Riemann Hypothesis: Greatest Unsolved Problem in History of Mathematics

Riemann Hypothesis: Greatest Unsolved Problem in History of Mathematics

Hello Steemians, in this article, I am going to give you a gentle introduction to the Riemann Hypothesis. I will explain it in an easy manner. Please read the article and give valuable suggestions.

(Bernhard Riemann)

Let’s start with the title, Riemann is from Friedrich Bernhard Riemann, is one of the famous mathematicians who put forward this hypothesis. The hypothesis is any argument that is assumed to be true but not proven yet. When we prove hypothesis, it becomes a theorem. It is amongst the greatest unsolved problems in mathematics. Despite the effort of many mathematicians, it has not been proven for last 350 years. If proven there are lots of implications, distribution of prime numbers being one.
The clay institute of mathematics has declared that anyone who either prove or disprove Riemann Hypothesis will be given 1 million dollars. So, welcome again to the first step towards earning 1 million dollars.

Without delay, I am going to write the hypothesis and try to break it down.

Riemann Hypothesis:

All non-trivial zeros of zeta function have the real part ½.

Now, first let us start with seventh word i.e. function. The interest you pay at bank depends upon the loan taken from the bank. The rate of population growth depends upon the existing population. The area of the circle depends upon the radius of circle.

In the above examples, the value of the variable let’s say y depends upon the value of some variable x. In this case, the y is called the function of x and it is denoted by;

y = f(x)

The x here is independent variable and y is called dependent variable as it depends upon the value of x.

Definition: If every element of set A is associated with unique element of set B then the relation from set A to set B is called function.

Some examples of functions are:

f(x) = x

f(x) = x^2-1 (x^2 = x*x)

f(x) = x^2+ 3x +2

f(x) = √(5-x)

The set of all the values that can be assigned to value of x is called domain of function and the set of corresponding values f(x) is called range of function.

In first function; x can take any real value, so, the domain of function is set of real numbers and the corresponding values can also be any real number so, the range of function is also set of real numbers.

In fourth function; When x is greater than 5, the value of f(x) doesn’t exist. So, the domain of function is all the real values less than or equal to 5. When something is square rooted, the answer cannot be negative number so all the positive numbers including 0 is the range of this function.

Now, this is the basic concept of function. Now moving on;

The zeros of function is the values of x when the values of function becomes zero. In the above examples;

f(x) = x; when x=0; f(x) = 0; so, 0 is the zero of this function.

f(x) = x^2 -1 when x =1, -1; f(x) = 0; so, 1 and -1 are the zeros of this function.

f(x) = √(5-x) when x =5; f(x) = 0; so, 5 is the zero of this function.

Now to the zeta function.

ζ(s) = 1/1^s +1/2^s + 1/3^s + 1/4^s …

For zeros of zeta function;

0 = 1/1^s +1/2^s + 1/3^s + 1/4^s …

This is true for all negative even numbers, i.e. when s = -2, -4, -6, -8, -10…, the above equality holds. These are called the trivial zeros of zeta function but the hypothesis is about non-trivial zeros. Zeros other than these are called non-trivial zeros.

Now, I realize that I must introduce you the complex number. When the square root of negative number comes, we leave them as solution doesn’t exist. These can be incorporated if we use complex number system.

The complex number is in the form of a + ib,

where, i= √(-1)
and, a and b are real numbers.

The real part of complex number is ‘a’
The imaginary part of complex number is ‘b’

Then we can restate hypothesis as;

All the non-trivial zeros of zeta function is in the form of ½ + it. Where ‘t’ is real number.

This very result is check for the first 10,000,000,000,000 solutions. This is true but this number or any number of check cannot hold the term “All” so the proof is still needed.

Summary:

All – All the solution not 1, not 100 not 100000000000 not 10^100000 or any finite number

non-trivial – other than negative even numbers in context.

zeros – the values for which functional values becomes zero.

zeta function- The strange looking function stated above

Real part – The part a of the complex number a + ib.

References

Content

1. Derbyshire, J. (2003). Prime Obsession. ‎Joseph Henry Press .

2. Thomas, G. B. (n.d.). Thomas' Calculus. Pearson Publication.

3. Sarnak, P. (n.d.). http://www.claymath.org/millennium-problems/riemann-hypothesis. Retrieved from http://www.claymath.org.

Image

1. http://www.sapaviva.com/georg-bernhard-riemann-2/

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