# PhD course by Francesco Cellarosi in Bologna

Between April 11 and May 12, 2022, Francesco Cellarosi will deliver the PhD course “Randomness in Number Theory: dynamical and probabilistic methods”, at the Department of Mathematics of Università di Bologna and on-line via MS Teams.

### Schedule

(All times are 4-6pm Italian time. Room name within the Math Building in parenthesis)

• April 11  (Seminario VIII Piano)
• April 13  (Aula VII Piano)
• April 20  (Bombelli)
• April 21  (Seminario VIII Piano)
• April 26  (Seminario VIII Piano)
• April 28  (Seminario VIII Piano)
• May 3  (Seminario VIII Piano)
• May 5  (Seminario VII Piano)
• May 10  (Bombelli)
• May 12  (Seminario VIII Piano)

### Program

The course will focus on recent advances concerning the study of random behaviour of number-theoretical sequences. We will mainly focus on the distribution of square-free integers and their generalisations (e.g. k-free, B-free). We will discuss Sarnak’s conjecture on the disjointness of the Mobius function $\mu(n)$ from sequences generated by zero-entropy dynamical systems. We will prove that $\mu^2(n)$ (the indicator of square-free integers) is completely deterministic and study the statistics of its patterns in long intervals. We will also discuss some very recent progress on the distribution of square-free integers in small intervals.

Time permitting, we may discuss the limiting distribution of quadratic Weyl sums and their generalisations (e.g. classical Jacobi theta functions). Quadratic Weyl sums are a special kind of exponential sums that appear naturally in number theory, mathematical physics, and representation theory. They can be interpreted as deterministic walks (with a random ‘seed’) in the complex plane. Generalising Sarnak’s equidistribution of horocycles under the action of the geodesic ow, we can study the limiting distribution of such Weyl sums. A stochastic process of number-theoretical origin can be defined using such sums. Understanding the behaviour of trajectories of the geodesic ow in a homogeneous space, we can study this process, that shares only some of its properties with those of the Brownian motion.

To attend the course please register here. You will receive an email with all the information and material about the course, and your email will be inserted in the course team on MS Teams.