -1 the ζ(s) = 1 + 2 + 3 + 4 + 5... and there is no way we can have a finite value to this series.
But, there was a great Indian mathematician Srinivasa Ramanujan who in fact assigned a finite value to this sum,
The formula is known as Ramanujan's formula.
1 + 2 + 3 + 4 + .. = -1/12. (See the proof here).
In general, the value of s can be a real number or complex number.
i.e s = x + iy;
Now the zeta function holds good for numbers in complex plain, the only condition is that the real part x should not equal to 1.
So, the zeta function is defined for all values of s (real and complex) minus the value 1+(real value)*i.
Now we are ready to dive into the million dollar mathematical problem. Take a look at the figure given below.
For what value s, the value of zeta function is zero ?
i.e If ζ(s) = 0, then s = ?
If you can answer the above question a million dollar is yours.
Now we know some of the solutions like s = -2 , -4, -6... , these are the trivial solutions and it is know that there doesn't exist any zeroes for which real part of s > 1.
Hence the remaining zeroes lie in the region of 0 to 1, and we are interested in these zeroes, so the Riemann Hypothesis is that all the zeroes that exist between 0 and 1 lies on the line s = 1/2.
i.e ζ( 1/2 + i*(some value) ) = 0;
Now on the basis of many many examples tested on Riemann hypothesis turn out to be true and the largest value tested so far is about in range of 10^24 , all the values turn out to lie on line k = 1/2.
So, in nutshell you get a million dollars if you prove ζ( 1/2 + i*(some value) ) = 0;
or if you disprove it then also you will have the money in your pocket.
If you think this money is less then sit back and get ready for the next post which will prove that you can create something more by starting with something less...
The Banach–Tarski Paradox