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The Famous Basel Problem: The process that led Euler to the answer!

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Euler is considered to be one of the greatest mathematician of the 18 th century and many believe that he was among the greatest of all time. The answer to the Basel problem is considered to be among his many gems. It is the sum of the series shown below and the answer is astounding! What is π doing there? Mathematician at the time summed up the series to the first 10, 100, and 1000 terms and the slow rate of convergence gave very little clue to what the answer could be.  Euler realized that he would have to find a function that would converge much faster. After some trial and error he found the function shown below whose power series expansion is shown.   So if we integrate the above expression from 0 to 1 This is what he was looking for! To evaluate the integral in the left, he broke it into two parts. Then he evaluated each of the integrals in turn. Then the next integral Put x=1-t When x=1/2, t=1/2 When x=1, t=0
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Millennium Problem: Riemann Hypothesis

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A prime number is a number that is divisible only by itself and 1. Some examples being  2, 3, 5, 7, 9, 11, 13.…   The great mathematician Euclid proved centuries ago that there are an infinite number of them continuing without pattern ad infinitum. However, their real importance is given by what is known as the Fundamental Theorem of Arithmetic. It states that every number can be uniquely expressed as a product of prime numbers. Let’s take the example of 364 = 2 × 2 × 7 × 13. This is the only way that 364 can be obtained by prime numbers! Therefore, they are the atoms or building blocks of numbers. But in 1859, the great German mathematician Bernhard Riemann hypothesized that the spacing of the primes logically follows from other numbers, now known as the “nontrivial zeros” of the Riemann zeta function. The Riemann zeta function takes inputs not only real numbers but also  complex numbers — meaning they have both “real” and “imaginary” components — and yields other numbers
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How to make a cool million Dollars by solving a math problem?

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The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of June 2020, six of the problems remain unsolved. A correct solution to any of the problems results in a US $1,000,000 prize sometimes called a Millennium Prize being awarded by the institute.  Solved Problem: -  Poincare conjecture. When the Clay institute first announced these seven problems, it was generally perceived by the mathematics community that these seven problems would remain unsolved for the foreseeable future. To their surprise, Grigori Perelman a recluse Russian mathematician solved the Poincare conjecture and in July 2010 he famously declined the 1 million dollar prize!   Furthermore, in August 2006, he declined the Fields medal which is considered even more prestigious than the Nobel Prize as it is given every four years to a mathematician under the age of 40. The Nobel Prize is given every year with no age limit. In a seri
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