BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Emmanuel Kowalski (ETH)
DTSTART:20200423T111000Z
DTEND:20200423T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/1
DESCRIPTION:Title: Equidistribution from the Chinese remainder theorem\nby Emmanu
el Kowalski (ETH) as part of Tel Aviv number theory seminar\n\n\nAbstract\
nSuppose that we choose arbitrarily a subset of residue classes modulo eac
h prime\, and use them with the Chinese Remainder Theorem to define subset
s of residue classes modulo all squarefree moduli. Under extremely general
conditions\, it follows that the fractional parts of these sets become eq
uidistributed modulo 1 for almost all moduli. The talk will discuss the pr
ecise statement of this general principle\, as well as some generalization
s and applications.(Joint work with K. Soundararajan)\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niclas Technau (Tel Aviv)
DTSTART:20200430T111000Z
DTEND:20200430T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/2
DESCRIPTION:Title: How random is a uniformly distributed sequence?\nby Niclas Tec
hnau (Tel Aviv) as part of Tel Aviv number theory seminar\n\n\nAbstract\nH
ow random is a uniformly distributed sequence? Fine-scale statistics provi
de an answer to this question. Our focus is on the pair and triple correla
tion statistics of sequences on the unit circle. In particular\, we report
on recent progress concerning the fractional parts of $n^\\alpha$ (joint
work with Nadav Yesha) and $\\alpha n^2$ (joint work with Aled Walker)\, w
here $\\alpha$ is a fixed positive number.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuomas Sahlsten (Manchester)
DTSTART:20200514T111000Z
DTEND:20200514T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/3
DESCRIPTION:Title: Quantum chaos and random surfaces of large genus\nby Tuomas Sa
hlsten (Manchester) as part of Tel Aviv number theory seminar\n\n\nAbstrac
t\nWe give an introduction to our recent work with Etienne Le Masson\, Joe
Thomas and Cliff Gilmore on spatial delocalisation of Eigenfunctions of t
he Laplacian for random surfaces of large genus. In particular we describe
the $L^p$ norms of Eigenfunctions in terms of purely geometric conditions
of hyperbolic surfaces\, which are shown to be almost surely satisfied in
large genus in the Weil-Petersson model for random surfaces. The work is
motivated by analogous large random graph results by Bauerschmidt\, Knowle
s and Yau and the delocalisation of cusp forms on arithmetic surfaces of l
arge level (related to Quantum Unique Ergodocity).\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Luethi (Tel Aviv)
DTSTART:20200507T111000Z
DTEND:20200507T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/4
DESCRIPTION:Title: Equidistribution of simultaneous supersingular reductions of compl
ex multiplication elliptic curves\nby Manuel Luethi (Tel Aviv) as part
of Tel Aviv number theory seminar\n\n\nAbstract\nUnder certain congruence
conditions\, the elliptic curves defined over the complex numbers with co
mplex multiplication (CM) by a given order can be reduced to supersingular
curves (SSC) defined over a finite field of prime characteristic. The (fi
nite) set of isomorphism classes of SSC curves carries a natural probabili
ty measure. It was shown by Philippe Michel via progress on the subconvexi
ty problem that the reductions of CM curves equidistribute among the SSC c
urves when the discriminant of the order diverges along the congruence con
ditions. We will describe a proof of equidistribution in the product of th
e simultaneous reductions with respect to several distinct primes of CM cu
rves of a given order using a recent classification of joinings for certai
n diagonalizable actions by Einsiedler and Lindenstrauss. This is joint wo
rk with Menny Aka\, Philippe Michel\, and Andreas Wieser.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (MPIM)
DTSTART:20200604T111000Z
DTEND:20200604T121000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/5
DESCRIPTION:Title: The negative Pell equation and the 8-rank of the class group\n
by Peter Koymans (MPIM) as part of Tel Aviv number theory seminar\n\n\nAbs
tract\nAbstract: Recently substantial progress has been made in the study
of 2-parts of class groups of quadratic number fields\, most notably by Al
exander Smith. In this talk we give an introduction to the topic. We start
with a classical result due to Gauss known as genus theory\, which descri
bes the \n2-torsion of the class group. We will then give a description of
the 4-torsion and 8-torsion of the class group. Finally we sketch how one
can apply these techniques to improve the current lower bounds on the num
ber of squarefree integers $d$ such that the negative Pell equation $x^2 -
dy^2 = -1$ is soluble in integers $x$ and $y$. This last part is joint wo
rk Stephanie Chan\, Djorjdo Milovic and Carlo Pagano.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianya Liu (Shandong)
DTSTART:20200521T111000Z
DTEND:20200521T121000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/6
DESCRIPTION:Title: The disjointness conjecture for skew products\nby Jianya Liu (
Shandong) as part of Tel Aviv number theory seminar\n\n\nAbstract\nThe dis
jointness conjecture of Sarnak states that the Mobius function is disjoint
with dynamical systems of zero entropy. In this talk I will describe how
to establish this conjecture for a class of skew products. This is joint w
ork with Wen Huang and Ke Wang.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Doron Puder (Tel Aviv)
DTSTART:20200611T111000Z
DTEND:20200611T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/7
DESCRIPTION:Title: The spectral gap of random hyperbolic surfaces\nby Doron Puder
(Tel Aviv) as part of Tel Aviv number theory seminar\n\n\nAbstract\nOn a
compact hyperbolic surface\, the Laplacian has a spectral gap between 0 an
d the next smallest eigenvalue if and only if the surface is connected. Th
e size of the spectral gap measures how "highly connected" the surface is.
We study the spectral gap of a random covering space of a fixed surface\
, and show that for every $\\varepsilon>0$\, with high probability as the
degree of the cover tends to $\\infty$\, the smallest new eigenvalue is at
least $3/16-\\varepsilon$. The number $3/16$ is\, mysteriously\, the same
spectral gap that Selberg obtained for congruence modular curves. \nOur m
ain tool is a new method to analyze random permutations "sampled by surfac
e groups". \nI intend to give some background to the result and discuss so
me ideas from the proof.\nThis is based on joint works with Michael Magee
and Frédéric Naud.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Par Kurlberg (KTH)
DTSTART:20200618T111000Z
DTEND:20200618T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/8
DESCRIPTION:Title: Distribution of lattice points on hyperbolic circles\nby Par K
urlberg (KTH) as part of Tel Aviv number theory seminar\n\n\nAbstract\nWe
study the distribution of lattice points lying on expanding circles in the
hyperbolic plane. The angles of lattice points arising from the orbit of
the modular group ${\\rm PSL}_2(\\mathbb{Z})$\, and lying on hyperbolic ci
rcles centered at i\, are shown to be equidistributed for generic radii (a
mong the ones that contain points). We also show that angles fail to equid
istribute on a thin set of exceptional radii\, even in the presence of gro
wing multiplicity. Surprisingly\, the distribution of angles on hyperbolic
circles turns out to be related to the angular distribution of euclidean
lattice points lying on circles in $\\mathbb{R}^2$\, along a thin subsequ
ence of radii. \n\nThis is joint work with D. Chatzakos\, S. Lester and I.
Wigman.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ade Irma Suriajaya (Kyushu)
DTSTART:20201022T110000Z
DTEND:20201022T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/9
DESCRIPTION:Title: Zeros of derivatives of the Riemann zeta function and relations to
the Riemann hypothesis\nby Ade Irma Suriajaya (Kyushu) as part of Tel
Aviv number theory seminar\n\n\nAbstract\nSpeiser in 1935 showed that the
Riemann hypothesis is equivalent to the first derivative of the Riemann z
eta function having no non-real zeros to the left of the critical line. Th
is result shows a relation between the distribution of zeros of the Rieman
n zeta function and that of its derivative. Implications of the Riemann hy
pothesis to distribution of zeros of higher order derivatives are known bu
t we are still yet to find an equivalence condition. Zeros of the derivati
ves of the Riemann zeta function in various setups were later studied by S
pira\, Berndt\, and also Levinson and Montgomery. Among those results\, a
quantitative version of Speiser's 1935 result was proven by Levinson and M
ontgomery by showing that the number of non-real zeros of the Riemann zeta
function does not differ much to those of its first derivative in 0 < Re(
s) < 1/2 (the left-half of the critical strip). I expect that this is a fo
rmulation which is applicable to all derivatives. In this talk\, I will in
troduce a few important results in this direction and new results obtained
. Further\, many results known for the Riemann zeta function have been gen
eralized to Dirichlet L-functions and some are even extended to more gener
al zeta and L-functions. I hope to give a brief introduction to what is kn
own for Dirichlet L-functions and the difficulties in studying its higher
order derivatives. I also hope to give an overview of common tools used in
this study.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Efthymios Sofos (Glasgow)
DTSTART:20201029T120000Z
DTEND:20201029T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/10
DESCRIPTION:Title: Schinzel's Hypothesis with probability 1 and rational points\
nby Efthymios Sofos (Glasgow) as part of Tel Aviv number theory seminar\n\
n\nAbstract\nSchinzel's Hypothesis states that every integer polynomial sa
tisfying certain congruence conditions represents infinitely many primes.
It is one of the main problems in analytic number theory but is completely
open\, except for polynomials of degree 1. We describe our recent proof o
f the Hypothesis for 100% of polynomials (ordered by size of coefficients)
. Furthermore\, we give applications in Diophantine geometry. Joint work w
ith Alexei Skorobogatov\, preprint: https://arxiv.org/abs/2005.02998.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Ostafe (UNSW)
DTSTART:20201203T120000Z
DTEND:20201203T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/11
DESCRIPTION:Title: On some gcd problems and unlikely intersections\nby Alina Ost
afe (UNSW) as part of Tel Aviv number theory seminar\n\n\nAbstract\nLet $a
\,b$ be multiplicatively independent positive integers. Bugeaud\, Corvaja
and Zannier (2003) proved that $a^n-1$ and $b^n-1$ have only a small comm
on divisor\, namely \n$$\n\\gcd(a^n-1\,b^n-1)\\le \\exp(\\varepsilon n)\n$
$\nfor any fixed $\\varepsilon>0$ and sufficiently large $n$. Ailon and R
udnick (2004) were the first to consider the function field analogue and p
roved a much stronger result in this setting. These results triggered a fl
oodgate of various extensions and generalisations\, from the number case\,
to function fields in both zero and positive characteristics. For example
\, in the function field case besides powering there is another natural op
eration: iteration of functions. \n\nIn this talk I will survey some of th
ese results and their connections to some unlikely intersection problems f
or parametric curves. I will also discuss similar questions for linear rec
urrence sequences over function fields and pose some open questions.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel El-Baz (TU Graz)
DTSTART:20201105T120000Z
DTEND:20201105T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/12
DESCRIPTION:Title: A pair correlation problem and counting lattice points via the ze
ta function\nby Daniel El-Baz (TU Graz) as part of Tel Aviv number the
ory seminar\n\n\nAbstract\nThe pair correlation function is a local measur
e of the randomness of a sequence. The behaviour of the pair correlation o
f sequences of the form $(\\{a_n \\alpha\\})$ for almost every real number
$\\alpha$ where $(a_n)$ is a sequence of integers is by now relatively we
ll-understood. In particular\, a connection to additive combinatorics was
made by relating that behaviour to the additive energy of the sequence $(a
_n)$.\n Zeev Rudnick and Niclas Technau have recently started investiga
ting the case of $(a_n)$ being a sequence of real numbers. This talk is ba
sed on joint work with Christoph Aistleitner and Marc Munsch in which we p
ursue this line of research.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary Soroker (Tel Aviv)
DTSTART:20201112T120000Z
DTEND:20201112T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/13
DESCRIPTION:Title: Random Polynomials\, Probabilistic Galois Theory\, and Finite Fie
ld Arithmetic\nby Lior Bary Soroker (Tel Aviv) as part of Tel Aviv num
ber theory seminar\n\n\nAbstract\nWe will discuss recent advances on the f
ollowing two question: \nLet $A(X) = \\sum \\pm X^i$ be a random polynomia
l of degree n with coefficients taking \nthe values -1\,1 independently ea
ch with probability 1/2.\n\nQ1: What is the probability that A is irreduci
ble as the degree goes to infinity\n\nQ2: What is the typical Galois of A?
\n\nOne believes that the answers are YES and THE FULL SYMMETRIC GROUP\, r
espectively.\nThese questions were studied extensively in recent years\, a
nd we will survey \nthe tools developed to attack these problem and partia
l results.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoav Gath (Technion)
DTSTART:20201119T120000Z
DTEND:20201119T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/14
DESCRIPTION:Title: The lattice point counting problem on Heisenberg groups\nby Y
oav Gath (Technion) as part of Tel Aviv number theory seminar\n\n\nAbstrac
t\nEuclidean lattice point counting problems\, the classical example of wh
ich is the Gauss circle problem\, are an important topic in classical anal
ysis and have been the driving force behind much of the developments in th
e area of analytic number theory in the 20th century. While it is well kno
wn that homogeneous groups provide a natural setting to generalize many qu
estions of Euclidean harmonic analysis\, it was only recently that analogu
es of the Euclidean lattice point counting problem were considered for a c
ertain family of 2-step nilpotent homogeneous groups. I will present the l
attice point counting problem for Cygan-Koranyi norm balls on the Heisenbe
rg groups\, which is the analogue of the lattice point counting problem fo
r Euclidean balls. I will describe recently obtained results relating to t
he distribution and moments of the error term on the Heisenberg groups\, a
nd discuss the similarities (and stark differences) between the Euclidean
and Heisenberg case.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nadav Yesha (Haifa)
DTSTART:20201210T120000Z
DTEND:20201210T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/15
DESCRIPTION:Title: Poisson statistics for sequences modulo one\nby Nadav Yesha (
Haifa) as part of Tel Aviv number theory seminar\n\n\nAbstract\nA natural
way to test for the randomness of a sequence of points in $\\mathbb R/\\ma
thbb Z$ is to consider its local statistics such as the $k$-level correlat
ions and the nearest-neighbour gap distribution\, and compare them to thos
e of a sequence of uniform independent random points (Poisson statistics).
\nIn this talk I will describe recent results concerning two important ex
amples of such sequences:\n\n- The sequence $\\{x^n\\}$\, where in a joint
work with Aistleitner\, Baker and Technau we showed that for almost all $
x>1$\, all the correlations and hence the normalized gaps have a Poissonia
n limit distribution.\n\n- The sequence $\\{n^x\\}$\, where in a joint wor
k with Technau we showed Poissonian $k$-level correlations for almost all
$x$ sufficiently large (depending on $k$).\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erez Nesharim (Hebrew University)
DTSTART:20201217T120000Z
DTEND:20201217T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/16
DESCRIPTION:Title: Diophantine approximation over function fields and the t-adic Lit
tlewood conjecture\nby Erez Nesharim (Hebrew University) as part of Te
l Aviv number theory seminar\n\n\nAbstract\nAbstract: The Littlewood conje
cture in simultaneous approximation and the p-adic Littlewood conjecture a
re famous open problems in the intersection of number theory and dynamics.
In a joint work with Faustin Adiceam and Fred Lunnon we show that an anal
ogue of the p-adic Littlewood conjecture over $\\mathbb F_3((1/t))$ is fal
se. The counterexample is given by the Laurent series whose coefficients a
re the regular paper folding sequence\, and the method of proof is by redu
ction to the non vanishing of certain Hankel determinants. The proof is co
mputer assisted and it uses substitution tilings of $\\mathbb Z^2$ and a g
eneralisation of Dodgson's condensation algorithm for computing the determ
inant of any Hankel matrix.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noam Kimmel (TAU)
DTSTART:20210107T120000Z
DTEND:20210107T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/17
DESCRIPTION:Title: On covariance of eigenvalue counts and lattice point problems
\nby Noam Kimmel (TAU) as part of Tel Aviv number theory seminar\n\n\nAbst
ract\nWe explore the covariance of error terms coming from Weyl's conjectu
re regarding the number of Dirichlet eigenvalues up to size $X$.\nWe also
consider this problem in short intervals\, i.e. the error term of the numb
er of eigenvalues in the window $[X\, X+S]$ for some $S(X)$.\nWe look at t
hese error terms for planar domains where the Dirichlet eigenvalues can be
explicitly calculated.\nIn these cases\, the error term is closely relate
d to the error term from the classical lattice points counting problem of
expanding planar domains.\nWe give a formula for the covariance of such er
ror terms\, for general planar domains.\nWe also give a formula for the co
variance of error terms in short intervals\, for sufficiently large interv
als.\nGoing back to the Dirichlet eigenvalue problem\, we give results reg
arding the covariance of the error terms in short intervals of 'generic' r
ectangles.\nWe also explore a specific example\, namely we compute the cov
ariance between the error terms of an equilateral triangle and various rec
tangles.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ezra Waxman (TU Dresden)
DTSTART:20210304T120000Z
DTEND:20210304T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/18
DESCRIPTION:Title: Artin Twin Primes and Poisson Binomial Distributions\nby Ezra
Waxman (TU Dresden) as part of Tel Aviv number theory seminar\n\n\nAbstra
ct\nWe say that a prime $p$ is an Artin prime for $g$ if $g$ is a primitiv
e root mod $p$. For appropriately chosen integers $g$ and $d$\, we presen
t a conjecture for the asymptotic number of prime pairs $(p\,p+d)$ such th
at both $p$ and $p+d$ are Artin primes for $g$. Our model suggests that t
he distribution of Artin prime pairs\, amongst the ordinary prime pairs\,
is largely governed by a Poisson binomial distribution. Time permitting\,
we moreover present a conjecture for the variance of Artin primes across
short intervals of ordinary primes\, obtained via similar heuristic method
s (Joint work with Magdaléna Tinková and Mikuláš Zindulka).\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jingwei Guo (University of Science and Technology of China (USTC))
DTSTART:20210311T120000Z
DTEND:20210311T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/19
DESCRIPTION:Title: Some improved remainder estimates in Weyl’s law\nby Jingwei
Guo (University of Science and Technology of China (USTC)) as part of Tel
Aviv number theory seminar\n\n\nAbstract\nOne of the most important objec
ts in spectral geometry is the counting function for the eigenvalues $\\la
mbda_j$ for the Dirichlet Laplacian associated with planar domains. The si
mplest domains are squares\, disks and ellipses. \nIt is well-known that f
or each of these domains its eigenvalue counting function\n $\\#\\{ \\lam
bda_j\\le\\mu^2 \\}$ \n has an asymptotic containing two main terms $a \\
mu^2 -b \\mu$ \nand a remainder of size $o(\\mu)$. To improve the estimat
e of the remainder term had been one of the most attractive problems in sp
ectral geometry for decades.\n\n\nI will introduce background briefly and
explain how to transfer the above problem into problems of counting lattic
e points\, to which tools from analysis and analytic number theory can be
applied. I will mention our progresses for disks\, annuli and balls in hig
h dimensions\, joint with Wolfgang Mueller\, Weiwei Wang and Zuoqin Wang.\
n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalie Evans (Queen Mary)
DTSTART:20210408T110000Z
DTEND:20210408T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/20
DESCRIPTION:Title: Correlations of almost primes\nby Natalie Evans (Queen Mary)
as part of Tel Aviv number theory seminar\n\n\nAbstract\nThe Hardy-Littlew
ood generalised twin prime conjecture states an asymptotic formula for the
number of primes $p\\le X$ such that $p+h$ is prime for any non-zero even
integer $h$. While this conjecture remains wide open\, Matomäki\, Radziw
ill and Tao proved that it holds on average over $h$\, improving on a prev
ious result of Mikawa. In this talk we will discuss an almost prime analog
ue of the Hardy-Littlewood conjecture for which we can go beyond what is k
nown for primes. We will describe some recent work in which we prove an as
ymptotic formula for the number of almost primes $n=p_1p_2 \\le X$ such th
at $n+h$ has exactly two prime factors which holds for a very short averag
e over $h$.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Etai Leumi (Tel Aviv)
DTSTART:20210422T110000Z
DTEND:20210422T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/21
DESCRIPTION:Title: The LCM problem for function fields\nby Etai Leumi (Tel Aviv)
as part of Tel Aviv number theory seminar\n\n\nAbstract\nCilleruelo conje
ctured that for any irreducible polynomial $f$ with integer coefficients\,
and degree greater than one\, the least common multiple of the values of
$f$ \nat the first $N$ integers satisfies \n$$\\log {\\rm lcm} (f(1)\,...
\,f(N))\\sim (\\deg(f)-1 )N\\log N$$ as $N$ tends to infinity. \nHe prove
d this only for $\\deg(f)=2$. No example in higher degree is known. We stu
dy the analogue of this conjecture for function fields\, \nwhere we replac
e the integers by the ring of polynomials over a finite field. In that set
ting we are able to establish some instances of the conjecture for higher
degrees. \nThe examples are all "special" polynomials $f(X)$\, which hav
e the property that the bivariate polynomial $f(X)-f(Y)$ factors into line
ar terms in the base field.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Misha Sodin (Tel Aviv)
DTSTART:20210429T110000Z
DTEND:20210429T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/22
DESCRIPTION:Title: Equidistribution of zeroes of power series and binary correlation
s of coefficients.\nby Misha Sodin (Tel Aviv) as part of Tel Aviv numb
er theory seminar\n\n\nAbstract\nAbstract:\nWe will discuss global and loc
al equidistribution of zeroes of power series \nwith coefficients $r(n)/\\
sqrt{n!}$ where $r(n)$ is a sequence of complex-valued multipliers having
binary correlations and no gaps in the spectrum.\nWe apply our approach to
several examples of the sequence r of very different\norigin\, in particu
lar various sequences of arithmetic origin such as the Möbius function wh
ere we see connections to Chowla’s conjecture\, random multiplicative\nf
unctions\, and the function $e(xn^2)$ where the Diophantine nature of x pl
ays a role.\n\nThe talk will be based on joint work with Alexander Boriche
v and Jacques Benatar\n(arXiv:1908.09161\, arXiv:2104.04812)\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Leuthi (Tel Aviv)
DTSTART:20210506T110000Z
DTEND:20210506T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/23
DESCRIPTION:Title: Random walks on homogeneous spaces\, Spectral Gaps\, and Khinchin
e's theorem on fractals\nby Manuel Leuthi (Tel Aviv) as part of Tel Av
iv number theory seminar\n\n\nAbstract\nKhinchine's classical theorem in E
uclidean space gives a zero one law describing the approximability of typi
cal points by rational points. In 1984\, Mahler asked how well points on t
he middle third Cantor set can be approximated by rational numbers\, both
from within and from outside Cantor's set. His question fits into an attem
pt to determine conditions under which subsets of Euclidean space inherit
the Diophantine properties of the ambient space. \nFor certain fractals si
gnificant progress has been made regarding the Diophantine properties of t
ypical points\, albeit\, almost all known results have been of "convergenc
e type". In this talk\, we will discuss the first instances where a comple
te analogue of Khinchine’s theorem for fractal measures is obtained. Our
results hold for fractals generated by rational similarities of Euclidean
space that have sufficiently small Hausdorff co-dimension. The main new i
ngredient is an effective equidistribution theorem for associated fractal
measures on the space of unimodular lattices. The latter is established us
ing a spectral gap property of a type of Markov operators associated with
the generating similarities. This is joint work with Osama Khalil.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Marklof (Bristol)
DTSTART:20211028T110000Z
DTEND:20211028T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/24
DESCRIPTION:Title: How random are the roots of quadratic congruences?\nby Jens M
arklof (Bristol) as part of Tel Aviv number theory seminar\n\n\nAbstract\n
In 1963 Christopher Hooley showed that the roots of a quadratic congruence
mod m\, appropriately normalized and averaged\, are uniformly distributed
mod 1. In this lecture\, which is based joint work with Matthew Welsh (Br
istol)\, we will study pseudo-randomness properties of the roots on finer
scales and prove for instance that the pair correlation density converges
to an intriguing limit. A key step in our approach is to translate the pro
blem to convergence of certain geodesic random line processes in the hyper
bolic plane.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura Monk (MPIM)
DTSTART:20211125T120000Z
DTEND:20211125T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/25
DESCRIPTION:Title: Geometry and spectrum of random hyperbolic surfaces\nby Laura
Monk (MPIM) as part of Tel Aviv number theory seminar\n\n\nAbstract\nThe
aim of this talk is to describe typical compact hyperbolic surfaces: resul
ts will be stated for most surfaces rather than every single one of them.
In order to motivate this idea\, I will first present examples introduced
in literature as limiting cases of famous theorems\, and argue that they m
ight be seen as "atypical". This will allow us to appreciate the contrast
with a fast-growing family of new results in both geometry and spectral th
eory\,\nwhich are established with probability close to one in various set
tings\, while being false for these atypical surfaces. In particular\, I w
ill discuss results on the distribution of eigenvalues and the geometry of
long geodesics\, as well as ongoing research on spectral gaps.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Par Kurlberg (KTH)
DTSTART:20211104T120000Z
DTEND:20211104T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/26
DESCRIPTION:Title: Cancellation in exponential sums over small multiplicative subgro
ups of Z/pZ\nby Par Kurlberg (KTH) as part of Tel Aviv number theory s
eminar\n\n\nAbstract\nWe will sketch the proof of a breakthrough result (f
rom around 2005) by Bourgain\, Chang\, Glibichuk\, and Konyagin who proved
that there is cancellation in exponential sums formed by summing $\\exp(2
\\pi i h/p)$ for $h$ ranging over elements in a "small" multiplicative s
ubgroup $H$ of the finite field $Z/pZ$. The result was discussed in the fi
rst talk of the semester\, for showing that the digits of $1/p$ are unifor
mly distributed if the period is not too small. The proof uses ideas from
additive combinatorics\, in particular the "sum-product theorem" and the B
alog-Gowers-Szemeredi theorem (roughly\, subsets of $Z/pZ$ with "large add
itive energy" must contain "large" subsets $S$ with property that the sum
set $S+S$ is "small").\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sa'ar Zehavi (TAU)
DTSTART:20211021T110000Z
DTEND:20211021T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/27
DESCRIPTION:Title: Sectorial equidistribution of the roots of $x^2=-1 \\mod p$\n
by Sa'ar Zehavi (TAU) as part of Tel Aviv number theory seminar\n\nLecture
held in Schreiber 309\, TAU.\n\nAbstract\nThe equation $x^2 + 1 = 0 \\mod
p$ has solutions whenever $p = 2$ or $4n+1$. A famous theorem of Fermat s
ays that these primes are exactly the ones that can be described as a sum
of two squares. That the roots of the former equation are equidistributed
is a famous theorem of Duke\, Friedlander and Iwaniec from 1995. We examin
e what happens to the distribution when one adds a restriction on the prim
es which has to do with the angle in the plane formed by their correspondi
ng representation as a sum of squares. This simple arithmetic question has
a solution which involves multiple disciplines of number theory\, but the
talk does not assume any previous background.\n\nThe talk will be deliver
ed in person\, no online access.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Sartori (Tel Aviv)
DTSTART:20211111T120000Z
DTEND:20211111T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/28
DESCRIPTION:Title: On the least primitive root modulo a prime\nby Andrea Sartori
(Tel Aviv) as part of Tel Aviv number theory seminar\n\nLecture held in S
chreiber 309\, TAU.\n\nAbstract\nGiven a prime $p$\, the generators of the
multiplicative group of the integers modulo $p$ are called primitive root
s. In 1930 Vinogradov conjectured that the smallest generator\, the least
primitive root\, is smaller than any power of $p$. This talk will be a gen
eral introduction to the subject. I will discuss the classic results of Vi
nogradov and Burgess towards this conjecture and describe some more recent
improvements for primes such that $p–1$ does not have small odd prime
factors.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ofir Gorodetsky (Oxford)
DTSTART:20211209T120000Z
DTEND:20211209T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/29
DESCRIPTION:Title: Sums of two squares are strongly biased towards quadratic residue
s\nby Ofir Gorodetsky (Oxford) as part of Tel Aviv number theory semin
ar\n\n\nAbstract\nChebyshev famously observed empirically that more often
than not\, there are more primes of the form 3 mod 4 up to x than primes o
f the form 1 mod 4. This was confirmed theoretically much later by Rubinst
ein and Sarnak in a logarithmic density sense. Our understanding of this i
s conditional on the generalized Riemann Hypothesis as well as Linear Inde
pendence of the zeros of L-functions.\n\nWe investigate similar questions
for sums of two squares in arithmetic progressions. We find a significantl
y stronger bias than in primes\, which happens for almost all integers in
a natural density sense. Because the bias is more pronounced\, we do not n
eed to assume Linear Independence of zeros\, only a Chowla-type Conjecture
on non-vanishing of L-functions at 1/2.\nThe bias is stronger because it
arises from a multiplicative contribution of squares as opposed to additiv
e contribution (as in the case of primes). \n\nTo illustrate\, we have und
er GRH that the number of sums of two squares up to x that are 1 mod 3 is
greater than those that are 2 mod 3 for all but o(x) integers.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Lester (King's College London)
DTSTART:20211216T120000Z
DTEND:20211216T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/30
DESCRIPTION:Title: Spacing statistics for lattice points on circles\nby Steve Le
ster (King's College London) as part of Tel Aviv number theory seminar\n\n
\nAbstract\nIn this talk I will describe the distribution of lattice point
s lying on circles. A striking result of Kátai and Környei shows that al
ong a density one subsequence of admissible radii the angles of lattice po
ints lying on circles are uniformly distributed in the limit as the radius
tends to infinity. Their result goes further\, proving that uniform distr
ibution persists even at very small scales\, meaning that the angles are u
niformly distributed within quickly shrinking arcs. A more refined problem
is to understand how the lattice points are spaced together at the local
scale\, e.g. given a circle containing $N$ lattice points determine the nu
mber of gaps between consecutive angles of size less than $1/N$. I will d
iscuss some recent joint work with Pär Kurlberg in which we compute the n
earest neighbor spacing of the angles along a density one subsequence of a
dmissible radii.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Verónica Becher (Universidad de Buenos Aires)
DTSTART:20220224T120000Z
DTEND:20220224T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/31
DESCRIPTION:Title: Poisson generic real numbers.\nby Verónica Becher (Universid
ad de Buenos Aires) as part of Tel Aviv number theory seminar\n\n\nAbstrac
t\nYears ago\, Zeev Rudnick defined the Poisson generic real numbers as th
ose where the number of occurrences of the long strings \nin the initial
segments of their fractional expansions in some base follow the Poisson di
stribution. \nPeres and Weiss proved that almost all real numbers\, with r
espect to Lebesgue measure\, are Poisson generic. \nThey also showed that
Poisson genericity implies Borel normality\, but the two notions do not co
incide\, witnessed by the famous \nChampernowne constant. I will discuss
these results and present a construction of a Poisson generic real for a
fixed parameter lambda.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xin Zhang (Hong Kong University)
DTSTART:20220310T120000Z
DTEND:20220310T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/32
DESCRIPTION:Title: Traces of elements in thin subgroups of the modular group\nby
Xin Zhang (Hong Kong University) as part of Tel Aviv number theory semina
r\n\n\nAbstract\nFor a subgroup of the modular group\, we ask which intege
rs occur as the trace of an element of the subgroup? For the modular group
itself\, every integer occurs. The question is particularly interesting f
or "thin" groups\, which are certain subgroups of infinite index. We use c
ircle method to prove a local-global theorem on the set of traces\, when t
he subgroup contains a parabolic element. This yields fine information on
the length spectrum (the set of lengths of closed geodesics) of the hyperb
olic surface associated to this group. This is joint work with Alex Kontor
ovich.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary Soroker (Tel Aviv)
DTSTART:20220303T120000Z
DTEND:20220303T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/33
DESCRIPTION:Title: Distribution of rational points on elliptic curves\nby Lior B
ary Soroker (Tel Aviv) as part of Tel Aviv number theory seminar\n\nLectur
e held in Schreiber 309\, TAU.\n\nAbstract\nHilbert's irreducibility theor
em may be formulated as the statement the rational points on the line cann
ot be covered by rational points coming from finitely many covers.\nWhen o
ne wants to replace the line\, by more complicated varieties\, such as ell
iptic curves or algebraic groups\, such a naive statement fails\, due to t
he existences of isogenies. It turns out the correct generalization is for
ramified covers. \n\nIn the talk\, we will discuss some recent progress o
n these problems.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ezra Wazman (University of Haifa)
DTSTART:20220324T120000Z
DTEND:20220324T130000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/34
DESCRIPTION:Title: The Correction Factor in Artin's Primitive Root Conjecture: Class
ically and over Fq[T]\nby Ezra Wazman (University of Haifa) as part of
Tel Aviv number theory seminar\n\nLecture held in Schreiber 309\, TAU.\n\
nAbstract\nIn 1927\, E. Artin proposed a conjecture for the natural densit
y of primes $p$ for which $g$ is a primitive root mod $p$. By observing
numerical deviations from Artin’s originally predicted asymptotic\, Der
rick and Emma Lehmer (1957) identified the need for an additional correcti
on factor\; leading to a modified conjecture which was eventually proved t
o be correct by Hooley (1967) under the assumption of GRH. An appropriate
analogue of Artin's primitive root conjecture may also be formulated over
a global function field\, where Bilharz provided a proof that is correct
under the assumption that $g$ is a "geometric" element. In this talk we
discuss the correction factor that emerges when one removes the assumption
that $g$ is geometric\; thereby completing the proof of Artin's primitiv
e root conjecture for arbitrary function fields in one variable over a fin
ite field.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Wigman (King's College London)
DTSTART:20220407T110000Z
DTEND:20220407T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/35
DESCRIPTION:Title: The Robin problem for rectangles\nby Igor Wigman (King's Coll
ege London) as part of Tel Aviv number theory seminar\n\nLecture held in S
chreiber 309\, TAU.\n\nAbstract\nRobin boundary conditions are used in hea
t conductance theory to interpolate between a perfectly insulating boundar
y\, described by Neumann boundary conditions\, and a temperature fixing bo
undary\, described by Dirichlet boundary conditions. The Neumann and Diric
hlet spectrum of a square are just sums of two squares\, hence have a dire
ct arithmetic significance. The Robin spectrum is more mysterious and unti
l now the number theory behind it was not explored. We study the statistic
s and the arithmetic properties of the Robin spectrum of a rectangle. Base
d on a joint work with Zeev Rudnick.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacques Benatar (Tel Aviv)
DTSTART:20220428T110000Z
DTEND:20220428T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/36
DESCRIPTION:Title: Extremal bounds for Dirichlet polynomials with multiplicative coe
fficients\nby Jacques Benatar (Tel Aviv) as part of Tel Aviv number th
eory seminar\n\n\nAbstract\nI will discuss some recent work\, joint with A
lon Nishry and Brad Rodgers\, concerning the distribution of Dirichlet and
trigonometric polynomials generated by multiplicative coefficients $f(n)$
.\n\nIn the first part of the talk we will explore some old and new result
s for deterministic sequences $f(n)$ (Möbius\, Legendre symbol\,...)\, st
opping along our journey to marvel at a variety of wild and thorny conject
ures. The second half of the talk will be devoted to Steinhaus random mult
iplicative coefficients $f(n)=X(n)$.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noam Kimmel (Tel Aviv)
DTSTART:20220512T110000Z
DTEND:20220512T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/37
DESCRIPTION:Title: The Least Common Multiple of Polynomial sequences\nby Noam K
immel (Tel Aviv) as part of Tel Aviv number theory seminar\n\n\nAbstract\n
The prime number theorem can be stated as saying that the logarithm of the
least common multiple (LCM) of the first $N$ integers is asymptotically e
qual to $N$\, as was known to Chebyshev. \n\nMotivated by this formulation
\, we look at a generalization - the least common multiple of polynomial s
equences. The case of a polynomial in one variable was first studied by Ci
lleruelo in 2011\, who determined the asymptotics of the quadratic case\,
and has since been explored by various other researchers. The conclusion i
s that for an irreducible polynomial $F(x)$ of degree $d\\geq 2$\, \n$\\l
og LCM (F(1)\,\\dots F(N))$ grows roughly as $N \\log N$\, though we stil
l do not know the asymptotics\, conjectured to be $(d-1)N \\log(N)$. \n\n
In this talk we consider polynomials in two variables. We discover that al
ready in the quadratic case\, there is a range of asymptotic behaviours. W
e show that for "generic" quadratic polynomials\, the growth of log LCM of
the values of $F$ up to $N$ has order of magnitude $N\\log\\log N/\\sqrt{
\\log N}$ (which as we shall explain\, is the answer for a suitable random
model)\, \nbut for certain degenerate cases such as $(x+y)^2$ or $x^2+y^2
$\, the answers are different.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel Aviv)
DTSTART:20220519T110000Z
DTEND:20220519T120000Z
DTSTAMP:20260422T225921Z
UID:TAUNumbertheory/38
DESCRIPTION:Title: eigenvalue statistics for hyperbolic surfaces of large genus\
nby Zeev Rudnick (Tel Aviv) as part of Tel Aviv number theory seminar\n\n\
nAbstract\nAn outstanding conjecture in quantum chaos is that the statisti
cs of the energy levels of "generic" chaotic systems with time reversal sy
mmetry are described by those of the Gaussian Orthogonal Ensemble (GOE) in
Random Matrix Theory. Conjectural examples are the eigenvalues of the Lap
lacian on a "generic" hyperbolic surface. This conjecture has proved to be
extremely difficult\, with no single case being proved\, the closest case
being some results for the Riemann zeros which seem to have similar stati
stics\, those of the Gaussian Unitary Ensemble. It has long been desired t
o improve the situation by averaging over a suitable ensemble of chaotic s
ystems. I will describe a version of such ensemble averaging on the moduli
space of compact hyperbolic surfaces\, equipped with the Weil-Petersson m
easure\, using the pioneering work of Maryam Mirzakhani. For a suitable qu
antity\, we obtain a small confirmation of GOE statistics.\n
LOCATION:https://researchseminars.org/talk/TAUNumbertheory/38/
END:VEVENT
END:VCALENDAR