Our graduate students
Haddad, Tony
-
Faculty of Arts and Science - Department of Mathematics and Statistics
André-Aisenstadt Office 6253
Courriels
Research area
Student supervision Expand all Collapse all
Prime number races
Theses and supervised dissertations / 2020-08
Haddad, Tony
Abstract
Abstract
Under the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, Rubinstein
and Sarnak proved that the values of x which have more prime numbers less than
or equal to x of the form 4n + 3 than primes of the form 4n + 1 have a logarithmic density
of approximately 99.59%. In general, the study of the difference #{p < x : p in A} − #{p < x : p in B}
for two subsets of the primes A and B is called the prime number race between A and B. In
this thesis, we will analyze the prime number race between the primes p such that 2p + 1 is
also prime (these primes are called the Sophie Germain primes) and the primes p such that
2p − 1 is also prime. To understand this, we first present Rubinstein and Sarnak’s analysis
to understand where the bias between primes that are 1 (mod 4) and the ones that are
3 (mod 4) comes from and give a conjecture on the distribution of Sophie Germain primes.