Counting Primes

Since the probability that a number is prime is roughly the number of primes is close to Indeed, if is the number of primes the recently proved quasi-Riemann hypothesis implies for some constants and If the Riemann hypothesis is true, any will do. But here’s what I want to talk about today. For all where can actually be counted, But in 1914, Littlewood shocked the world by sketching a proof that changes sign infinitely often! Littlewood’s proof didn’t say when first exceeds He gave his student Stanley Skewes the task of figuring that out. In 1933, Skewes gave an enormous upper bound for where this first happens, assuming the Riemann hypothesis: This is called Skewes’ number. Hardy called it “the largest number which has ever served any definite purpose in mathematics”. In 1955, still working on his advisor’s assignment, Skewes gave an even larger upper bound not assuming the Riemann hypothesis: This disgustingly large quantity is called the second Skewes number. I first read about…