# Seminars of final exam "Selected Topics in Number Theory for Advanced Cryptography"

**Monday 18 July 2022 – at 10.00 am**

The seminar will take place only online, through the ZOOM platform. To get the access codes please contact the secretary office (phd.maths [at] unitn.it)

**Giuliano Romeo **- PhD in Mathematics, Politecnico di Torino

Title: **On the proof of the ternary Goldbach's conjecture through the Circle Method**

**Abstract**:

Goldbach's conjecture is one of the most famous and difficult problems in Number Theory and in all mathematics. The binary (or strong) Goldbach's conjecture asserts that every even number greater than two can be written as a sum of two primes and it still remains an open problem. The ternary (or weak) Goldbach's conjecture, stating that every odd number greater than seven can be expressed as a sum of three odd primes, has been recently solved, in 2013, by H. Helfgott. The aim of this talk is to show how it is possible to convert this problem from an elementary point of view into a complex analysis framework and to present the main ideas behind the Circle Method, devised by Hardy, Littlewood and Ramanujan for the investigation of additive problems in Number Theory, which has been employed by Helfgott in the proof of the ternary Goldbach's conjecture.

**Contact person: **Giordano Santilli

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**Wednesday 20 July 2022 – at 3.00 pm
Seminar Room “-1” – Department of Mathematics**

The seminar will take place in presence and online, through the ZOOM platform.To get the access codes please contact the secretary office

**Giulia Cavicchioni **- PhD in Mathematics, University of Trento

Title: **Dirichlet L-series, the Generalized Riemann Hypothesis and its consequences**

**Abstract**:

The Riemann zeta function, whose non-trivial zeros are related to the distribution of primes, plays a crucial role in Analytic Number Theory. Riemann conjectured that the non-trivial zeros of the zeta have real parts equal to ½. This conjecture, the Riemann hypothesis, is considered to be one of the most important unsolved problems in mathematics. However, many generalizations of the Riemann zeta function, such as the Dirichlet L-functions, are known. In this talk, after having introduced the Dirichlet character, we will investigate some properties of the Dirichlet L-functions. We will study the distribution of the zeros of the L-functions, yielding a generalization of the Riemann hypothesis. The Generalized Riemann hypothesis conjectures that all the non-trivial zeros of the Dirichlet L-functions have real parts equal to ½. Finally we will discuss some consequences of the Generalized Riemann Hypothesis, such as a strengthening of the Prime Number Theorem, and Goldbach weak conjecture.

The seminar corresponds to the final exam of Selected Topics in Number Theory for Advanced Cryptography, a planned course within Giulia Cavicchioni’s first year PHD study programme.

**Contact person: **Giordano Santilli

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**Thursday 21 July 2022 – at 10.00 am**

**Seminar Room “-1” – Department of Mathematics**

The seminar will take place in presence and online, through the ZOOM platform.To get the access codes please contact the secretary office

**Giovanni Tognolini **- PhD in Mathematics, University of Trento

Title: **The Bombieri-Vinogradov Theorem**

**Abstract**:

The Bombieri-Vinogradov Theorem is a statement about the error term in the Prime Number Theorem for arithmetic progressions.

We give the proper context in which this theorem lies and introduce the necessary notation and nomenclature.

Therefore, we will be able to state the theorem, explain its importance and sketch the proof, giving emphasis on an analytic principle, namely the large sieve, which is useful not only for the proof of our theorem, but also for many other applications in Number Theory.

The seminar corresponds to the final exam of Selected Topics in Number Theory for Advanced Cryptography, a planned course within Giovanni Tognolini’s first year PHD study programme.

**Contact person: **Giordano Santilli

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**Friday 22 July 2022 – at 10.00 am**

**Seminar Room “-1” – Department of Mathematics**

The seminar will take place in presence and online, through the ZOOM platform.To get the access codes please contact the secretary office

**Chiara Spadafora**- PhD in Mathematics, University of Trento

Title: **The theory of Binary Quadratic Forms**

**Abstract**:

The objective of this seminar is to give an introduction to the theory of binary quadratic forms, starting from basic notions, e.g. the discriminant of a form and forms equivalence. The representation problem and the minimum problem will be introduced and a first strategy to solve the representation problem will be discussed.

Although naive, this strategy gives rise to three important algorithmic problems:

(i) how to construct all the forms with the same discriminant,

(ii) how to decide whether two forms are equivalent,

(iii) how to find the automorphism group of a form.

The subsequent part will be devoted to integral positive definite forms, problems (ii) and (iii) will be solved and a reduction algorithm together with a strategy to solve the representation problem will be presented.

**Contact person: **Giordano Santilli