Prime numbers are the building blocks of mathematics. They are the foundation of all formulas, conjectures and proofs – in one way or another, everything can be linked back to prime numbers. So it’s unnerving to many mathematicians that the very nature of prime numbers, and indeed the very nature of mathematics itself, is shrouded by chaos: Prime numbers lack order.
While Euclid proved that there are an infinite amount of prime numbers back in 300 BC, even after 2000 years, we cannot find a pattern for the sequence of these infinite prime numbers. This has remained the biggest, and most important, unsolved problem in mathematics.
However, there is still hope. A hypothesis was proposed by Bernhard Riemann in 1859 (The Riemann hypothesis), which, if solved, could give the answer we are looking for.
Breaking down The Riemann hypothesis?
Proving the Riemann Zeta Function is a divergent series for s=1 via the integral test:
Proving the Riemann Zeta Function is a convergent series for s >1 :
Considering complex numbers and complex exponents:
Extending the domain:
References:
Figure 1:
Wikipedia contributors. (2021c, July 20).Riemann zeta function. Wikipedia. https://en.wikipedia.org/wiki/Riemann_zeta_function#/media/File:Zeta_polar.svg
Intuitive Understanding Of Euler’s Formula – BetterExplained. (n.d.). Better Explained. Retrieved 23 July 2021, from https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/
Visualizing the Riemann zeta function and analytic continuation. (2016, December 9). [Video]. YouTube. https://www.youtube.com/watch?v=sD0NjbwqlYw
Riemann Hypothesis - Numberphile. (2014, March 11). [Video]. YouTube. https://www.youtube.com/watch?v=d6c6uIyieoo
Wikipedia contributors. (2021, July 16). Riemann hypothesis. Wikipedia. https://en.wikipedia.org/wiki/Riemann_hypothesis
Wikipedia contributors. (2021a, May 31). Particular values of the Riemann zeta function. Wikipedia. https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function
Why does the harmonic series diverge? - Quora. (n.d.). Quora. Retrieved 23 July 2021, from https://www.quora.com/Why-does-the-harmonic-series-diverge
The Zeta Function: Convergence. (2015, December 5). [Video]. YouTube. https://www.youtube.com/watch?v=0QIvHhxeCjk
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