Power-free values and number of divisors of polynomials

The project Power-free values and number of divisors of polynomials focuses on two classical problems from analytic number theory. The first one is the problem of estimating the number of power-free values of polynomials of two variables, including when the two arguments are prime numbers. This field is important for estimating for example square-free values of polynomials, which have importance in cryptography, without which the security of the modern digital world is unthinkable. The second topic of this project is the problem for the average number of divisors of polynomials, for which there are conjectured magnitudes of growth, but up to now none of these conjectures have been fully verified for polynomials of degree higher than two. At the same time the divisor problem has many implications, including in the recent breakthrough problem for small gaps between primes. The first problem with which this project deals is to estimate the number of power-free values of certain polynomials of two variables, where only lower bounds of the expected order of magnitude are available. There is a conjecture of Erdos for existence of infinitely many power-free values of polynomials in one variable at prime arguments, which was completely resolved only recently. We would like to provide more instances of two-variable polynomials, for which we can extend this conjecture of Erdos. Until now we gave one such example and we are not aware of other such examples in the literature. For this, among straightforward tools from analytic number theory, we will need strong non-trivial estimates of the number of solutions of congruences or Diophantine equations in a box of restricted size, like the ones obtained by Baier and Browning. Our first goal connected to the divisor problem is to estimate the growth of the average number of divisors of reducible quadratic polynomials and provide an explicit upper bound for this sum, which would have applications in Diophantine number theory. Then we would like to investigate this average sum for polynomials of higher degree, at least for polynomials of special type. This is an area in which there is insufficient progress. We would build on methods from analytic number theory initiated by Hooley, Heath-Brown and Browning.

The project "Power-free values and number of divisors of polynomials" focused on couple of classical problems from analytic number theory concerning properties of polynomials with integer arguments. The first one is the problem of estimating the number of power-free values of polynomials of many variables, including when the arguments are prime numbers. This topic is especially important for example in the case of square-free values of polynomials, which could have applications in cryptography and digital security. The other main topic of this project is the problem for the average number of divisors of polynomials, for which there are conjectured magnitudes of growth, but up to now none of these conjectures have been fully verified for polynomials of degree higher than two. We were especially interested in providing upper bounds with explicit constants for such sums over quadratic polynomials, which have applications in the classical Diophantine number theory. At the same time the divisor problem has also modern implications, including in the breakthrough problem for small gaps between primes. The outcome of the project amounts to several publications, participation and presentations at international conferences, hosting visits of first-class mathematicians and initiating fruitful collaborations. The research goals set up by this project were achieved to a very big extent and some extra related questions were also considered. Still there are more conjectures whose proofs remain out of reach for the current state of art in this field of Number theory. That is why we believe that this research area will continue to be attractive for further development.