Date: Monday, February 04, 2013
Location: 4096 East Hall (3:00 PM to 5:00 PM)
Title: Unbounded ranks of elliptic curves, highly biased prime number races and the explicit formula
Abstract: In 1853, Chebyshev remarked that there are more primes of the form 4n+3 than of the form 4n+1 in the interval [1,x], for many values of x. Rubinstein and Sarnak established under technical hypotheses that the logarithmic density of x for which Chebyshev's assertion is true is of 0.9959.... They also studied an even more biased race, and showed that the density of x such that Li(x)>\pi(x) is of 0.99999973... Since their 1994 paper, many other densities have been computed and none of these numbers were found to exceed this last value. A natural question to ask is whether this is the highest value one will ever find, or if on the contrary there exists highly biased prime number races whose associated density can be arbitrarily close to 1. Our goal is to discuss recent results on highly biased prime number races in two contexts. We will first establish a conditional equivalence between the existence of highly biased elliptic curve races and the existence of elliptic curves of arbitrarily large analytic rank. We will then show that highly biased prime number races do exist in the context of primes in arithmetic progressions, and describe how to construct such races. Finally we will describe how to weaken the technical hypotheses which are omnipresent in these types of problems. The central object on which this theory is built is the explicit formula, and if time allows, we will describe the techniques used in the proofs, which involve ideas from probability theory and from the theory of almost periodic functions.
Speaker: Daniel Fiorilli
Institution: UM
Event Organizer: mityab@umich.edu
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