Amazing stuff!
"Just as molecules are composed of atoms, in math, every natural number can be broken down into its prime factors—those that are divisible only by themselves and 1. Mathematicians want to understand how primes are distributed along the number line, in the hope of revealing an organizing principle for the atoms of arithmetic.
“At first sight, they look pretty random,” ... “But actually, there’s believed to be this hidden structure within the prime numbers.” ...
For 165 years, mathematicians seeking that structure have focused on the Riemann hypothesis. Proving it would offer a Rosetta Stone for decoding the primes—as well as a $1 million award from the Clay Mathematics Institute. Now, in a preprint posted online on 31 May, Maynard and Larry Guth of the Massachusetts Institute of Technology have taken a step in this direction by ruling out certain exceptions to the Riemann hypothesis. The result is unlikely to win the cash prize, but it represents the first progress in decades on a major knot in math’s biggest unsolved problem, and it promises to spark new advances throughout number theory. ...
In the late 1700s, at the age of 16, German mathematician Carl Friedrich Gauss saw that the frequency of prime numbers seems to diminish as they get bigger and posited that they scale according to a simple formula: the number of primes less than or equal to X is roughly X divided by the natural logarithm of X. Gauss’s estimate has stood up impressively well. To the best mathematicians can tell, the actual number of primes bounces slightly above and below this curve up to infinity. That known primes follow such a simple formula so closely suggests the primes aren’t completely random; there must be some deep connections governing where they appear. ..."
In the late 1700s, at the age of 16, German mathematician Carl Friedrich Gauss saw that the frequency of prime numbers seems to diminish as they get bigger and posited that they scale according to a simple formula: the number of primes less than or equal to X is roughly X divided by the natural logarithm of X. Gauss’s estimate has stood up impressively well. To the best mathematicians can tell, the actual number of primes bounces slightly above and below this curve up to infinity. That known primes follow such a simple formula so closely suggests the primes aren’t completely random; there must be some deep connections governing where they appear. ..."
From the (very short) abstract (for a 48 pages long paper):
"We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length N taking values of size close to N3/4, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate N(σ,T)≤T30(1−σ)/13+o(1) and asymptotics for primes in short intervals of length x17/30+o(1)."
New large value estimates for Dirichlet polynomials (open access)
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