Mod-gaussian Convergence: New Limit Theorems in Probability and Number Theory

We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the Katz-Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős-Kac Theorem. DOI: https://doi.org/10.1515/FORM.2011.030 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-53391 Accepted Version Originally published at: Jacod, J; Kowalski, E; Nikeghbali, A (2011). Mod-Gaussian convergence: new limit theorems in probability and number theory. Forum Mathematicum, 23(4):835-873. DOI: https://doi.org/10.1515/FORM.2011.030 ar X iv :0 80 7. 47 39 v1 [ m at h. N T ] 2 9 Ju l 2 00 8 MOD-GAUSSIAN CONVERGENCE: NEW LIMIT THEOREMS IN PROBABILITY AND NUMBER THEORY J. JACOD, E. KOWALSKI, AND A. NIKEGHBALI Abstract. We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the Katz-Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős-Kác Theorem. We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the Katz-Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős-Kác Theorem.