The Million-Dollar Equation: too hard for Sheldon??

Solve one equation and earn a million dollars! We will explorer the secrets behind the Riemann Hypothesis - the most famous open problem in mathematics - and what it would tell us about prime numbers.

I should have mentioned one additional property, namely zeros are mirrored along the line 1/2, even though non of them are found and maybe even non of them even exist. This way, every zero not on the line would giva a harmonic with Re(s) greater than 1/2, thereby breaking the estimates for the prime counting function.



Results discussed / references
List of the 7 million-dollar Millenium Problems: https://en.wikipedia.org/wiki/Millennium_Prize_Problems
How to extend the Riemann zeta function: https://www.youtube.com/watch?v=sD0NjbwqlYw
Current best approximations for pi(x): https://www.sciencedirect.com/science/article/pii/S0022314X15001924 (Corollary 2)
"Implementation of Riemann’s Explicit Formula for Rational and Gaussian Primes in Sage": http://ism.uqam.ca/~ism/pdf/Hutama-scientific%20report.pdf
"Some Calculations Related to Riemann's Prime Number Formula": https://www.ams.org/journals/mcom/1970-24-112/S0025-5718-1970-0277489-3/S0025-5718-1970-0277489-3.pdf
"The Riemann hypothesis is true up to 3*10^12": https://arxiv.org/pdf/2004.09765.pdf
Consequences of different zero-free regions on the growth of |pi(x)-li(x)|: A.E. Ingham: The Distribution of Prime Numbers, Cambridge University Press
Scene from Big Bang Theory: S12E6 The Imitation Perturbation

Riemann Hypothesis 1000000 million dollar mathematics math maths problem harmonics waves prime numbers primes prime number theorem zeta function zeros unsolved prime counting function Millennium sheldon Big Bang Theory