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Abstracts for PLeaSANT (Participative learning seminar analytic number theory)

Alternatively have a look at the program.

Elementary theory of prime numbers

Posted in
Speaker: 
Franceso Papallardi
Affiliation: 
Università degli Studi Roma Tre
Date: 
Mon, 2017-10-02 17:30 - 18:30
Location: 
MPIM Seminar Room

Chebychev's Theorem states that the order of magnitude of the prime counting function $\pi(x)$ is $x/\log x$. This result was "the state of the art" until the proof of the Prime Number Theorem by Hadamard and de la Vallée-Poussin in 1895. We shall outline a proof of Chebychev's estimates and deduce from his estimate Mertens' Theorem which provides an asymptotic formula for $\sum_{p\le x}p^{-1}$ as $x\rightarrow\infty$.

Artin’s conjecture and the ternary Goldbach problem

Posted in
Speaker: 
Efthymios Sofos
Affiliation: 
University of Leiden/MPI
Date: 
Thu, 2017-10-05 16:30 - 18:00
Location: 
MPIM Lecture Hall

Fix 3  non-square integers a_1,a_2,a_3, none of which being -1.
I will talk about the problem of representating a large odd integer as a sum of 3 primes p_1,p_2,p_3
such that every p_i has a_i as a primitive root. 

Solutions of Diophantine equations with all variables being primes with prescribed primitive roots have not been studied before.

Square-free values of a polynomial (and the analytic rank of an elliptic curve)

Posted in
Speaker: 
Julie Desjardins
Affiliation: 
MPIM
Date: 
Wed, 2017-10-11 16:30 - 17:30
Location: 
MPIM Lecture Hall

Let f be a primitive separable polynomial of positive degree in Z[x]. It is conjectured that a positive proportion of the
values are square-free. This is known as the square-free conjecture. In 1953, Erdös showed that for cubics there are
infinitely many square-free values, and in 1967, Hooley gave the result about positive density when $\deg=3$. Although
we know that is suffices to check the conjecture for the factors, it is still not know for irreducible polynomials of degree
greater than $3$. In $h$ variables, Greaves proves the conjecture for degree $\leq 3h$.

An introduction to the Hardy--Littlewood method

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Speaker: 
Efthymios Sofos
Affiliation: 
University of Leiden/MPI
Date: 
Wed, 2017-10-18 16:30 - 17:30
Location: 
MPIM Lecture Hall

We will go through the conditional (under the generalised Riemann hypothesis) proof of Hardy and Littlewood
that every sufficiently large odd integer is the sum of three primes.

Alfred Tarski’s great algorithm. Decidability of elementary algebra and geometry

Posted in
Speaker: 
Yuri Matiyasevich
Affiliation: 
Steklov Insitute of Mathematics, St. Petersburg
Date: 
Mon, 2017-10-23 12:00 - 14:00
Location: 
MPIM Lecture Hall

On of the most important decidability results is Tarski Theorem. It allows us to determine whether a closed first-order
formula with real variables is true or not. As a corollary we get the decidability of elementary geometry. The original
proof given by Tarski was very involved. After that it was simplified by many researchers. In my talk I am to present
a version of this algorithm which is 

  • easy to understand;
  • easy to implement on a computer; 
  • very inefficient (compared to modern advanced algorithms).
 

p-adic interpolation of modular forms and Galois representations

Posted in
Speaker: 
Andrea Conti
Affiliation: 
MPI
Date: 
Wed, 2017-11-08 16:30 - 18:00
Location: 
MPIM Lecture Hall

In a seminal paper, Serre studied p-adic limits of modular forms with the aim of constructing
p-adic L-functions for totally real number fields. Since then, the theme of p-adic interpolation
of automorphic forms has played an important role in many achievements of contemporary
number theory, such as Wiles's proof of Fermat's Last Theorem. I will give a brief overview
of Serre's method and of the results of Hida, Coleman-Mazur, Buzzard and Chenevier in the
construction of p-adic families of modular forms and of their associated Galois representations.

On the Liouville function in short intervals

Posted in
Speaker: 
Alisa Sedunova
Affiliation: 
MPI
Date: 
Wed, 2017-11-15 16:30 - 18:00
Location: 
MPIM Lecture Hall
We present a brilliant proof due to Matomäki-Radziwiłł of the asymptotic behaviour of Liouville function in short intervals. We also sketch a 
proof of a more general result concerning a similar problem for a 
real-valued multiplicative function. 

The Erdős–Kac theorem

Posted in
Speaker: 
Efthymios Sofos
Affiliation: 
University of Leiden/MPIM
Date: 
Mon, 2017-11-27 14:00 - 15:00
Location: 
MPIM Lecture Hall
The Erdős–Kac theorem states that the number of distinct prime factors of a positive integer n follows
the normal distribution with average loglog(n) and variance loglog(n).A proof via the Central Limit
Theorem will be given.
 
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