Problems on Twin Primes, Goldbach's Conjecture, the Riemann Hypothesis and zeros of $L$-functions in Number Theory (Women in Mathematics)

Abstract

Number Theory has a very long history that dates back thousands of years. The main goal of this study is to understand properties of numbers which essentially can be reduced to understanding prime numbers. Although we have the outstanding Prime Number Theorem, more precise information about the distribution of prime numbers is mostly unknown. For example, it is also not known if there are infinitely many pairs of prime numbers having difference 2, the so-called twin prime pairs. Recent breakthroughs in Analytic Number Theory have succeeded in showing the infinitude of prime pairs with small gaps, which is the contribution of Yitang Zhang, one of this year's Fields medalists, James Maynard, and also Terrence Tao. The 280-year-old Goldbach's conjecture and the Riemann hypothesis which is now over 160 years old are also among the most famous yet important unsolved problems in Analytic Number Theory. The Riemann Hypothesis is a conjecture about the location of zeros of the Riemann zeta function. The importance of this problem not only in Number Theory but also many other areas of Mathematics and even Physics is reflected in many known equivalent statements. In Analytic Number Theory alone, we know the equivalence between the Riemann Hypothesis and many prime distribution related problems. Its equivalence to Goldbach related problems is also known. It is important to note that Goldbach's conjecture itself is an independent problem to the Riemann Hypothesis and neither is stronger than the other. In this talk, I would like to introduce a few interesting recent results in this direction.