I will present a recent joint work with Ya.G. Sinai. We investigate the ``randomness" of the classical Möbius function by means of a statistical mechanical model for square-free numbers and we prove some new results, including a non-standard limit theorem where the Dickman-De Bruijn distribution appears. Although we use a probabilistic approach, this work is inspired by a conjecture by P. Sarnak, and by a number of recent results relating Number Theory and Ergodic Theory.

In this talk, I will give new proofs for the hardness amplification of fficiently samplable predicates and of weakly verifiable puzzles. More oncretely, in the first part of the talk, I will give a new proof of Yao's XOR-Lemma as well as related theorems in the cryptographic setting. This proof seems simpler than previous ones, yet immediately generalizes to statements similar in spirit such as the extraction lemma used to obtain pseudo-random generators from one-way functions [Hastad, Impagliazzo, Levin, Luby, SIAM J. on Comp. 1999].

We prove a complexity dichotomy theorem for all non-negatively weighted counting Constraint Satisfaction Problems (#CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms and the celebrated dichotomy theorem for unweighted #CSP. Our dichotomy theorem gives a succinct criterion for tractability. If a set F of constraint functions satisfies the criterion, then the #CSP problem defined by F is solvable in polynomial time; if F does not satisfy the criterion, then the problem is #P-hard.