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BEGIN:VEVENT
SUMMARY:Alexander Mangerel (CRM\, Montreal)
DTSTART;VALUE=DATE-TIME:20200716T170000Z
DTEND;VALUE=DATE-TIME:20200716T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/1
DESCRIPTION:Title: Squarefree Integers in Arithmetic Progressions to Smooth/Friable Mod
uli\nby Alexander Mangerel (CRM\, Montreal) as part of ViBraNT (Virtua
l Brazilian Number Theory seminar)\n\n\nAbstract\nI will discuss how to ob
tain an asymptotic formula (with power-savings error term) for the count o
f squarefree integers in an arithmetic progression when the modulus does n
ot have any large prime factors\, using a blend of cohomological technique
s and p-adic methods. For this collection of moduli our results go beyond
the best existing admissible range obtained recently by Nunes.\n\nThis is
joint work with C. Perret-Gentil.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jing Zhao (Max Planck)
DTSTART;VALUE=DATE-TIME:20200730T170000Z
DTEND;VALUE=DATE-TIME:20200730T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/2
DESCRIPTION:Title: Discrete negative moments of $\\zeta'(\\rho)$\nby Jing Zhao (Max
Planck) as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\
nAbstract\nI shall talk about a recent result of a joint work with Winston
Heap and Junxian Li. We proved lower bounds for the discrete negative 2kt
h moments of the derivative of the Riemann zeta function\, which agrees wi
th a conjecture of Gonek and Hejhal. We also proved a general formula for
the discrete twisted 2nd moment of the Riemann zeta function. This agrees
with a conjecture of Conrey and Snaith.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ramon Nunes (UFC)
DTSTART;VALUE=DATE-TIME:20200723T170000Z
DTEND;VALUE=DATE-TIME:20200723T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/3
DESCRIPTION:Title: Moments of k-free numbers in arithmetic progressions.\nby Ramon
Nunes (UFC) as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n
\n\nAbstract\nWe will discuss the moments of distribution of $k$-free numb
ers in arithmetic progressions for which we show estimates improving on pr
evious results by Hall and the author. We will present conjectures due mai
nly to Montgomery and according to which our results are nearly optimal. T
he key new idea is to complement Hall's argument based on the so-called fu
ndamental lemma of Montgomery and Vaughan with some elementary estimates o
n the region where the previous approach is wasteful.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winston Heap (Max Planck)
DTSTART;VALUE=DATE-TIME:20200806T170000Z
DTEND;VALUE=DATE-TIME:20200806T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/4
DESCRIPTION:Title: Random multiplicative functions and a model for the Riemann zeta fun
ction\nby Winston Heap (Max Planck) as part of ViBraNT (Virtual Brazil
ian Number Theory seminar)\n\n\nAbstract\nWe look at a weighted sum of ran
dom multiplicative functions and view this as a model for the Riemann zeta
function. We investigate various aspects including its high moments\, dis
tribution and maxima.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrés Chirre (NTNU)
DTSTART;VALUE=DATE-TIME:20200924T170000Z
DTEND;VALUE=DATE-TIME:20200924T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/5
DESCRIPTION:Title: The behavior of the argument of the Riemann zeta-function\nby An
drés Chirre (NTNU) as part of ViBraNT (Virtual Brazilian Number Theory se
minar)\n\n\nAbstract\nIn this talk we will review some recent results rela
ted to the argument function of the Riemann zeta function\, assuming the R
iemann hypothesis. The use of bandlimited approximations and the resonance
method will help us to describe the behavior of this oscillatory function
. Finally\, we will extend these results to the antiderivatives of the arg
ument function that encode\, in a certain way\, information about the argu
ment function.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (Warwick)
DTSTART;VALUE=DATE-TIME:20200813T170000Z
DTEND;VALUE=DATE-TIME:20200813T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/6
DESCRIPTION:Title: Moments of Weyl sums\, restriction estimates\, and diophantine equat
ions\nby Sam Chow (Warwick) as part of ViBraNT (Virtual Brazilian Numb
er Theory seminar)\n\n\nAbstract\nWe discuss the role played by moment est
imates for Weyl sums in counting solutions to diophantine equations\, and
the analogous role played by restriction estimates in the combinatorial th
eory of diophantine equations. Additionally\, we sketch some modern techni
ques used to prove such estimates.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleksiy Klurman (Max Planck and University of Bristol)
DTSTART;VALUE=DATE-TIME:20200827T170000Z
DTEND;VALUE=DATE-TIME:20200827T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/7
DESCRIPTION:Title: Monotone chains in multiplicative sets\nby Oleksiy Klurman (Max
Planck and University of Bristol) as part of ViBraNT (Virtual Brazilian Nu
mber Theory seminar)\n\n\nAbstract\nIt is a rather difficult task to show
that given a general sequence $a(1)\,a(2)\\dots$ and admissible set of int
egers $h_1\,h_2\\dots h_k$ each possible arrangement $a(n+h_1)\\le a(n+h_2
)\\le\\dots a(n+h_k)$ occurs for infinitely many integers $n.$\nIn this ta
lk\, we describe how recent advances in multiplicative number theory and t
heory of automorphic forms allow us to shed some light on such questions r
elated to the coefficients of Hecke cusp forms\n(based on a joint work wit
h A. Mangerel).\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Harper (Warwick)
DTSTART;VALUE=DATE-TIME:20200820T170000Z
DTEND;VALUE=DATE-TIME:20200820T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/8
DESCRIPTION:Title: Multiplicative chaos in number theory\nby Adam Harper (Warwick)
as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract
\nMultiplicative chaos is the general name for a family of probabilistic o
bjects\, which can be thought of as the random measures obtained by taking
the exponential of correlated Gaussian random variables. Multiplicative c
haos turns out to be closely connected with various problems in analytic n
umber theory\, including the value distribution of the Riemann zeta functi
on on the critical line\, the moments of character sums\, and various mode
l versions of these problems. I will try to give a gentle introduction to
these issues and connections\, presenting both results and open problems w
ithout assuming too much background knowledge. (This will be a lightly upd
ated version of the talk I gave last year in Cetraro.)\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gady Kozma (Weizmann Institute of Science)
DTSTART;VALUE=DATE-TIME:20200910T170000Z
DTEND;VALUE=DATE-TIME:20200910T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/9
DESCRIPTION:Title: Random polynomials\, sieves and Dedekind zeta functions\nby Gady
Kozma (Weizmann Institute of Science) as part of ViBraNT (Virtual Brazili
an Number Theory seminar)\n\n\nAbstract\nWhat is the probability that a ra
ndom polynomial with coefficients +/-1 is irreducible over the rationals?
This fascinating problem\, still open\, has seen a lot of progress in the
last few years. We will survey this progress\, with particular emphasis on
new results\, joint with Lior Bary-Soroker and Dimitris Koukoulopoulos.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Gerspach (ETH\, Zürich)
DTSTART;VALUE=DATE-TIME:20200903T170000Z
DTEND;VALUE=DATE-TIME:20200903T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/10
DESCRIPTION:Title: Low pseudomoments of the Riemann zeta function and its powers\n
by Maxim Gerspach (ETH\, Zürich) as part of ViBraNT (Virtual Brazilian Nu
mber Theory seminar)\n\n\nAbstract\nThe pseudomoments of the Riemann zeta
function are the moments of the partial sums associated to zeta on the cri
tical line. Using probabilistic methods of Harper\, we provide bounds whic
h imply the order of magnitude of all pseudomoments. We also provide upper
and lower bounds for the pseudomoments of the powers of zeta that are alm
ost-matching when combined with previous bounds of Bondarenko\, Heap and S
eip\, and turn out to behave in a somewhat different manner. In this talk\
, I will mostly try to give a heuristic argument in support of the results
by relating these quantities to moments of random multiplicative function
s and to random Euler products.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucile Devin (University of Gothenburg)
DTSTART;VALUE=DATE-TIME:20201001T170000Z
DTEND;VALUE=DATE-TIME:20201001T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/11
DESCRIPTION:Title: Chebyshev’s bias and sums of two squares\nby Lucile Devin (Un
iversity of Gothenburg) as part of ViBraNT (Virtual Brazilian Number Theor
y seminar)\n\n\nAbstract\nStudying the secondary terms of the Prime Number
Theorem in Arithmetic Progressions\, Chebyshev claimed that there are mor
e prime numbers congruent to 3 modulo 4 than to 1 modulo 4. We will explai
n and qualify this claim following the framework of Rubinstein and Sarnak.
Then we will see how this framework can be adapted to other questions on
the distribution of prime numbers. This will be illustrated by a new Cheby
shev-like claim : there are “more” prime numbers that can be written
as a sum of two squares with the even square larger than the odd square th
an the other way around.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitris Koukoulopoulos (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20201015T170000Z
DTEND;VALUE=DATE-TIME:20201015T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/12
DESCRIPTION:Title: How concentrated can the divisors of a typical integer be?\nby
Dimitris Koukoulopoulos (Université de Montréal) as part of ViBraNT (Vir
tual Brazilian Number Theory seminar)\n\n\nAbstract\nThe Delta function me
asures the concentration of the sequence of divisors of an integer. Specif
ically\, given an integer $n$\, we write $\\Delta(n)$ for the maximum over
$y$ of the number of divisors of $n$ lying in the dyadic interval $[y\,2y
]$. It was introduced by Hooley in 1979 because of its connections to vari
ous problems in Diophantine equations and approximation. In 1981\, Maier a
nd Tenenbaum proved that $\\Delta(n)>1$ for almost all integers $n$\, thus
settling a 1948 conjecture due to Erdös. In subsequent work\, they prove
d that $(\\log\\log n)^{c+o(1)}\\le \\Delta(n)\\le (\\log\\log n)^{\\log2+
o(1)}$\, where $c=(\\log2)/\\log(\\frac{1-1/\\log 27}{1-\\log3})\\approx 0
.33827$ for almost all integers $n$. In addition\, they conjectured that $
\\Delta(n)=(\\log\\log n)^{c+o(1)}$ for almost all $n$. In this talk\, I w
ill present joint work with Ben Green and Kevin Ford that disproves the Ma
ier-Tenenbaum conjecture by replacing the constant $c$ in the lower bound
by another constant $c'=0.35332277\\dots$ that we believe is optimal. We a
lso prove analogous results about permutations and polynomials over finite
fields by reducing all three cases to an archetypal probabilistic model.\
n
LOCATION:https://researchseminars.org/talk/NumberTheory2/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Granville (Universite de Montréal)
DTSTART;VALUE=DATE-TIME:20200917T170000Z
DTEND;VALUE=DATE-TIME:20200917T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/13
DESCRIPTION:Title: Heuristics and computations for primes in short intervals\; and sie
ves and Siegel zeros\nby Andrew Granville (Universite de Montréal) as
part of ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract\n
We describe joint work with Allysa Lumley in which we try to get an idea o
f the range of values the number of primes can take in an interval of leng
th y near to x. Our understanding is limited by our limited understanding
of the sieve and\, if we have time\, we will explain how that understandi
ng cannot be improved without showing that there are no Siegel zeros\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Bloom (Cambridge)
DTSTART;VALUE=DATE-TIME:20201022T170000Z
DTEND;VALUE=DATE-TIME:20201022T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/14
DESCRIPTION:Title: Additive structure in dense sets of integers\nby Thomas Bloom (
Cambridge) as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n\
n\nAbstract\nHow much additive structure can we guarantee in sets of integ
ers\, knowing only their density? The study of which density thresholds ar
e sufficient to guarantee the existence of various kinds of additive struc
tures is an old and fascinating subject with connections to analytic numbe
r theory\, additive combinatorics\, and harmonic analysis.\n\nIn this talk
we will discuss recent progress on perhaps the most well-known of these t
hresholds: how large do we need a set of integers to be to guarantee the e
xistence of a three-term arithmetic progression? In recent joint work with
Olof Sisask we broke through the logarithmic density barrier for this pro
blem\, establishing in particular that if a set is dense enough such that
the sum of reciprocals diverges\, then it must contain a three-term arithm
etic progression\, establishing the first case of an infamous conjecture o
f Erdos.\n\nWe will give an introduction to this problem and sketch some o
f the recent ideas that have made this progress possible. We will also dis
cuss a recent application to the density threshold of a set containing no
square differences.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Peluse (Oxford)
DTSTART;VALUE=DATE-TIME:20201008T170000Z
DTEND;VALUE=DATE-TIME:20201008T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/15
DESCRIPTION:Title: An asymptotic version of the prime power conjecture for perfect dif
ference sets\nby Sarah Peluse (Oxford) as part of ViBraNT (Virtual Bra
zilian Number Theory seminar)\n\n\nAbstract\nA subset D of a finite cyclic
group Z/mZ is called a "perfect difference set" if every nonzero element
of Z/mZ can be written uniquely as the difference of two elements of D. If
such a set exists\, then a simple counting argument shows that m=n^2+n+1
for some nonnegative integer n. Singer constructed examples of perfect dif
ference sets in Z/(n^2+n+1)Z whenever n is a prime power\, and it is an ol
d conjecture that these are the only such n for which a perfect difference
set exists. In this talk\, I will discuss a proof of an asymptotic versio
n of this conjecture: the number of n less than N for which Z/(n^2+n+1)Z c
ontains a perfect difference set is ~N/log(N).\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Ford (University of Illinois at Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20201203T170000Z
DTEND;VALUE=DATE-TIME:20201203T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/16
DESCRIPTION:Title: Divisors of integers\, permutations and polynomials\nby Kevin F
ord (University of Illinois at Urbana-Champaign) as part of ViBraNT (Virtu
al Brazilian Number Theory seminar)\n\n\nAbstract\nWe describe a probabili
stic model that describes the statistical behavior of the divisors of inte
gers\, divisors of permutations and divisors of polynomials over a finite
field. We will discuss how this can be used to obtain new bounds on the c
oncentration of divisors of integers\, improving a result of Maier and Ten
enbaum. This is joint work with Ben Green and Dimitris Koukoulopoulos.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Maynard (Oxford)
DTSTART;VALUE=DATE-TIME:20201029T170000Z
DTEND;VALUE=DATE-TIME:20201029T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/17
DESCRIPTION:Title: Primes in arithmetic progressions to large moduli\nby James May
nard (Oxford) as part of ViBraNT (Virtual Brazilian Number Theory seminar)
\n\n\nAbstract\nI'll talk about some recent work extending the Bombieri-Vi
nogradov Theorem to moduli larger than x^{1/2} provided the moduli have a
conveniently sized divisor. In different formulations\, this allows us to
handle moduli as large as x^{3/5}\, or allows for complete uniformity with
respect to the residue class as in the original Bombieri-Vinogradov theor
em.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (University of Turku)
DTSTART;VALUE=DATE-TIME:20201119T170000Z
DTEND;VALUE=DATE-TIME:20201119T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/18
DESCRIPTION:Title: Almost primes in almost all very short intervals\nby Kaisa Mato
mäki (University of Turku) as part of ViBraNT (Virtual Brazilian Number T
heory seminar)\n\n\nAbstract\nBy probabilistic models one expects that\, a
s soon as $h \\to \\infty$ with $X \\to \\infty$\, short intervals of the
type $(x- h \\log X\, x]$ contain primes for almost all $x \\in (X/2\, X]$
. However\, this is far from being established. In the talk I discuss rela
ted questions and in particular describe how to prove the above claim when
one is satisfied with finding $P_2$-numbers (numbers that have at most tw
o prime factors) instead of primes.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maksym Radziwill (Caltech)
DTSTART;VALUE=DATE-TIME:20201112T170000Z
DTEND;VALUE=DATE-TIME:20201112T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/19
DESCRIPTION:Title: The Fyodorov-Hiary-Keating conjecture\nby Maksym Radziwill (Cal
tech) as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAb
stract\nI will discuss recent progress on the Fyodorov-Hiary-Keating conje
cture\non the distribution of the local maximum of the Riemann zeta-functi
on. This is joint\nwork with Louis-Pierre Arguin and Paul Bourgade.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Εfthymios Sofos (University of Glasgow)
DTSTART;VALUE=DATE-TIME:20201105T170000Z
DTEND;VALUE=DATE-TIME:20201105T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/20
DESCRIPTION:Title: Schinzel Hypothesis with probability 1 and rational points\nby
Εfthymios Sofos (University of Glasgow) as part of ViBraNT (Virtual Brazi
lian Number Theory seminar)\n\n\nAbstract\nJoint work with Alexei Skorobog
atov\, preprint: https://arxiv.org/abs/2005.02998. Schinzel's Hypothesis s
tates that every integer polynomial satisfying certain congruence conditio
ns represents infinitely many primes. It is one of the main problems in an
alytic number theory but is completely open\, except for polynomials of de
gree 1. We describe our recent proof of the Hypothesis for 100% of polynom
ials (ordered by size of coefficients). We use this to prove that\, with p
ositive probability\, Brauer--Manin controls the Hasse principle for Chât
elet surfaces.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sandoel Vieira (IMPA)
DTSTART;VALUE=DATE-TIME:20201126T170000Z
DTEND;VALUE=DATE-TIME:20201126T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/21
DESCRIPTION:Title: M\\L is not closed\nby Sandoel Vieira (IMPA) as part of ViBraNT
(Virtual Brazilian Number Theory seminar)\n\n\nAbstract\nIn this talk we
will describe joint work with C. G. Moreira\, C. Matheus and D. Lima in wh
ich we proved that $M\\setminus L$ is not a closed subset of $\\mathbb{R}$
. For that\, we show that $1+3/\\sqrt{2}$ is a point of the Lagrange spect
rum $L$ which is accumulated by a sequence of elements of the complement $
M\\setminus L$ of the Lagrange spectrum in the Markov spectrum $M$.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Ramaré (Aix-Marseille)
DTSTART;VALUE=DATE-TIME:20210114T170000Z
DTEND;VALUE=DATE-TIME:20210114T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/22
DESCRIPTION:Title: An additive question in multiplicative number theory\nby Olivie
r Ramaré (Aix-Marseille) as part of ViBraNT (Virtual Brazilian Number The
ory seminar)\n\n\nAbstract\nWhile studying the representation of a congrue
nce class or a ray-class by a product of three small primes\, we stumbled
on an auxiliary additive combinatorics question involving sum-free sets in
finite abelian groups that seems to be new. The aim of the talk is to pre
sent this question.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cathy Swaenepoel (Paris Diderot)
DTSTART;VALUE=DATE-TIME:20210121T170000Z
DTEND;VALUE=DATE-TIME:20210121T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/23
DESCRIPTION:Title: Prime numbers with preassigned digits\nby Cathy Swaenepoel (Par
is Diderot) as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n
\n\nAbstract\nBourgain (2015) estimated the number of prime numbers with a
proportion c>0 of preassigned digits in base 2 (c is an absolute constant
not specified). We present a generalization of this result in any base $g
\\geq2$ and we provide explicit admissible values for the proportion c dep
ending on g. Our proof\, which adapts\, develops and refines Bourgain’s
strategy\, is based on the circle method and combines techniques from harm
onic analysis together with results on zeros of Dirichlet L-functions\, no
tably a very strong zero-free region due to Iwaniec.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kyle Pratt (Oxford)
DTSTART;VALUE=DATE-TIME:20210128T170000Z
DTEND;VALUE=DATE-TIME:20210128T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/24
DESCRIPTION:Title: Landau-Siegel zeros and central values of L-functions\nby Kyle
Pratt (Oxford) as part of ViBraNT (Virtual Brazilian Number Theory seminar
)\n\n\nAbstract\nResearchers have tried for many years to eliminate the po
ssibility of Landau-Siegel zeros---certain exceptional counterexamples to
the Generalized Riemann Hypothesis. Often one thinks of these zeros as bei
ng a severe nuisance\, but there are many situations in which their existe
nce allows one to prove spectacular\, though illusory\, results. I will re
view some of this history and some of these results. In the latter portion
of the talk I will discuss recent work\, joint with H. M. Bui and Alexand
ru Zaharescu\, in which we show that the existence of Landau-Siegel zeros
has implications for the behavior of $L$-functions at the central point.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Munsch (Graz)
DTSTART;VALUE=DATE-TIME:20210204T170000Z
DTEND;VALUE=DATE-TIME:20210204T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/25
DESCRIPTION:Title: Pair correlation of sequences: metric results and a modified additi
ve energy\nby Marc Munsch (Graz) as part of ViBraNT (Virtual Brazilian
Number Theory seminar)\n\n\nAbstract\nThe uniform distribution of a seque
nce $\\{x_n\\}_{n\\geq 1}$ measures the pseudo-random behavior at a global
scale. At a more localized\nscale\, we can study the pair correlation for
sequences in the unit interval. Pseudo-random behavior with respect to th
is statistic is called Poissonian behavior. The metric theory of pair corr
elations of sequences of the form $(a_n\\alpha)_{n \\geq 1}$ has been pio
neered by Rudnick\, Sarnak and Zaharescu. Recently\, a general framework w
as developed which gives a criterion for Poissonian pair correlation of su
ch sequences for almost $\\alpha \\in (0\,1)$\, in terms of the additive e
nergy of the integer sequence $\\{a_n\\}_{n \\geq 1}$. In the present talk
we will discuss a similar framework in the more delicate case where $\\{a
_n\\}_{n \\geq 1}$ is a sequence of reals. We give a criterion involving a
modified version of the additive energy expressed via a diophantine inequ
ality. We give several concrete applications of our method and present som
e open problems. This is joint work with Christoph Aistleitner and Daniel
EL-Baz.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joni Teräväinen (Oxford)
DTSTART;VALUE=DATE-TIME:20210211T170000Z
DTEND;VALUE=DATE-TIME:20210211T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/26
DESCRIPTION:Title: Higher order uniformity of the Möbius function\nby Joni Teräv
äinen (Oxford) as part of ViBraNT (Virtual Brazilian Number Theory semina
r)\n\n\nAbstract\nI will discuss recent work where we prove that the Möbi
us function is orthogonal to a wide class of phase functions (including al
l polynomial phases) on almost all very short intervals. I will also discu
ss applications to superpolynomial word complexity for the Liouville seque
nce and to a new averaged version of Chowla's conjecture. This is joint wo
rk with Kaisa Matomäki\, Maksym Radziwiłł\,Terence Tao and Tamar Ziegle
r.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andreas Weingartner (South Utah university)
DTSTART;VALUE=DATE-TIME:20210218T170000Z
DTEND;VALUE=DATE-TIME:20210218T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/27
DESCRIPTION:Title: An extension of the Siegel-Walfisz theorem\nby Andreas Weingart
ner (South Utah university) as part of ViBraNT (Virtual Brazilian Number T
heory seminar)\n\n\nAbstract\nWe extend the Siegel-Walfisz theorem to a fa
mily of integer\nsequences that are characterized by constraints on the si
ze of the\nprime factors. Besides prime powers\, this family includes smoo
th\nnumbers\, almost primes and practical numbers.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Caroline Turnage-Butterbaugh (Carleton college)
DTSTART;VALUE=DATE-TIME:20210225T170000Z
DTEND;VALUE=DATE-TIME:20210225T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/28
DESCRIPTION:Title: Gaps between zeros of the Riemann zeta-function\nby Caroline Tu
rnage-Butterbaugh (Carleton college) as part of ViBraNT (Virtual Brazilian
Number Theory seminar)\n\n\nAbstract\nLet $0 < \\gamma_1 \\le \\gamma_2 \
\le \\cdots $ denote the\nordinates of the complex zeros of the Riemann ze
ta-function function\nin the upper half-plane. The average distance betwee
n $\\gamma_n$ and\n$\\gamma_{n+1)$ is $2\\pi / \\log \\gamma_n$ as $n\\to
\\infty$. An\nimportant goal is to prove unconditionally that these distan
ces\nbetween consecutive zeros can much\, much smaller than the average fo
r\na positive proportion of zeros. We will discuss the motivation behind\n
this endeavor\, progress made assuming the Riemann Hypothesis\, and\nrecen
t work with A. Simonič and T. Trudgian to obtain an unconditional\nresult
.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brad Rodgers (Queen’s University)
DTSTART;VALUE=DATE-TIME:20210415T170000Z
DTEND;VALUE=DATE-TIME:20210415T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/29
DESCRIPTION:Title: The distribution of random polynomials with multiplicative coeffici
ents\nby Brad Rodgers (Queen’s University) as part of ViBraNT (Virtu
al Brazilian Number Theory seminar)\n\n\nAbstract\nA classic paper of Sale
m and Zygmund investigates the distribution of trigonometric polynomials w
hose coefficients are chosen randomly (say +1 or -1 with equal probability
) and independently. Salem and Zygmund characterized the typical distribut
ion of such polynomials (gaussian) and the typical magnitude of their sup-
norms (a degree N polynomial typically has sup-norm of size $\\sqrt{N \\lo
g N}$ for large N). In this talk we will explore what happens when a weak
dependence is introduced between coefficients of the polynomials\; namely
we consider polynomials with coefficients given by random multiplicative f
unctions. We consider analogues of Salem and Zygmund's results\, exploring
similarities and some differences.\n\nSpecial attention will be given to
a beautiful point-counting argument introduced by Vaughan and Wooley which
ends up being useful.\n\nThis is joint work with Jacques Benatar and Alon
Nishry.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sandro Bettin (University of Genova)
DTSTART;VALUE=DATE-TIME:20210311T170000Z
DTEND;VALUE=DATE-TIME:20210311T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/30
DESCRIPTION:Title: Modularity and distribution of quantum knots invariants\nby San
dro Bettin (University of Genova) as part of ViBraNT (Virtual Brazilian Nu
mber Theory seminar)\n\n\nAbstract\nWe consider Zagier's modularity conjec
ture for the colored Jones\npolynomials of hyperbolic knots. We prove this
conjecture in some\ncases and show that\, in the case of the 4_1 knot\, o
ne can also deduce\na law of large for the values of the colored Jones pol
ynomial at roots\nof unity. This is joint work with Sary Drappeau.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniele Mastrostefano (Warwick)
DTSTART;VALUE=DATE-TIME:20210318T170000Z
DTEND;VALUE=DATE-TIME:20210318T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/31
DESCRIPTION:Title: The partial sum of a random multiplicative function on integers wit
h a large prime factor\nby Daniele Mastrostefano (Warwick) as part of
ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract\nLet $f(n)
$ be a Rademacher random multiplicative function. We prove that\, for any
$\\epsilon>0$ and as $x\\rightarrow +\\infty$\, we almost surely have\n\n$
\\sum_{n\\leq x\,\\\; \\\\ P(n)>\\sqrt{x}} f(n)\\ll\\sqrt{x}(\\log\\log x)
^{1/4+\\epsilon}\,$\n\nwhere $P(n)$ stands for the largest prime factor of
$n$. \nThis is close to be sharp and gives an indication of the size of t
he largest fluctuations of the full partial sum.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marco Aymone (UFMG)
DTSTART;VALUE=DATE-TIME:20210408T170000Z
DTEND;VALUE=DATE-TIME:20210408T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/34
DESCRIPTION:Title: Some oscillation theorems in analytic and probabilistic Number Theo
ry\nby Marco Aymone (UFMG) as part of ViBraNT (Virtual Brazilian Numbe
r Theory seminar)\n\n\nAbstract\nThis talk will be divided into two indepe
ndent parts. In the first part of the talk I will discuss the prime number
race mod 4: Usually one assumes standards conjectures as GRH to deduce so
me results that captures the intuition behind the Tchébyshev bias -- I wi
ll do the other way around. In the second part of the talk I will discuss
a recent work with Winston Heap and Jing Zhao on sign changes of the parti
al sums of a random multiplicative function.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Hughes (University of York)
DTSTART;VALUE=DATE-TIME:20210325T170000Z
DTEND;VALUE=DATE-TIME:20210325T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/35
DESCRIPTION:Title: A Random Matrix Model for Gram's Law\nby Chris Hughes (Universi
ty of York) as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n
\n\nAbstract\nIt is well known that the counting function for the Riemann\
nzeta zeros\, N(T)\, has a smooth main term and a much smaller\ndiscontinu
ous correction term\, S(T). Gram's Law is the observation\nthat between co
nsecutive points where the smooth part of the counting\nfunction is an int
eger\, there typically is exactly one zeta zero. This\n"Law" doesn't hold
all the time\, and we will use random matrix theory\nto model the proporti
on of time the law holds for. The flavour of\nrandom matrix theory that no
rmally models the Riemann zeros is the\nunitary group. However\, studying
Gram's Law requires the special\nunitary group\, where many of the useful
techniques for random unitary\nmatrices fail to hold. Much of this work wa
s done jointly with my\nformer PhD student Catalin Hanga.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Youness Lamzouri (Institut Elie Cartan de Lorraine)
DTSTART;VALUE=DATE-TIME:20210506T170000Z
DTEND;VALUE=DATE-TIME:20210506T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/36
DESCRIPTION:Title: Zeros of linear combinations of L-functions near the critical line<
/a>\nby Youness Lamzouri (Institut Elie Cartan de Lorraine) as part of ViB
raNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract\nIn this talk
\, I will present a recent joint work with Yoonbok Lee\, where we investig
ate the number of zeros of linear combinations of $L$-functions in the vic
inity of the critical line. More precisely\, we let $L_1\, \\dots\, L_J$ b
e distinct primitive $L$-functions belonging to a large class (which conje
cturally contains all $L$-functions arising from automorphic representatio
ns on $\\text{GL}(n)$)\, and $b_1\, \\dots\, b_J$ be real numbers. Our mai
n result is an asymptotic formula for the number of zeros of $F(\\sigma+it
)=\\sum_{j\\leq J} b_j L_j(\\sigma+it)$ in the region $\\sigma\\geq 1/2+1/
G(T)$ and $t\\in [T\, 2T]$\, uniformly in the range $\\log \\log T \\leq G
(T)\\leq (\\log T)^{\\nu}$\, where $\\nu\\asymp 1/J$. This establishes a g
eneral form of a conjecture of Hejhal in this range. The strategy of the p
roof relies on comparing the distribution of $F(\\sigma+it)$ to that of an
associated probabilistic random model.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chantal David (Concordia University)
DTSTART;VALUE=DATE-TIME:20210422T170000Z
DTEND;VALUE=DATE-TIME:20210422T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/37
DESCRIPTION:Title: One-Level density for cubic characters over the Eisenstein field\nby Chantal David (Concordia University) as part of ViBraNT (Virtual Bra
zilian Number Theory seminar)\n\n\nAbstract\nWe show that the one-level de
nsity for $L$-functions associated with the cubic residue symbols $\\chi_n
$\, with $n \\in \\mathbb{Z}[\\omega]$ square-free\, satisfies the Katz-Sa
rnak conjecture for all test functions whose Fourier transforms are suppor
ted in $(-13/11\, 13/11)$\, under GRH. This is the first result extending
the support outside the trivial range $(-1\, 1)$ for a family of cubic $L$
-functions. This implies that a positive proportion of the $L$-functions a
ssociated with these characters do not vanish at the central point $s = 1/
2$. A key ingredient is a bound on an average of generalized cubic Gauss s
ums at prime arguments\, whose proof is based on the work of Heath-Brown a
nd Patterson.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louis Chen (National University of Singapore)
DTSTART;VALUE=DATE-TIME:20210527T170000Z
DTEND;VALUE=DATE-TIME:20210527T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/39
DESCRIPTION:Title: A probabilistic approach to the Erdös-Kac theorem for additive fun
ctions\nby Louis Chen (National University of Singapore) as part of Vi
BraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract\nWe present
a new approach to assessing the rates of convergence to the Gaussian and P
oisson distributions in the Erdös-Kac theorem for additive arithmetic fun
ctions of a random integer. Our approach is probabilistic\, working direct
ly on spaces of random variables without any use of Fourier analytic metho
ds. Of the methods we used is Stein’s method. Our results generalize the
existing ones in the literature. This talk is based on joint work with Ar
turo Jaramillo and Xiaochuan Yang.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anders Södergren (Chalmers)
DTSTART;VALUE=DATE-TIME:20210610T170000Z
DTEND;VALUE=DATE-TIME:20210610T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/40
DESCRIPTION:Title: Can a random lattice and its dual be independent?\nby Anders S
ödergren (Chalmers) as part of ViBraNT (Virtual Brazilian Number Theory s
eminar)\n\n\nAbstract\nIn this talk I will discuss Rogers' mean value form
ula in the space of unimodular lattices as well as a recent generalization
of Rogers' formula. In particular\, I will describe a formula for mean va
lues of products of Siegel transforms with arguments taken from both a lat
tice and its dual lattice. The main application is a result on the joint d
istribution of the vector lengths in a random lattice and its dual lattice
in the limit as the dimension of the lattices tends to infinity\, and pro
vides a partial affirmative answer to the question in the title. This is j
oint work with Andreas Strömbergsson.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Fiorilli (CNRS)
DTSTART;VALUE=DATE-TIME:20210513T170000Z
DTEND;VALUE=DATE-TIME:20210513T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/41
DESCRIPTION:Title: Higher moments of primes in intervals and in arithmetic progression
s\nby Daniel Fiorilli (CNRS) as part of ViBraNT (Virtual Brazilian Num
ber Theory seminar)\n\n\nAbstract\nSince the work of Selberg and of Barban
\, Davenport and\nHalberstam\, the variances of primes in intervals and in
arithmetic\nprogressions has been widely studied and continue to be an ac
tive topic\nof research. However\, much less is known about higher moments
. Hooley\nestablished a bound on the third moment in progressions\, which
was\nlater sharpened by Vaughan for a variant involving a major arcs\napp
roximation. Little is known for moments of order four or higher\,\nother t
han the conjecture of Hooley and the conditional result of\nMontgomery-Sou
ndararajan. In this talk I will discuss recent joint work\nwith Régis de
la Bretèche on weighted moments in short intervals and on\nweighted momen
ts of moments in progressions. In particular we will show\nhow to deduce s
harp unconditional omega results on all weighted even\nmoments in certain
ranges.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vorrapan Chandee (KSU)
DTSTART;VALUE=DATE-TIME:20210603T170000Z
DTEND;VALUE=DATE-TIME:20210603T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/42
DESCRIPTION:Title: The sixth moment of Dirichlet L-functions without average in the t-
aspect\nby Vorrapan Chandee (KSU) as part of ViBraNT (Virtual Brazilia
n Number Theory seminar)\n\n\nAbstract\nWe prove an asymptotic for the six
th moment of Dirichlet L-functions averaged over primitive characters modu
lo q\, over all moduli q <= Q. Unlike the previous work of Conrey\, Iwanie
c\, and Soundararajan\, we do not need to include an average on the critic
al line\, thus requiring treatment of the "unbalanced" sums. This is a joi
nt work with Xiannan Li\, Kaisa Matomaki\, and Maksym Radziwill.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sávio Ribas (UFOP)
DTSTART;VALUE=DATE-TIME:20210520T170000Z
DTEND;VALUE=DATE-TIME:20210520T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/44
DESCRIPTION:Title: Some direct and inverse zero-sum problems\nby Sávio Ribas (UFO
P) as part of ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstr
act\nIn this talk\, we will introduce the main zero-sum problems in additi
ve combinatorics. In particular\, we will define the Davenport and the Erd
ös-Ginzburg-Ziv constants\, among other similar constants for finite grou
ps. We will also present their main results so far and Gao's conjecture th
at connects some of these constants (which has already been proven for abe
lian groups). In addition\, we will present the similar weighted problems
and the inverse problems. This is a joint work with D.V. Avelar\, F.E. Bro
chero Martínez\, A. Lemos and B.K. Moryia.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ricardo Misturini (UFRGS)
DTSTART;VALUE=DATE-TIME:20210617T170000Z
DTEND;VALUE=DATE-TIME:20210617T180000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/45
DESCRIPTION:Title: Law of the Iterated Logarithm for a Random Dirichlet Series\nby
Ricardo Misturini (UFRGS) as part of ViBraNT (Virtual Brazilian Number Th
eory seminar)\n\n\nAbstract\nWe consider the random Dirichlet series F(σ)
obtained when\, in each term of the sum that defines the Riemann Zeta fun
ction ζ(σ)\, we put + or - signs chosen independently and uniformly at r
andom. This series converges when σ > 1/2. We study the behavior of F(σ)
when σ goes to 1/2\, providing a Law of the Iterated Logarithm\, which d
escribes the magnitude of the fluctuations of F(σ). This is a joint work
with Marco Aymone and Susana Frómeta.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Shparlinski (UNSW)
DTSTART;VALUE=DATE-TIME:20221004T160000Z
DTEND;VALUE=DATE-TIME:20221004T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/46
DESCRIPTION:Title: Maximal Operators and Restriction Bounds for Weyl Sums\nby Igor
Shparlinski (UNSW) as part of ViBraNT (Virtual Brazilian Number Theory se
minar)\n\n\nAbstract\nhttps://w3.impa.br/~goncalves/assets/files/Shparlins
kiAbstract.pdf\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Ramaré (Marseillle)
DTSTART;VALUE=DATE-TIME:20221011T160000Z
DTEND;VALUE=DATE-TIME:20221011T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/47
DESCRIPTION:Title: The Levin-Fainleib pathway\, around and further\nby Olivier Ram
aré (Marseillle) as part of ViBraNT (Virtual Brazilian Number Theory semi
nar)\n\n\nAbstract\nWe shall describe three occurrences of a device introd
uced by Levin and Fainleib in 1967 and go on to present a recent extension
of one of these\, which results from a collaboration with Alisa Sedunova
and Ritika Sharma.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winston Heap
DTSTART;VALUE=DATE-TIME:20221018T160000Z
DTEND;VALUE=DATE-TIME:20221018T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/48
DESCRIPTION:Title: Mean values of long Dirichlet polynomials\nby Winston Heap as p
art of ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract\nWe
first survey some applications of mean value results for Dirichlet polyno
mials over primes in the theory of the Riemann zeta function. This include
s Selberg's central limit theorem/value distribution and the pair correla
tion of zeros. We highlight a common obstacle in these areas which is that
of longer sums and how they usually require the assumption of the twin-pr
ime conjectures when computing their mean values. We then give some exampl
es showing how\, on the assumption of the Riemann hypothesis\, one can com
pute asymptotics for moments of long sums without such conjectures.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Mangerel (Durham)
DTSTART;VALUE=DATE-TIME:20221025T160000Z
DTEND;VALUE=DATE-TIME:20221025T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/49
DESCRIPTION:Title: Large order Dirichlet characters and an analogue of a conjecture of
Vinogradov\nby Alexander Mangerel (Durham) as part of ViBraNT (Virtua
l Brazilian Number Theory seminar)\n\n\nAbstract\nLet $q$ be a large prime
. It is an old and classical problem to understand\nthe distribution of qu
adratic residues and non-residues modulo $q$. According\nto an old and fam
ous conjecture of I.M. Vinogradov\, the least quadratic\nnon-residue n mod
ulo q should satisfy $n \\leq q^c$ for any positive $c > $0\, when\n$q$ is
large enough. This statement would be implied by non-trivial upper\nbound
s for averages of the Legendre symbol $\\left( \\frac{n}{q}\\right)$ with
$n \\leq q^c.$ Currently\nthe best such results\, due essentially to Burge
ss\, are satisfactory only\nwhen $c > 1/4$\, due to the potential obstruct
ion\, difficult to rule out\, that\n$\\left( \\frac{n}{q}\\right) = +1$ fo
r many initial integers n.\n\nIn this talk I will discuss a generalisation
of Vinogradov's conjecture to\nother primitive Dirichlet characters \\chi
modulo q\, seeking the least n for\nwhich $\\chi(n)$ is not $1$. I will e
xplain some recent work of mine that shows\,\nusing techniques from additi
ve combinatorics\, that when the order d of $\\chi$\ngrows with q the afor
ementioned obstruction does not occur\, that the\nanalogue of Vinogradov's
conjecture for $\\chi$ does hold\, and that moreover\n$\\chi(n) = 1$ with
$n \\leq q^c$ is in fact a rare event for all $c > 0$. I will\nalso discu
ss some results related to showing cancellation in short sums of\n$\\chi(n
)$ with $n \\leq q^c$ for $c > 0$ arbitrarily small\, going beyond Burgess
'\nestimate.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Xu (Stanford)
DTSTART;VALUE=DATE-TIME:20221101T160000Z
DTEND;VALUE=DATE-TIME:20221101T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/50
DESCRIPTION:Title: Central limit theorems for random multiplicative functions\nby
Max Xu (Stanford) as part of ViBraNT (Virtual Brazilian Number Theory semi
nar)\n\n\nAbstract\nIn joint work with Kannan Soundararajan\, we consider
the behavior of random multiplicative functions when summed over subsets o
f the integers in [1\, N]\, and give several examples of sets where such s
ums satisfy a central limit theorem. In contrast\, as we know from the wor
k of Harper\, the partial sums over all integers in [1\, N] do not satisfy
a central limit theorem.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cynthia Bortolotto (Zurich)
DTSTART;VALUE=DATE-TIME:20221115T160000Z
DTEND;VALUE=DATE-TIME:20221115T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/51
DESCRIPTION:Title: Weyl sums with Multiplicative Coefficients and Joint Equidistributi
on\nby Cynthia Bortolotto (Zurich) as part of ViBraNT (Virtual Brazili
an Number Theory seminar)\n\n\nAbstract\nIn 1964\, Hooley proved that for
an irreducible polynomial p in Z[x]\, the ratios v/n for v roots of the po
lynomial p modulo n\, are equidistributed modulo 1. We prove joint equidis
tribution of these roots of polynomial congruences and polynomial values.
As part of the proof\, we generalize a result of Montgomery and Vaughan re
garding exponential sums with multiplicative coefficients to the setting o
f Weyl sums.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia)
DTSTART;VALUE=DATE-TIME:20221129T160000Z
DTEND;VALUE=DATE-TIME:20221129T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/52
DESCRIPTION:Title: The negative Pell equation\nby Carlo Pagano (Concordia) as part
of ViBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract\nIn 19
95\, Peter Stevenhagen made a conjecture about the number of square-free p
ositive integers d up to X such that the negative Pell equation attached t
o d admits a solution in the integers. I will present a joint work with Pe
ter Koymans resolving this conjecture.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christoph Aisleitner (TU Graz)
DTSTART;VALUE=DATE-TIME:20221108T160000Z
DTEND;VALUE=DATE-TIME:20221108T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/53
DESCRIPTION:Title: A first guide to uniform distribution mod 1 and discrepancy\nby
Christoph Aisleitner (TU Graz) as part of ViBraNT (Virtual Brazilian Numb
er Theory seminar)\n\n\nAbstract\nThe foundations for the theory of unifor
m distribution modulo 1 were laid in Hermann Weyl's seminal paper of 1916.
Originally motivated by questions from Diophantine approximation\, it tur
ned out that the concept of uniform distribution is connected with many ot
her mathematical areas\, including exponential sums\, ergodic theory\, and
numerical analysis. This is a survey talk\, where we will sketch some of
the basic concepts and results of the theory. Keywords are: Kronecker sequ
ences\, pseudorandomness\, Erdös-Turan inequality\, Koksma inequality\, a
nd Roth's theorem. In the end we give a brief exposition on how this machi
nery was applied in very recent work of the author (jointly with Bence Bor
da and Manuel Hauke\, arXiv:2210.14095) on the distribution of partial quo
tients of reduced fractions with fixed denominator.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleksiy Klurman (Bristol)
DTSTART;VALUE=DATE-TIME:20221122T160000Z
DTEND;VALUE=DATE-TIME:20221122T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/54
DESCRIPTION:Title: Automatic semigroups\nby Oleksiy Klurman (Bristol) as part of V
iBraNT (Virtual Brazilian Number Theory seminar)\n\n\nAbstract\nThe main g
oal of the talk is to discuss recent progress in our understanding of the
following general phenomena: how does multiplicative structure correlate w
ith "automaticity"? No preliminary background is required. This is based o
n a joint work with J. Konieczny.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Destagnol (Paris-Saclay)
DTSTART;VALUE=DATE-TIME:20221206T160000Z
DTEND;VALUE=DATE-TIME:20221206T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/55
DESCRIPTION:Title: The Loughran--Smeets conjecture for some Châtelet type varieties\nby Kevin Destagnol (Paris-Saclay) as part of ViBraNT (Virtual Brazilia
n Number Theory seminar)\n\n\nAbstract\nSerre in 1990 started a research p
rogram aiming to understand the probability that a randomly chosen diophan
tine equation has a solution over Q. Most cases of the problem are still o
pen today\, even when the equations satisfy the Hasse principle but the Lo
ughran--Smeets conjectures give predictions for that probability in certai
n cases.\nI will report on joint work with Efthymios Sofos regarding this
problem for x^2-Dy^2=P_1(t)...P_R(t)z^2 where D is a square-free integer a
nd P_i are fixed integer polynomials of any degree in n variables\, with n
relatively large compared to the degrees of the P_i.\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayla Gafni (Mississipi)
DTSTART;VALUE=DATE-TIME:20221213T160000Z
DTEND;VALUE=DATE-TIME:20221213T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/56
DESCRIPTION:by Ayla Gafni (Mississipi) as part of ViBraNT (Virtual Brazili
an Number Theory seminar)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Greenfeld (Princeton)
DTSTART;VALUE=DATE-TIME:20230110T160000Z
DTEND;VALUE=DATE-TIME:20230110T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/57
DESCRIPTION:by Rachel Greenfeld (Princeton) as part of ViBraNT (Virtual Br
azilian Number Theory seminar)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Vaaler (Austin)
DTSTART;VALUE=DATE-TIME:20230124T160000Z
DTEND;VALUE=DATE-TIME:20230124T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/58
DESCRIPTION:by Jeffrey Vaaler (Austin) as part of ViBraNT (Virtual Brazili
an Number Theory seminar)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stevan Gajovic (Max Planck)
DTSTART;VALUE=DATE-TIME:20230131T160000Z
DTEND;VALUE=DATE-TIME:20230131T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/59
DESCRIPTION:by Stevan Gajovic (Max Planck) as part of ViBraNT (Virtual Bra
zilian Number Theory seminar)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kaisa Matomäki (Turku)
DTSTART;VALUE=DATE-TIME:20230207T160000Z
DTEND;VALUE=DATE-TIME:20230207T170000Z
DTSTAMP;VALUE=DATE-TIME:20230208T070611Z
UID:NumberTheory2/60
DESCRIPTION:by Kaisa Matomäki (Turku) as part of ViBraNT (Virtual Brazili
an Number Theory seminar)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/NumberTheory2/60/
END:VEVENT
END:VCALENDAR