Turku Number Theory Seminar

The Turku Number Theory Seminar features talks which are mainly given by the members of the number theory research group of the University of Turku. 

The seminar currently takes place in the seminar room M2 on the second floor of Quantum, unless stated otherwise. The Quantum building hosts the Department of Mathematics and Statistics of the University of Turku.

Below you can see the titles and the abstracts of past and upcoming talks, as well as a map showing where the seminar is held.

NEXT TALK

Tuesday 7/10/2025, 11:00am  Note that this talk is exceptionally on Tuesday!

Speaker: Jori Merikoski, University of Turku

• Title: The divisor function along arithmetic progressions and binary cubic polynomials

• Abstract: We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic progressions to almost all moduli of exponent 2/3. Second, we show an asymptotic formula for the divisor function along the nonhomogeneous binary cubic polynomial $XY^2+1$. The proof is based on the Deligne bound for correlations of Kloosterman sums. This is joint work with Lasse Grimmelt.

PREVIOUS TALKS

Wednesday 24/9/2025, 11:00am 

Speaker: Tony Haddad, University of Turku

• Title: A coupling for prime factors and three applications to divisors 

• Abstract: In 2002, Arratia introduced a coupling between a random integer and a Poisson-Dirichlet process such that the distance between the prime factors of the integer and the components of the process is small in expectation. Last year, in collaboration with Dimitris Koukoulopoulos, we proved a conjecture of Arratia by refining this coupling to bring the prime factors even closer to the components. In this presentation, I will present this refined coupling and three applications to the theory of divisors: the Dirichlet law for random k-factorizations, counting integers having a divisor within a given interval and the average size of the "middle" divisor of an integer. 

Wednesday 17/9/2025, 11:00am 

Speaker: Kaisa Matomäki, University of Turku

• Title: Linnik's problem for multiplicative functions 

• Abstract: Linnik's famous theorem states that there exists a positive constant $C$ such that for any sufficiently large integer $q$ and any $a$ co-prime to $q$, there exists a prime number $p \leq q^C$ such that $p \equiv a \pmod{q}$. Thanks to works of Jutila, Heath-Brown, Xylouris and others, we now know that one may take $C = 5$. In this talk I will discuss a variant of Linnik's problem for the Möbius function and other multiplicative functions. This is joint work with Joni Teräväinen.  

Wednesday 10/9/2025, 11:00am 

Speaker: Jori Merikoski, University of Turku

• Title: The greatest prime factor and uniform equidistribution of quadratic polynomials 

• Abstract: We show that the greatest prime factor of n^2+h is at least n^{1.312} infinitely often. This provides an unconditional proof for the exponent previously known under the Selberg eigenvalue conjecture. Furthermore, we get the same exponent uniformly in h < n^{1+o(1)} under a natural hypothesis on real characters. The same uniformity is obtained for the equidistribution of the roots of quadratic congruences modulo primes. This is joint work with L. Grimmelt, as an application of our work on weighted averages of SL(2,R) automorphic kernel, which replaces the previously used sums of Kloosterman sums methods.

Monday 9/6/2025, 11:00am  Note the day and time change!

Speaker: Martin Čech, Charles University

• Title: Halász theorem revisited

• Abstract: Halász theorem states that given a 1-bounded multiplicative function f(n), it's average is 0 unless f(n) is in some sense close to n^it for some real number t. The original proof of the theorem was somewhat complicated, but a few years ago, Granville, Harper and Soundararajan gave a simpler and intuitive proof. During the talk, we will consider Halász theorem from a more analytic point of view and will attempt to give a somewhat similar, but an even simpler and more intuitive proof. The talk will be based on work in progress, joint with Yu-Chen Sun.

Tuesday 20/5/2025, 12:15pm

Speaker: Mikko Jaskari, University of Turku

• Title: On bounds of the prime number theorem

• Abstract: The von Mangoldt formula is a clear way to connect zeros of the Riemann zeta function with prime numbers by a certain 'zero sum'. We see that bounding the real part of the zeros as far as possible from the 1-line helps us to bound the error of the prime number theorem. I will discuss about the ongoing project with Andrew Fiori. Our goal is to provide tools which can be used to evaluate explicit bounds for zero sums for L-functions and our special interest is the zeros near the 1-line.

Tuesday 6/5/2025, 12:15pm

Speaker: Niklas Miller, Aalto University

• Title: The kissing number problem for the cross-polytope

• Abstract:  Newton and Gregory debated in 1694 whether or not it is possible to bring 13 spheres of equal radii into contact with a given one with the same radius. As conjectured by Newton, the answer is negative, a fact that was proven only in 1953 by Schütte and van der Waerden. More generally, the kissing number problem asks for the maximum number of translates of a given convex body that can be brought into contact with a central one. In this talk, new asymptotic upper and lower bounds for the kissing number of the n-dimensional cross-polytope, i.e., the convex hull of the +/- pairs of standard basis vectors in R^n, are presented. The lattice kissing number problem for the 4-dimensional cross-polytope (the hexadecachoron) is solved: it is shown that the kissing configuration consisting of the shortest non-zero vectors of the lattice D_4^+, and its odd-coordinate version, are the only lattices producing a kissing configuration of 40 vectors.

Tuesday 22/4/2025, 12:15pm

Speaker: Sebastian Zuniga Alterman, University of Turku

• Title: Averages of arithmetic functions and sets with sieve structure

• Abstract: We study some averages of arithmetic functions of a certain class, essentially generalising the divisor function, which involve the support of a sequence with sieve structure.

We are inspired by the Erdös divisor bound on polynomials (1951) as well as by Wolke's and Shiu's results in the early 70's. We discuss the smoothest conditions that one can consider for obtaining such bounds and derive some applications. This is work in progress.

Tuesday 8/4/2025, 12:15pm

Speaker: Stelios Sachpazis, University of Turku

• Title: Large oscillations of the argument of Dirichlet polynomials

• Abstract: The last two centuries have witnessed a lot of work on the values of the Riemann ζ function inside the critical strip. Part of this work consists of results concerned with its large values. In one of the earliest such results from 1977, Montgomery adopted an approach which also allowed him to deliver large oscillations of arg ζ(σ+it) when 1/2<σ<1. In 2022, Yang and Xu deviated from the study of the large values of ζ and applied the so-called resonance method of Soundararajan to deliver large values for Dirichlet polynomials. Inspired by their work, one might try to seek results similar to those regarding the Riemann zeta function for Dirichlet polynomials. As we mentioned, in the "universe" of the Riemann zeta function, we do not only have results on large values of ζ in the critical strip; We also have results on large oscillations of its argument, like the one that Montgomery had obtained in 1977. To this end, we are going to see what we can say about large oscillations of Dirichlet polynomials of unitary, completely multiplicative coefficients when 1/2<σ<1. This talk is based on on-going work with Mikko Jaskari.

Tuesday 25/3/2025, 12:15pm

Speaker: Sarvagya Jain, University of Turku

• Title: Additive Problems with Primes from a Thin Bohr Set

• Abstract: Additive problems involving special subsets of primes have driven significant developments in additive number theory, including the circle method and the transference principle. Let ∣∣x∣∣ denote the distance of x from the nearest integer. For an irrational number α, Matomäki studied the subset of primes satisfying ∣∣αp∣∣ < 1/p^τ, and showed that for 0 < τ < 1/3, there are infinitely many primes in this subset. In this talk, I will discuss ongoing work in which I explore primes within this subset for 0 < τ < 1/9 and outline a strategy employing the circle method for addressing the analogues classical additive problems using such primes.

Speaker: Lillian Pierce, Duke University

• Title: Polynomial sieve methods and thin sets

• Abstract: Many problems in number theory can be framed as questions about counting solutions to a Diophantine equation (say, within a certain “box”). If there are very few, or very many variables, certain methods gain an advantage, but sometimes there is extra structure that can be exploited as well. For example: let f be a given polynomial with integer coefficients in n variables. How many values of f are a perfect square? A perfect cube? Or, more generally, a value of a different polynomial of interest, say g(y)? These questions arise in a variety of specific applications, and also in the context of a general conjecture of Serre on counting points in thin sets. We will describe how sieve methods can exploit this type of structure, and in particular how a polynomial sieve method allows enough flexibility so that the variables in the polynomials f and g can “mix.” We will furthermore describe new work that is insensitive to the singularity of the underlying hypersurface, joint with Dante Bonolis and Katharine Woo.

Tuesday 25/2/2025, 12:15pm

Speaker: Jesse Jääsaari, University of Turku

• Title: Sixth moment of twisted L-functions

• Abstract: In 1978 Heath-Brown obtained a non-trivial upper bound for the twelfth moment of the Riemann zeta function. The q-aspect analogues of this result for Dirichlet L-functions with certain special moduli have since been established. In this talk I will discuss on-going work concerning generalisations of these results for the sixth moment of twisted L-functions.

Tuesday 11/2/2025, 12:15pm

Speaker: Kaisa Matomäki, University of Turku

• Title: Mollifying revisited

• Abstract: This is a sort of continuation of a talk I gave in February 2024 but the audience is not expected to remember anything from the previous talk. This time I will first briefly discuss (again) how mollified moments have been used to study non-vanishing of Dirichlet L-functions at the central point $s = 1/2$. Then I will describe our very general results concerning optimality of mollifiers, and present applications to the case of non-vanishing of Dirichlet L-functions at the central point. The talk is based on on-going joint work with Martin Čech.

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TALKS OF 2024

Tuesday 10/12/2024, 11:00am

Speaker: Kaisa Matomäki, University of Turku

• Title: On Levinson's method

• Abstract: The celebrated Riemann hypothesis asserts that all the non-trivial zeroes of the Riemann zeta function lie on the critical line $\Re s = 1/2$. Unconditionally it's only known that positive proportion of the non-trivial zeros lie on that line. Levinson's method with suitable mollifiers gives best known lower bounds for this proportion. Based on literature, I will describe Levinson's method and discuss associated mollifiers.

Tuesday 3/12/2024, 11:00am

Speaker: Lasse Grimmelt, University of Oxford

• Title: Twisted correlations of the divisor function via discrete averages of SL(2,R) Poincaré series

• Abstract: This talk is based on joint work with Jori Merikoski. We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with a high uniformity in the modulus. It is obtained by using the spectral methods of SL(2,R) automorphic forms to study Poincaré series over congruence subgroups while keeping track of interactions between multiple orbits. The approach offers increased flexibility over the widely used sums of Kloosterman sums techniques. We give applications to correlations of the divisor function twisted by periodic functions and the fourth moment of Dirichlet L-functions on the critical line.

Tuesday 26/11/2024, 11:00am

Speaker: Mikko Jaskari, University of Turku

• Title: On resonance method and extreme values of Dirichlet polynomials with multiplicative coefficients

• Abstract: The resonance method introduced by K. Soundararajan is an efficient way to evaluate extreme values of the Riemann zeta function and L-functions. We will discuss how this method can be used to do the same thing for Dirichlet polynomials. We look into the recent result of Max Wenqiang Xu and Daodao Yang and I will possibly discuss an ongoing project with Stelios Sachpazis.

Tuesday 19/11/2024, 11:00am

Speaker: Jesse Jääsaari, University of Turku

• Title: Sup-norms for non-compact arithmetic surfaces in the depth aspect

• Abstract: In this talk we discuss on-going work concerning a variant of the sup-norm problem for Maass cusp forms of level N in the so-called depth aspect, where one takes N=p^n with p a fixed prime and n varying. We will sketch a proof giving a power-saving improvement over the local bound for these sup-norms.

Tuesday 12/11/2024, 11:00am

Speaker: Hamed Mousavi, University of Bristol

• Title: Ergodic theory near to the endpoint

• Abstract: In this talk, we explore ergodic averages near the endpoint. We demonstrate that a maximal estimate is both a necessary and sufficient condition for proving pointwise convergence. If time permits, we will also discuss the well-known divergence criteria established by Lavictoire.

Tuesday 5/11/2024, 11:00am

Speaker: Sebastian Zuniga Alterman, University of Turku

• Title: From explicit estimates for the primes to explicit estimates for the Möbius function

• Abstract: We present the best up to date explicit bounds for the average and the logarithmic average of the Möbius function. In order to do that, we rely on a convolution identity, some numerical calculations and an inequality of Balazard. We also discuss about further improvements. This is a submitted joint work with O. Ramaré.

Thursday 31/10/2024, 11:00am

Speaker: Joni Teräväinen, University of Turku

• Title: On the exceptional set in the abc conjecture

• Abstract: The well known abc conjecture asserts that for any coprime triple of positive integers satisfying a+b=c, we have c<K_{epsilon} rad(abc)^{1+epsilon}, where rad is the squarefree radical function. 

In this talk, I will discuss a proof giving the first power-saving improvement over the trivial bound for the number of exceptions to this conjecture. The proof is based on a combination of various methods for counting rational points on curves, and an involved combinatorial analysis to patch these cases together. 

This is joint work with Tim Browning and Jared Lichtman.

Wednesday 23/10/2024, 11:00am

Note the different day. This week's talk is on Wednesday.

Speaker: Sarvagya Jain, University of Turku

• Title: Negative Values of Truncations of L(1,χ)

• Abstract: While the class number formula guarantees the positivity of L(1,χ) for primitive Dirichlet characters, it's tempting to assume that the truncations of this sum are also always non-negative. However, Granville and Soundararajan have shown this conjecture to be false. In this talk, we will delve into their work. We'll also discuss the broader context of their investigation: the problem of determining how negative the sum $\sum_{n\leq x}f(n)/n$ can be for a general multiplicative function $f$.

Thursday 17/10/2024, 11:00am   This week's talk was postponed for a later date.

Thursday 10/10/2024, 11:00am

Speaker: Stelios Sachpazis, University of Turku

• Title: Chowla's conjecture under Landau-Siegel zeroes

• Abstract: In 1965, Chowla made the following conjecture.

Chowla's conjecture: If $\lambda$ is the Liouville function and $k\geqslant 2$ is a fixed integer, then for any fixed distinct non-negative integers $h_1,\ldots,h_k$, we have that

$$\sum_{n\leqslant x}\lambda(n+h_1)\cdots\lambda(n+h_k)=o(x)\quad \text{as} \quad x \to\infty.$$

An unconditional answer to this conjecture is yet to be found, and in this talk, we are taking a conditional approach towards it. In particular, we will discuss Chowla's conjecture under the existence of Landau-Siegel zeroes. We will start by introducing the notion of a Landau-Siegel zero and then we will heuristically explain why the existence of these zeroes is a useful assumption when "attacking" Chowla's conjecture. We will continue by describing how one can establish a bound for the sums $\sum_{n\leqslant x}\lambda(n+h_1)\cdots \lambda(n+h_k)$ using the presence of Landau-Siegel zeroes. The estimate that we will present succeeds the previous respective works of Germán and Kátai, Chinis, and Tao and Teräväinen. This talk is based on joint work with Mikko Jaskari.

Wednesday 21/8/24, 11:00am

Speaker: Ping Xi, Xi’an Jiaotong University

• Title: The Brun-Titchmarsh Theorem

• Abstract: The classical Brun-Titchmarsh theorem gives an upper bound, which is of correct order of magnitude, for the number of primes in an individual arithmetic progression. We will discuss our recent work on sharpening this theorem with better constants by combining Dirichlet polynomials, character/exponential sums, $\ell$-adic cohomology and spectral theory of automorphic forms. If time permits, we also mention its connection with the Landau-Siegel zero and subconvex bounds for Dirichlet L-functions. This is a joint work with Junren Zheng.

Thursday 8/8/24, 11:00am

Speaker: Oleksiy Klurman, University of Bristol

• Title: Counting sign changes

• Abstract: The aim of this talk is to discuss a simple way of producing sign changes of weighted multiplicative sums. We illustrate its applicability  by studying the number of sign changes of partial sums of "typical" real Dirichlet character and the number of sign changes of partial sums of random multiplicative functions as well as counting real zeros of Fekete polynomials. This is based on a joint work with Y. Lamzouri and M. Munsch.

Wednesday 12/6/24, 11:00am

Speaker: Javier Pliego, Università degli Studi di Genova

• Title: On Vu's theorem in Waring's problem

• Abstract: Answering a question of Nathanson about thin basis, Vu showed in 2000 the existence for k>1 and some s=s(k) of subsequences X_{k} satisfying that for every sufficiently large natural number n the number of solutions of n=x_{1}^{k}+...+x_{s}^{k} with x_{i}\in X_{k}, which we denote by R_{s}(X_{k},n), satisfies the relation R_{s}(X_{k},n) ≈ log(n). Soon after the previous paper was published, Wooley (2003) improved the constraint on the number of variables. In this talk we shall discuss new results concerning problems within this circle of ideas.

Wednesday 24/4/24, 12:00pm

Speaker: Sarvagya Jain, University of Turku

• Title: Distribution of Smooth Numbers in Short Intervals

• Abstract: A positive integer is considered y-smooth if all its prime factors are less than or equal to y. There are several works in the literature about the distribution of y-smooth numbers in intervals of the form [x, x+h], where h is significantly smaller than x. Notably, Matomäki and Radziwiłł proved that when y is a fixed power of x, almost all intervals [x, x+h] contain the expected proportion of y-smooth numbers, provided h tends to infinity with x. In this talk, I will focus on the ideas behind understanding the distribution of y-smooth numbers in short intervals, especially for smaller values of y.

Wednesday 17/4/24, 12:00pm

Speaker: Martin Čech, University of Turku

• Title: One-level density of families of L-functions

• Abstract: We will introduce the one-level density of zeros in families of L-functions, which describes the distribution of zeros close to the central point. By the conjectures of Katz and Sarnak, this distribution is related to the similar statistic of eigenvalues of random matrix ensembles.

We will then describe joint work with L. Devin, D. Fiorilli, K. Matomäki and Anders Södergren, which focuses on the family of Maass cusp forms.

Wednesday 10/4/24, 12:00pm

Speaker: Mikko Jaskari, University of Turku

• Title: On Chowla's conjecture and Siegel zeros

• Abstract: The Liouville function maps every integer that is a product of an odd number of prime numbers to -1 and every integer that is a product of an even number of prime numbers to +1. Chowla's conjecture then gives us an assumption on how the values of the Liouville function for consecutive integers should behave. If we assume the existence of Siegel zeros, we are able to prove certain bounds related to Chowla's conjecture. In this talk I will give some introduction on how to approach Chowla's conjecture under the assumption of Siegel zeros.

Wednesday 3/4/24, 12:00pm

Note the different room! Quantum, 2nd floor, Seminar Room M1

Speaker: Sebastian Zuniga-Alterman, University of Turku

• Title: Sieves and weighted sieves with switching

• Abstract: This talk consists of two parts. We first give a short introduction to sieves in general. Second, we discuss about weighted sieves and how the switching principle can be used, under some set of conditions, to detect (p,P_S) numbers for determined sets, where p is a prime and P_S is a natural number with at most S prime factors. We give some applications for particular weights and discuss how some weights are better than others in those examples. This is a joint work with K. Matomäki.

Wednesday 27/3/24, 12:00pm

Speaker: Yu-Chen Sun, University of Turku

• Title: Prime gaps and the least prime in an arithmetic progression

• Abstract: Prime gaps is one of the central topics in number theory. There are many breakthrough papers on both small and large gaps in recent years. The lower bound for the least prime in an arithmetic progression is closely related to large prime gaps. In this talk, we will firstly introduce the recent progress in prime gaps and ideas of proofs, then present our theorem concerning a combination of small and large prime gaps, and an improvement on a lower bound for the least prime in an arithmetic progression. This is a joint work with Hao Pan

Wednesday 20/3/24, 12:00pm

Speaker: Jesse Jääsaari, University of Turku

• Title: On the Real Zeros of Half-integral Weight Hecke Cusp Forms

• Abstract: We will discuss on-going work concerning the distribution of zeros of half-integral weight Hecke cusp forms of large weight on the surface $\Gamma_0(4)\backslash \H$. In particular, we are interested in the "real" zeros lying on the geodesic segments $\Re(s)=-1/2$ and $\Re(s)=0$. We will give estimates for the number of these zeros as the weight tends to infinity.

Wednesday 13/3/2024, 12:00pm

Speaker: Mengdi Wang, University of Turku

• Title: Local Fourier uniformity of divisor function

• Abstract:  Let d_k(n) be the k-fold divisor function, counting the ways n can be expressed as the product of k natural numbers, and let α be an irrational frequency. We expect that d_k(n) will exhibit orthogonality to e(nα) when the supported interval is not too short. In this talk, we are going to investigate the current research on this problem and introduce some general ideas on studying local Fourier uniformity of multiplicative functions.

Wednesday 6/3/2024, 12:00pm

Speaker: Pranendu Darbar, NTNU Trondheim

• Title: Mesoscopic fluctuation of zeros of L-functions over function fields via Selberg’s theorem

• Abstract: In this presentation, I will talk about the Selberg Central Limit (SCL) theorem as applied to shifted values of the logarithm of L-functions over function fields across different scales: microscopic (low-lying zero heights), mesoscopic, and macroscopic. Additionally, exploring the SCL for linear combinations of L-functions with varying shifts will unveil insights into their inter-dependencies, leading to the generation of a Gaussian process. It also produces a zero repulsion result for the corresponding L-function in the mesoscopic scale. This is a joint work with Fatma Cicek and Allysa Lumley.

Wednesday 28/2/2024, 12:00pm

Speaker: Stelios Sachpazis, University of Turku

• Title: Primes in arithmetic progressions and Siegel zeroes

• Abstract: Let $x\geqslant 1$ and let $q$ and $a$ be two coprime positive integers. As usual, 

$$\psi(x;q,a):=\sum_{n\leqslant x:\,n\equiv a (\text{mod}\,q)}\Lambda(n),$$

where $\Lambda$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of ''extreme'' Siegel zeroes and established an asymptotic formula for $\psi(x;q,a)$ beyond the limitations of GRH, with moduli $q$ beyond $\sqrt{x}$ yielding non-trivial information. In particular, they obtained a meaningful asymptotic for $q\leqslant x^{1/2+1/231}$.

We will see how one can relax the ''extremity'' of the exceptional zeroes and replace it by simply the definition of a Siegel zero. We will also discuss an idea to improve the Friedlander-Iwaniec regime and reach the range $q\leqslant x^{1/2+1/82-\varepsilon}$.This talk is based on on-going work.

Wednesday 21/2/2024   Winter Break

Wednesday 14/2/2024   Unfortunately, this week's speaker had to cancel their talk.

Wednesday 7/2/2024, 12:00pm

Speaker: Kaisa Matomäki, University of Turku

• Title: Mollifying Dirichlet L-functions and their derivatives

• Abstract: I will first discuss how mollified moments have been used to study non-vanishing of Dirichlet L-functions at the central point $s = 1/2$. Then I will describe some observations concerning mollifiers. In the final part I will briefly show how some of these observations can be used to improve on the known non-vanishing proportion of $L'(1/2, \chi)$ (and of higher derivatives). The talk is based on on-going joint work with Martin Čech.

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TALKS OF 2023

Thursday 14/12/23, 11:00am

Speaker: Mengdi Wang, University of Turku

• Title: Fourier uniformity of Liouville function in short intervals on average

• Abstract: The local Fourier uniformity conjecture asserts that on average, the Liouville function exhibits higher-order Fourier uniformity in very short intervals. Formally, as $H$ tends to infinity with $X$, the following estimate holds:

$$\int_X^{2X} \sup_{g} |\sum_{x<n\leq x+H} \lambda(n) F(g(n))\Gamma| dx=o(HX),$$

where $g:\mathcal Z\to G$ is a polynomial and $F(g(n)\Gamma)$ is a nilsequence. In this talk, we are going to elucidate recent advancements in this conjecture, with a primary focus on the paper "Phase Relations and Pyramids" authored by Miguel Walsh.

Thursday 7/12/23, 11:00am

Speaker: Kaisa Matomäki, University of Turku

• Title: Introduction to Harman's sieve method

• Abstract: Classical sieve methods based only on "type I information" are not able to detect primes. On the other hand, some combinatorial identities such as Vaughan's identity and Heath-Brown's identity allow one to detect primes when one has also sufficiently good "type II information". Harman's sieve allows one to obtain a lower bound for the number of primes when the arithmetic information is good enough but not sufficient for an application of those combinatorial identities.

In the talk I will explain what type I and type II information means, and how Harman's sieve works.

Thursday 30/11/23, 11:00am

Speaker: Andrew Granville, Université de Montréal

• Title: The ''pretentious'' approach to analytic number theory

• Abstract: Riemann’s 1859 monograph established that the distribution of primes can be understood through the study of the zeros of the Riemann zeta function (which all occur in its domain of meromorphic continuation). This extraordinary approach has subsequently been extended to many questions and is the main basis of the whole subject of analytic number theory. However time has highlighted important limitations to how far one can, in practice, push this approach, qualitatively (eg extending estimates to short intervals), quantitatively (eg in obtaining strong error terms), and in flexibility (eg in proving anaytic continuation for all Langland-Selberg L-functions). 

In 2009, Soundararajan and I identified an alternative approach avoiding the thorny issue of analytic continuation, building on the many ad hoc techniques used on various questions, in the hope that we could recover all of the results from the Riemann theory, prove new results that were not accessible previously, and develop new and rich perspectives for various open questions. These goals are now being realized in the work of many people. A substantial number of researchers have developed new methods based on these beginnings, sometimes beyond anything Sound and I had imagined. This in turn leads to many new questions and avenues, as we try to identify exactly what this theory is and how best to develop it.

In this talk, we will discuss and introduce this newish approach and identify some of the key questions that have been explored in this basic theory, including some exciting applications.

Thursday 23/11/23, 11:00am

Speaker: Sebastian Zuniga Alterman, University of Turku

• Title: Möbius function, an identity factory with applications I (joint work with O. Ramaré)

• Abstract: By using an identity relying a sum to an integral, we obtain a family of identities relying the averages $M(X)=\sum_{n\leq X}\mu(n)$ and $m(X)=\sum_{n\leq X}\frac{\mu(n)}{n}$. Further, by choosing some specific families, we study two summatory functions related to the Möbius function $\mu$, namely $\sum_{n\leq X}\frac{\mu(n)}{n^s}$ and $\sum_{n\leq X}\frac{\mu(n)}{n^s}\log(\frac{X}{n})$, where $s\in\mathbb{C}$ and $\Re s>0$. We also explore some applications and examples when $s$ is real.

Thursday 16/11/23, 11:00am

Speaker: Martin Čech, University of Turku

• Title: Meromorphic continuation of multiple Dirichlet series

• Abstract: The well-known conjectures of Keating and Snaith allow us to predict asymptotic formulas for moments in families of L-functions. We will consider the family of real Dirichlet L-functions, for which it is known that these asymptotic formulas can be proved provided the associated multiple Dirichlet series have a sufficiently distant meromorphic continuation. We will talk about the relation between the multiple Dirichlet series and classical approach, and give explicit description of the regions where the multiple Dirichlet series associated with the k-th moment can be defined.

Thursday 9/11/23, 11:00am

Speaker: Jesse Jääsaari, University of Turku

• Title: Mass Equidistribution for Saito-Kurokawa Lifts

• Abstract: In this talk we will discuss the distribution of mass of Siegel modular forms. In particular, we investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture in the weight aspect for Siegel cusp forms of degree 2. Assuming the Generalised Riemann Hypothesis (GRH) we establish QUE for Saito-Kurokawa lifts as the weight tends to infinity. As an application, we obtain equidistribution of the associated zero divisors under GRH. This is joint work with Steve Lester and Abhishek Saha.

Thursday 2/11/23, 11:00am

*Room change from this time onwards: Quantum, Seminar room M1, 2nd floor

Speaker: Andrew Granville, Université de Montréal

• Title: Sums of two squares one of which is a prime squared

• Abstract: There are ∼ π/2 ⋅ x/log x integers up to x that are the sum of two squares, one of which is the square of a prime. The error term surprisingly depends on the multiplication table problem, and we believe does not have a simple asymptotic but rather is a function of the fractional part of loglog x/log 2. Joint work with Cihan Sabuncu and Alisa Sedunova.

Thursday 26/10/23, 11:00am

Speaker: Mikko Jaskari, University of Turku

• Title: On explicit bounds for the prime number theorem

• Abstract: The prime number theorem is a famous theorem that captures the main asymptotic essence of the distribution of prime numbers. This theorem is a consequence of the fact that all the zeros of the Riemann zeta function have a real part smaller than 1. However, the prime number theorem gives just an approximation for counting primes. We will take a look at how we can obtain unconditional explicit bounds for prime counting functions by using the information we know about zeros of the zeta function.

Thursday 19/10/23, 11:00am

Speaker: Yu-Chen Sun, University of Turku

• Title: On additive complements of the squares

• Abstract: Let $\mathcal{S}=\{1^2,2^2,3^2,...\}$ be the set of squares and $\mathcal{W}=\{w_n\}_{n=1}^{\infty} \subset \mathbb{N}$ be an additive complement of $\mathcal{S}$ so that $\mathcal{S} + \mathcal{W} \subset \{n \in \mathbb{N}: n \geq N_0\}$ for some $N_0$. Let $\mathcal{R}_{\mathcal{S},\mathcal{W}}(n) = \#\{(s,w):n=s+w, s\in \mathcal{S}, w\in \mathcal{W}\}$. In this talk, we will introduce some progress on the lower bound of $\sum_{n=1}^N \mathcal{R}_{\mathcal{S},\mathcal{W}}(n)$ and the behaviour of $w_n$. This is a joint work with Yuchen Ding, Li-Yuan Wang and Yutong Xia.

Thursday 12/10/23, 11:00am

Speaker: Stelios Sachpazis, University of Turku

• Title: A strongly uniform estimate for primes in arithmetic progressions revisited

• Abstract: Let a and q be two coprime positive integers. In 1944, Linnik proved his celebrated theorem concerning the size of the smallest prime p(q,a) in the arithmetic progression a(mod q). In his attempt to prove this result, Linnik established an estimate for the sums of the von Mangoldt function Λ on arithmetic progressions. His work on p(q,a) was later simplified, but the simplified proofs relied in one form or another on the same advanced tools that he originally used. The last two decades, some more elementary approaches for Linnik’s theorem have appeared. Nonetheless, none of them furnishes an estimate of the same quantitative strength as the ones that the classical methods obtain for the sums of Λ on arithmetic progressions. In this talk, we will see how one can seal this gap and recover an analogue of Linnik’s estimate, that is a strongly uniform estimate for primes in arithmetic progressions, by largely elementary means. The ideas that I will describe build on methods from the treatment of Koukoulopoulos on multiplicative functions with small partial sums and his pretentious proof of the prime number theorem in arithmetic progressions.

Thursday 5/10/23, 11.00am

Speaker: Mengdi Wang, University of Turku

• Title: Counting polynomial patterns in sparse sets

• Abstract: In this talk, I plan to take counting linear forms in primes as an example to illustrate the general ideas of higher-order Fourier analysis for counting linear patterns of finite complexity in sparse sets. Subsequently, I will introduce some unconventional yet valuable generalization results that can be used to count polynomial patterns in sparse sets. This talk is based on joint work with Lilian Matthiesen.

Thursday 28/9/23, 11:00am

Speaker: Joni Teräväinen, University of Turku

• Title: On the Fourier uniformity problem for small sets

• Abstract: The Fourier uniformity conjecture of Tao states that the maximal exponential sum of the Liouville function over a short interval [x,x+H] exhibits cancellation for almost all x. This conjecture and its generalizations are closely related to Chowla's conjecture. One may consider a simplification of this problem where the maximal exponential sum is only taken over a subset of all the frequencies in [0,1]. We show that this relaxed Fourier uniformity conjecture holds if the frequency is restricted to lie in any closed set of measure zero. We also obtain some results in the case of more general polynomial and nilsequence twists. This is joint work with A. Kanigowski, M. Lemańczyk and F. Richter.

Thursday 21/9/23, 11:00am

Speaker: Kaisa Matomäki, University of Turku

• Title: Detecting primes in multiplicatively structured sequences

• Abstract: I will discuss a new sieve set-up which allows one to find prime numbers in sequences that have a suitable multiplicative structure and good "type I information". Among other things, the method gives a new L-function free proof of Linnik's theorem concerning the least prime in an arithmetic progression. The talk is based on on-going joint work with Jori Merikoski and Joni Teräväinen.