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Abstracts for Conference on "Asymptotic Counting and L-Functions"

Alternatively have a look at the program.

Arithmetic counting and zeta functions - a panorama

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Speaker: 
Valentin Blomer
Affiliation: 
University of Bonn
Date: 
Mon, 05/05/2025 - 10:00 - 11:00
Location: 
MPIM Lecture Hall

I present a variety of examples on the interplay of number theoretic counting problems and analytic properties of zeta functions. I take a closer look at the representation zeta function of SL(3, Z) and show how to obtain analytic continuation and a natural boundary for this function.

 

Vanishing of primitive root densities

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Speaker: 
Peter Stevenhagen
Affiliation: 
Leiden University
Date: 
Mon, 05/05/2025 - 11:30 - 12:30
Location: 
MPIM Lecture Hall

 

Artin's conjecture on average and short character sums

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Speaker: 
Igor Shparlinski
Affiliation: 
University of New South Wales
Date: 
Mon, 05/05/2025 - 14:30 - 14:55
Location: 
MPIM Lecture Hall

Let $N_a(x)$ denote the number of primes up to $x$ for which the integer $a$ is a primitive root. We show
that $N_a(x)$ satisfies the asymptotic predicted by Artin's conjecture for almost all
$1\le  a\le  \exp\left((\log \log x)^2\right)$. This improves on a result of Stephens (1969) which applies to the
range   $1\le  a\le \exp\left( 6 ( \log x \log  \log x)^{1/2}\right )$. A key ingredient in the proof is a new short

The multiplication table constant and sums of two squares 

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Speaker: 
Alisa Sedunova
Affiliation: 
Purdue University
Date: 
Mon, 05/05/2025 - 15:00 - 15:25
Location: 
MPIM Lecture Hall

Let $r_1(n)$ be the number of representations of n as the sum of a square and a square of a prime. We discuss the erratic behavior of $r_1$, which is similar to the one of the divisor function. We will show that the number of integers up to x that have at least one such representation is asymptotic to $(\pi/2) x / \log x$ minus a secondary term of size $x/(\log x)^{1+d+o(1)}$, where $d$ is the multiplication table constant. Detailed heuristics suggest very precise asymptotic for the secondary term as well.

Asymptotic Counting of RSA Integers

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Speaker: 
Sumaia Saad Eddin
Affiliation: 
Austrian Academy of Science
Date: 
Mon, 05/05/2025 - 15:30 - 15:55
Location: 
MPIM Lecture Hall

A few years ago, Pieter and I studied integers that can be expressed as the product of two constrained prime numbers, commonly known as RSA integers due to their relevance in cryptography. Our work focused on examining the asymptotic distribution of RSA integers and investigating earlier observations by Dummit, Granville, and Kisilevsky on the distribution of integers in arithmetic progressions, applied specifically to RSA integers.

Multiplicative functions are everywhere 

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Speaker: 
Oleksiy Klurman
Affiliation: 
University of Bristol
Date: 
Tue, 06/05/2025 - 10:00 - 11:00
Location: 
MPIM Lecture Hall

It has been known for more than a century that many classical problems in analytic number theory would follow from a deeper understanding of partial sums and correlations of multiplicative functions. In recent years, this subject has seen several transformative breakthroughs, leading to numerous applications in number theory, ergodic theory, and combinatorics. The aim of the talk is to discuss some of the ideas behind these developments.

Removing the Riemann Hypothesis from the pair correlation method for zeros of the Riemann zeta-function

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Speaker: 
Ade Irma Suriajaya
Affiliation: 
Kyushu University
Date: 
Tue, 06/05/2025 - 11:30 - 12:30
Location: 
MPIM Lecture Hall

Assuming the Riemann Hypothesis (RH), Montgomery (1973) proved a theorem concerning the pair correlation of nontrivial zeros of the Riemann zeta-function. One consequence of this theorem was that, under RH, at least 2/3 of the zeros are simple. In earlier papers, we have shown an unconditional version of this theorem of Montgomery and how to obtain the same proportion of simple zeros under a much weaker hypothesis than RH. We have furthermore found a connection to finding proportion of zeros on the critical line. This is joint work with Siegfred Alan C.

Prime components in Apollonian circle packings

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Speaker: 
Damaris Schindler
Affiliation: 
University of Göttingen
Date: 
Tue, 06/05/2025 - 14:30 - 14:55
Location: 
MPIM Lecture Hall

In this talk we discuss prime components and thickened prime components in Apollonian circle packings. In particular, we are interested in the set of curvatures that appear in these subsets of Apollonian circle packings and we prove first lower bounds on the number of curvatures of bounded height that appear in a thickened prime component. This is joint work with Elena Fuchs, Holley Friedlander, Piper Harris, Catherine Hsu, James Rickards, Katherine Sanden and Katherine Stange. 

 

The Game of Life and the Prime Number Theorem

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Speaker: 
Gunther Cornelissen
Affiliation: 
Utrecht University
Date: 
Tue, 06/05/2025 - 15:00 - 15:25
Location: 
MPIM Lecture Hall

We explain a curious coincidence between fixed points of certain cellular automata and of certain polynomial iterates, and show how this provides an analogue of the Prime Number Theorem (joint work with Jakub Byszewski).

 

Exploring the interplay of Euler-Kronecker constants, prime numbers, and Kummer's Conjecture

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Speaker: 
Neelam Kandhil
Affiliation: 
The University of Hong Kong
Date: 
Wed, 07/05/2025 - 10:00 - 11:00
Location: 
MPIM Lecture Hall

The Euler-Kronecker constant of a number field $K$ is defined as the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function associated with $K$ at $s=1$. Investigating the distribution of the normalized difference of the Euler-Kronecker constants of the prime cyclotomic field $\mathbb{Q}(\zeta_q)$ and its maximal real subfield, we connect it with Kummer's conjecture. This conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields.

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