Web Page of the Ottawa-Carleton Number Theory Seminar

Year 2015-2016

Abstract: Understanding mean values of multiplicative functions over number field play a key role in analytic number theory. Motivated by the recent work of Granville, Harper and Soundararajan we discuss the mean values of multiplicative functions over function field. In particular, we prove function field analogs of several classical results of Wirsing, Halasz and Hall. If time permits, we will also discuss spectrum of multiplicative functions over function field.

Abstract: An integer r is called squarefull (or powerful) if, for all primes, p dividing r, we have p² divides r. We connect several seemingly unrelated conjectures of Ankeny, Artin, Chowla and Mordell to a conjecture of Erdös on consecutive squarefull numbers. We then study the Erdös conjecture and relate it to the abc conjecture. We also derive by elementary methods several uncon- ditional results pertaining to the Erdös conjecture. This is a joint work with Prof. M.Ram Murty.

Abstract: Cramér's random model leads us to expect that the primes are distributed in a Poisson distribution around their mean spacing. It is conjectured that, for any given positive number λ and nonnegative integer m, the proportion of n ≤ x for which (n,n + λ logn] contains exactly m primes is asymptotically equal to λ^me^−λ/m! as x → ∞. It is also conjectured that, for any given numbers b > a ≥ 0, the proportion of n ≤ x for which d_n/logn ∈ (a,b], where d_n = p_n+1 − p_n and p_n is the nth smallest prime, is asymptotically equal to ∫_a^b e^−t dt as x → ∞.
By combining an Erdős–Rankin type construction, which produces large gaps between consecutive primes, with the Maynard–Tao breakthrough on bounded gaps between primes, we are able to show that the number of n ≤ x for which π(n + λlogn) − π(n) = m is at least x^{1 − o(1)}. We are also able to show that at least 25% of nonnegative real numbers are limit points of the sequence (d_n/logn) of normalized level spacings in the primes.
This includes joint work with William Banks and James Maynard. (Translated from L^AT_EX by H^EV^EA)

Title: The generalized circle problem, mean value formulas and Brownian motion.

Abstract: The generalized circle problem asks for the number of lattice points of an n-dimensional lattice inside a large Euclidean ball centered at the origin. In this talk I will discuss the generalized circle problem for a random lattice of large dimension n. In particular, I will present a result that relates the error term in the generalized circle problem to one-dimensional Brownian motion. The key ingredient in the discussion will be a new mean value formula over the space of lattices generalizing a formula due to C. A. Rogers. This is joint work with Andreas Strömbergsson.

Abstract: The correspondence between elliptic curves over Q up to isogeny and modular forms of weight 2 (due in one direction to Eichler and Shimura, in the other mostly to Wiles) is at the heart of modern arithmetic geometry. One would like to generalize to modular forms of weight greater than 2. Brown and Schnetz, motivated by ideas from physics, constructed certain varieties and observed that their numbers of points mod p sometimes matched the coefficients of modular forms. In this talk I will explain how I proved this in two of their examples in which no variety corresponding to the modular form was previously known. Time permitting, I will also describe how I have found varieties matching other modular forms by exploring nearby regions of an appropriate moduli space (conjectural for now, but provable by a finite computation).

Abstract: We will discuss the p-adic variation of the Eichler-Shimura isomorphism in the context of Shimura curves. In particular, we describe the finite slope part of the space of overconvergent modular symbols in terms of the finite slope part of the space of overconvergent modular forms. As an application we will explain how to attach Galois representations to certain overconvergent modular forms. This is joint work with Shan Gao.

Speaker: Giuseppe Melfi (The University of Applied Sciences of Western Switzerland, at Neuchâtel)

Abstract: A weird number is a positive integer n for which σ(n)>2n and such that n cannot be written as the sum of some of its proper divisors. Some structure properties of weird numbers in terms of their factorization are presented. In particular we give sufficient conditions to ensure that a positive integer is weird. As far as we know, the algorithms we present here are new, and could be suitable to be used in the search for odd weird numbers, whose existence is still an open question raised by Benkoski and Erdös in 1972.

Abstract: This talk is about various aspects of the Gauss factorial N_n! which is defined as the product of all positive integers up to N that are relatively prime to n. After a brief general discussion, I will present a variety of results which can be seen, roughly speaking, as extensions of Wilson's theorem. Of special interest will be the multiplicative order (and specifically order 1) of particular Gauss factorials modulo certain composites. The relevant results involve generalized Fermat numbers and a special and rare type of primes which we call Jacobi primes. Another class of results concerns the relationship between multiplicative orders modulo successive prime powers. These results are based on generalizations of binomial coefficient congruences of Gauss, Jacobi, and of Hudson and Williams. (Joint work with John B. Cosgrave)