# Number Theory

## An invitation to the algebraic geometry over idempotent semirings - lecture 2

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

### Mini-Course

This lecture is the second part of a mini-course, please see also

## An invitation to the algebraic geometry over idempotent semirings - Lecture 1

Idempotent semi-rings have been relevant in several branches of applied mathematics, like formal languages and combinatorial optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semi-rings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semi-rings and modules over them.

### Mini-Course

This lecture is the first part of a mini-course, please see also

## Generalized valuations and idempotization of schemes

Classical valuation theory has proved to be a valuable tool in number theory, algebraic geometry and singularity theory. For example, one can enrich spectra of rings with new points coming from valuations defined on them and taking values in totally ordered abelian groups.

Totally ordered groups are examples of idempotent semirings, and generalized valuations appear when we replace totally ordered abelian groups with more general idempotent semirings. An important example of idempotent semiring is the tropical semifield.

As an application of this set of ideas, we show how to associate an idempotent version of the structure sheaf of a scheme, which behaves particularly well with respect to idempotization of closed subschemes.

This is a joint work with Félix Baril Boudreau.

## The second moment of symmetric square L-functions over Gaussian integers

We prove an explicit formula for the first moment of Maass form symmetric square L-functions defined over Gaussian integers. As a consequence, we derive a new upper bound for the second moment. This is joint work with Dmitry Frolenkov.

## Exceptional Chebyshev's bias over finite fields

Chebyshev's bias is the surprising phenomenon that there is usually more primes of the form 4n+3 than of the form 4n+1 in initial intervals of the natural numbers. More generally, following work from Rubinstein and Sarnak, we know Chebyshev's bias favours primes that are not squares modulo a fixed integer q compared to primes which are squares modulo q. This phenomenon also appears over finite fields, where we look at irreducible polynomials modulo a fixed polynomial M. However, in the finite field case, there are a few known exceptions to this phenomenon, appearing as a result of multiplicative relations between zeroes of certain L-functions. In this work, we show, improving on earlier work by Kowalski, that those exceptions are rare. This is joint work with L. Devin, D. Keliher and W. Li.

## Euler's divergent series and primes in arithmetic progressions

Euler's divergent series $\sum_{n>0} n!z^n$ which converges only for $z = 0$ becomes an interesting object when evaluated with respect to a p-adic norm (which will be introduced in the talk). Very little is known about the values of the series. For example, it is an open question whether the value at one is irrational (or even non-zero). As individual values are difficult to reach, it makes sense to try to say something about collections of values over sufficiently large sets of primes. This leads to looking at primes in arithmetic progressions, which is in turn raises a need for an explicit bound for the number of primes in an arithmetic progression under the generalized Riemann hypothesis.

During the talk, I will speak about both sides of the story: why we needed good explicit bounds for the number of primes in arithmetic progressions while working with questions about irrationality, and how we then proved such a bound.

The talk is joint work with Tapani Matala-aho, Neea Palojärvi and Louna Seppälä. (Questions about irrationality with T. M. and L. S. and primes in arithmetic progressions with N. P.)

## A new explicit bound for the Riemann zeta function

I give a new explicit bound for the Riemann zeta function on the critical line. This is joint work with Dhir Patel and Andrew Yang. The context of this work highlights the importance of reliability and reproducibility of explicit bounds in analytic number theory.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar

## A logarithmic improvement in the Bombieri-Vinogradov theorem

We improve the best known to date result of Dress-Iwaniec-Tenenbaum, getting ($\log

x)^2$ instead of $\left(log x\right)^(5/2)$. We use a weighted form of Vaughan's identity, allowing a smooth truncation inside the procedure, and an estimate due to Barban-Vehov and Graham related to Selberg's sieve. We give effective and non-effective versions of the result.

This event is part of the PIMS CRG Group on L-Functions in Analytic Number Theory. More details can be found on the webpage here: https://sites.google.com/view/crgl-functions/crg-weekly-seminar?authuser=0

## An explicit error term in the prime number theorem for large x

In 1896, the prime number theorem was established, showing that π(x) ∼ li(x). Perhaps the most widely used estimates in explicit analytic number theory are bounds on |π(x)-li(x)| or the related error term |θ(x)-x|. In this talk we discuss methods one can use to obtain good bounds on these error terms when x is large. Moreover, we will explore the many ways in which these bounds could be improved in the future.

## Kummer Theory for Number Fields

Kummer theory is a classical theory about radical extensions of fields in the case where suitable roots of unity are present in the base field. Motivated by problems close to Artin's primitive root conjecture, we have investigated the degree of families of general Kummer extensions of number fields, providing parametric closed formulas. We present a series of papers that are in part joint work with Christophe Debry, Fritz Hörmann, Pietro Sgobba, and Sebastiano Tronto.