# 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.