Alessandro Zaccagnini

After my degree in Mathematics (1989) I earned my PhD in 1994 from the University of Turin, under the supervision of Alberto Perelli at Genoa. I have been a Research Fellow of the Department of Mathematics at Parma since 1993 until the end of 2005. I am currently Associate Professor of Mathematical Analysis at Parma.

In my PhD thesis I have established some approximate results for long standing conjectures of Hardy and Littlewood (1923) in the additive theory of numbers: according to their first conjecture, every large integer is either a perfect square or a sum of a perfect square and a prime number. More generally, it is conjectured that, for fixed k greater or equal to 2, every large number is either a perfect power or a sum of a prime and a k-th power. My results are upper bounds for the number of exceptions to the above conjecture in the interval [1,x], when k is larger than 2. I have also studied variants of the above conjecture: for example (with A. Perelli) I determined short intervals where "almost all" integers have the expected number of representations as sums of a prime and a k-th power, and upper bounds for the exceptional set under the Generalized Riemann Hypothesis. I also established similar results for sums of a prime number and a value of a polynomial with integral coefficients.

I worked on variants of the Goldbach problem, in particular to the Goldbach-Linnik problem with J. Pintz and A. Languasco.

I obtained results on primes in almost all short intervals and the relationship between bounds for the Selberg integral and Density Theorems for the zeros of the Riemann zeta function. With A. Perelli and A. Languasco we studied explicit quantitative relations between the Selberg integral and precise versions of Montgomery's Pair-Correlation Conjecture.

With A. Languasco we wrote a series of papers about the Mertens constant for arithmetic progressions, and another one on diophantine problems with prime variables.

Other papers concern the distribution of prime numbers in very short intervals (with D. Bazzanella and A. Languasco), and the Montgomery-Hooley Theorem (with A. Languasco and A. Perelli).

I also wrote a large number of papers and essays for the general public, in particular on the distribution of primes and their applications to ctyptography, in part with A. Languasco.