### Number Theory Seminar HS 2018

Unless otherwise stated, all talks start at 14.15 and take place in Seminarraum 05.001, Spiegelgasse 5,

Information for speakers

Date | Speaker | Title | Abstract |
---|---|---|---|

27.09.2018 | Extreme values of the Riemann Zeta function, and log-correlated
Gaussian fields | I will describe how the Riemann Zeta function on the critical line can be viewed as a pseudo-random Gaussian field with a correlation function with logarithmic growth. Such log-correlated random fields have recently attracted considerable interest in probability theory. Fyodorv, Hiary and Keating conjectured several striking results about the extreme values of the Riemann Zeta function based on this connection. In this talk I will explain how a certain approximate tree structure in Dirichlet polynomials can be used to prove one of their conjectures, giving the asymptotics of the maximum of the magnitude of the function in a typical interval of length O(1). | |

04.10.2018 | On Goormaghtigh's equation | In my talk I will present results that come from a joint work with M. Bennett and A. Gherga from The University of British Columbia. We studied Goormaghtigh's equation: \begin{equation}\label{eq} \frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n > 2. \end{equation} There are two known solutions $(x, y,m, n)=(2, 5, 5, 3), (2, 90, 13, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitely or infinitely many solutions, and not even if that is the case if we fix one of the variables. It is known that there are finitely many solutions if we fix any two variables. Moreover, there are effective results in all cases, except when the two fixed variables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionally satisfy $\gcd(m-1, n-1)>1$, then there is an effective finiteness result. My co-authors and me showed that if $n \geq 3$ is a fixed integer, then there exists an effectively computable constant $c (n)$ such that $\max \{ x, y, m \} < c (n)$ for all $x, y$ and $m$ that satisfy Goormaghtigh's equation with $\gcd(m-1,n-1)>1$. In case $n \in \{ 3, 4, 5 \}$, we solved the equation completely, subject to this non-coprimality condition. | |

01.11.2018 | Representation of integers by binary forms | Let $F(x, y)$ be an irreducible binary form with integer coefficients and of degree at least 3. By a well known result of Thue, the equation $F(x, y) = m$ has only finitely many solutions in integers $x$ and $y$. I will discuss some old and new quantitative results on the number of solutions of such equations. I will also talk about my joint work with Manjul Bhargava, where we use these bounds to show that many equations of the shape $F(x, y) = m$ have no solutions. | |

08.11.2018 | Solving polynomial-exponential equations. | Inspired by Schanuel's Conjecture, Boris Zilber has proposed a ``Nullstellensatz'' (also conjectural) asserting which sorts of polynomial-exponential equations in several variables have a complex solution. Last year Dale Brownawell and I published a proof in the situation which can be regarded as ``typical''. But it does not cover all situations for two variables, some of which involve simply stated problems in one variable like finding complex $z \neq 0$ with $e^z+e^{1/z}=1$. Recently Vincenzo Mantova and I have settled the general case of two variables. We describe our methods -- for example, to solve $$e^z+e^{\root 9 \of {1-z^9}}=1$$ one approach uses theta functions on ${\bf C}^{28}$. | |

22.11.2018 | On bounded automorphisms of fields with operators | Lascar showed that the group of automorphisms of the complex field which fix the algebraic closure of the prime field is simple. For this, he first showed that there are no non-trivial bounded automorphisms. An automorphism is bounded if there is a finite set $A$ such that the image of every element b is algebraic over $A$ together with $b$. The same result holds for a "universal" differentially closed field of characteristic zero, where we replace algebraic by differentially algebraic. Together with T. Blossier and C. Hardouin, we provided in https://arxiv.org/abs/1505.03669 a complete classification of bounded automorphisms in various fields equipped with operators, among others, for generic difference fields in all characteristics or for Hasse-Schmidt differential fields in positive characteristic. | |

29.11.2018 | Integrability of b-divisors on toroidal embeddings | Convex geometry has been widely and successfully used to explore the geometry of algebraic varieties. A well-known class of examples comes from the theory of toric varieties, where the combinatorics of a lattice polytope encrypts most of the geometric properties of the corresponding toric variety. In this talk, we will discuss how to use convex analytical methods on so called weakly embedded polyhedral complexes to obtain integrability results of toroidal b-divisors. The latter can be thought of as a limit of toroidal divisors defined over all possible toroidal compactifications of an algebraic variety. We will also mention applications of these results to Arakelov geometry. | |

06.12.2018 | Titchmarsh's divisor problem for multiplicative functions | The classical Titchmarsh divisor problem is concerned with the mean behaviour of the divisor function on the set of primes shifted by a non-zero constant. We consider analogues of this problem where the set of primes is replaced by other sets of multiplicative nature and also discuss further generalizations. This is joint work with S. Drappeau. | |

13.12.2018 | The twelfth moment of Dirichlet L-functions to smooth moduli | Heath-Brown proved a strong bound for the twelfth moment of the Riemann zeta function which is not sharp but that has the nice feature of giving the Weyl estimate as a corollary. In this talk we will take a look at the key points of Heath-Brown's proof and show how one can deduce an analogous result for Dirichlet L-functions with smooth moduli. Our result relies on bounds for some complicated exponential sums that we study by appealing to the theory of algebraic trace functions due to Katz and Fouvry-Kowalski-Michel. |