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BEGIN:VEVENT
SUMMARY:Emmanuel Kowalski (ETH)
DTSTART:20200423T111000Z
DTEND:20200423T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/1
DESCRIPTION:Title: Equidistribution from the Chinese remainder theorem\nby
Emmanuel Kowalski (ETH) as part of Tel Aviv number theory seminar\n\n\nAb
stract\nSuppose that we choose arbitrarily a subset of residue classes mod
ulo each prime\, and use them with the Chinese Remainder Theorem to define
subsets of residue classes modulo all squarefree moduli. Under extremely
general conditions\, it follows that the fractional parts of these sets be
come equidistributed modulo 1 for almost all moduli. The talk will discuss
the precise statement of this general principle\, as well as some general
izations and applications.(Joint work with K. Soundararajan)\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niclas Technau (Tel Aviv)
DTSTART:20200430T111000Z
DTEND:20200430T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/2
DESCRIPTION:Title: How random is a uniformly distributed sequence?\nby Nic
las Technau (Tel Aviv) as part of Tel Aviv number theory seminar\n\n\nAbst
ract\nHow random is a uniformly distributed sequence? Fine-scale statistic
s provide an answer to this question. Our focus is on the pair and triple
correlation 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 Walk
er)\, where $\\alpha$ is a fixed positive number.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuomas Sahlsten (Manchester)
DTSTART:20200514T111000Z
DTEND:20200514T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/3
DESCRIPTION:Title: Quantum chaos and random surfaces of large genus\nby Tu
omas Sahlsten (Manchester) as part of Tel Aviv number theory seminar\n\n\n
Abstract\nWe give an introduction to our recent work with Etienne Le Masso
n\, Joe Thomas and Cliff Gilmore on spatial delocalisation of Eigenfunctio
ns of the Laplacian for random surfaces of large genus. In particular we d
escribe the $L^p$ norms of Eigenfunctions in terms of purely geometric con
ditions of hyperbolic surfaces\, which are shown to be almost surely satis
fied in large genus in the Weil-Petersson model for random surfaces. The w
ork is motivated by analogous large random graph results by Bauerschmidt\,
Knowles and Yau and the delocalisation of cusp forms on arithmetic surfac
es of large level (related to Quantum Unique Ergodocity).\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Luethi (Tel Aviv)
DTSTART:20200507T111000Z
DTEND:20200507T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/4
DESCRIPTION:Title: Equidistribution of simultaneous supersingular reductions o
f complex multiplication elliptic curves\nby Manuel Luethi (Tel Aviv)
as part of Tel Aviv number theory seminar\n\n\nAbstract\nUnder certain con
gruence conditions\, the elliptic curves defined over the complex numbers
with complex multiplication (CM) by a given order can be reduced to supers
ingular curves (SSC) defined over a finite field of prime characteristic.
The (finite) set of isomorphism classes of SSC curves carries a natural pr
obability measure. It was shown by Philippe Michel via progress on the sub
convexity problem that the reductions of CM curves equidistribute among th
e SSC curves when the discriminant of the order diverges along the congrue
nce conditions. We will describe a proof of equidistribution in the produc
t of the simultaneous reductions with respect to several distinct primes o
f CM curves of a given order using a recent classification of joinings for
certain diagonalizable actions by Einsiedler and Lindenstrauss. This is j
oint work with Menny Aka\, Philippe Michel\, and Andreas Wieser.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Koymans (MPIM)
DTSTART:20200604T111000Z
DTEND:20200604T121000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/5
DESCRIPTION:Title: The negative Pell equation and the 8-rank of the class grou
p\nby Peter Koymans (MPIM) as part of Tel Aviv number theory seminar\n
\n\nAbstract\nAbstract: Recently substantial progress has been made in the
study of 2-parts of class groups of quadratic number fields\, most notabl
y by Alexander Smith. In this talk we give an introduction to the topic. W
e start with a classical result due to Gauss known as genus theory\, which
describes the \n2-torsion of the class group. We will then give a descrip
tion 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 number 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 j
oint work Stephanie Chan\, Djorjdo Milovic and Carlo Pagano.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianya Liu (Shandong)
DTSTART:20200521T111000Z
DTEND:20200521T121000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/6
DESCRIPTION:Title: The disjointness conjecture for skew products\nby Jiany
a Liu (Shandong) as part of Tel Aviv number theory seminar\n\n\nAbstract\n
The disjointness conjecture of Sarnak states that the Mobius function is d
isjoint with dynamical systems of zero entropy. In this talk I will descri
be how to establish this conjecture for a class of skew products. This is
joint work with Wen Huang and Ke Wang.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Doron Puder (Tel Aviv)
DTSTART:20200611T111000Z
DTEND:20200611T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/7
DESCRIPTION:Title: The spectral gap of random hyperbolic surfaces\nby Doro
n 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 betwe
en 0 and the next smallest eigenvalue if and only if the surface is connec
ted. The size of the spectral gap measures how "highly connected" the surf
ace is. We study the spectral gap of a random covering space of a fixed s
urface\, and show that for every $\\varepsilon>0$\, with high probability
as the degree of the cover tends to $\\infty$\, the smallest new eigenvalu
e is at least $3/16-\\varepsilon$. The number $3/16$ is\, mysteriously\, t
he same spectral gap that Selberg obtained for congruence modular curves.
\nOur main tool is a new method to analyze random permutations "sampled by
surface groups". \nI intend to give some background to the result and dis
cuss some ideas from the proof.\nThis is based on joint works with Michael
Magee and Frédéric Naud.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Par Kurlberg (KTH)
DTSTART:20200618T111000Z
DTEND:20200618T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/8
DESCRIPTION:Title: Distribution of lattice points on hyperbolic circles\nb
y Par Kurlberg (KTH) as part of Tel Aviv number theory seminar\n\n\nAbstra
ct\nWe study the distribution of lattice points lying on expanding circles
in the hyperbolic plane. The angles of lattice points arising from the or
bit of the modular group ${\\rm PSL}_2(\\mathbb{Z})$\, and lying on hyperb
olic circles centered at i\, are shown to be equidistributed for generic r
adii (among the ones that contain points). We also show that angles fail t
o equidistribute on a thin set of exceptional radii\, even in the presence
of growing multiplicity. Surprisingly\, the distribution of angles on hyp
erbolic circles turns out to be related to the angular distribution of euc
lidean lattice points lying on circles in $\\mathbb{R}^2$\, along a thin
subsequence of radii. \n\nThis is joint work with D. Chatzakos\, S. Lester
and I. Wigman.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ade Irma Suriajaya (Kyushu)
DTSTART:20201022T110000Z
DTEND:20201022T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/9
DESCRIPTION:Title: Zeros of derivatives of the Riemann zeta function and relat
ions to the Riemann hypothesis\nby Ade Irma Suriajaya (Kyushu) as part
of Tel Aviv number theory seminar\n\n\nAbstract\nSpeiser in 1935 showed t
hat the Riemann hypothesis is equivalent to the first derivative of the Ri
emann zeta function having no non-real zeros to the left of the critical l
ine. This result shows a relation between the distribution of zeros of the
Riemann zeta function and that of its derivative. Implications of the Rie
mann hypothesis to distribution of zeros of higher order derivatives are k
nown but we are still yet to find an equivalence condition. Zeros of the d
erivatives of the Riemann zeta function in various setups were later studi
ed by Spira\, Berndt\, and also Levinson and Montgomery. Among those resul
ts\, a quantitative version of Speiser's 1935 result was proven by Levinso
n and Montgomery by showing that the number of non-real zeros of the Riema
nn 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 formulation which is applicable to all derivatives. In this talk\, I
will introduce a few important results in this direction and new results o
btained. Further\, many results known for the Riemann zeta function have b
een generalized to Dirichlet L-functions and some are even extended to mor
e general zeta and L-functions. I hope to give a brief introduction to wha
t is known 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://stable.researchseminars.org/talk/TAUNumbertheory/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Efthymios Sofos (Glasgow)
DTSTART:20201029T120000Z
DTEND:20201029T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/10
DESCRIPTION:Title: Schinzel's Hypothesis with probability 1 and rational poin
ts\nby Efthymios Sofos (Glasgow) as part of Tel Aviv number theory sem
inar\n\n\nAbstract\nSchinzel's Hypothesis states that every integer polyno
mial satisfying certain congruence conditions represents infinitely many p
rimes. It is one of the main problems in analytic number theory but is com
pletely open\, except for polynomials of degree 1. We describe our recent
proof of the Hypothesis for 100% of polynomials (ordered by size of coeffi
cients). Furthermore\, we give applications in Diophantine geometry. Joint
work with Alexei Skorobogatov\, preprint: https://arxiv.org/abs/2005.0299
8.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Ostafe (UNSW)
DTSTART:20201203T120000Z
DTEND:20201203T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/11
DESCRIPTION:Title: On some gcd problems and unlikely intersections\nby Al
ina Ostafe (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 sma
ll common divisor\, namely \n$$\n\\gcd(a^n-1\,b^n-1)\\le \\exp(\\varepsilo
n n)\n$$\nfor any fixed $\\varepsilon>0$ and sufficiently large $n$. Ailo
n and Rudnick (2004) were the first to consider the function field analogu
e and proved a much stronger result in this setting. These results trigger
ed a floodgate 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 nat
ural operation: iteration of functions. \n\nIn this talk I will survey som
e of these results and their connections to some unlikely intersection pro
blems for parametric curves. I will also discuss similar questions for lin
ear recurrence sequences over function fields and pose some open questions
.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel El-Baz (TU Graz)
DTSTART:20201105T120000Z
DTEND:20201105T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/12
DESCRIPTION:Title: A pair correlation problem and counting lattice points via
the zeta function\nby Daniel El-Baz (TU Graz) as part of Tel Aviv num
ber theory seminar\n\n\nAbstract\nThe pair correlation function is a local
measure of the randomness of a sequence. The behaviour of the pair correl
ation of sequences of the form $(\\{a_n \\alpha\\})$ for almost every real
number $\\alpha$ where $(a_n)$ is a sequence of integers is by now relati
vely well-understood. In particular\, a connection to additive combinatori
cs was made by relating that behaviour to the additive energy of the seque
nce $(a_n)$.\n Zeev Rudnick and Niclas Technau have recently started in
vestigating the case of $(a_n)$ being a sequence of real numbers. This tal
k is based on joint work with Christoph Aistleitner and Marc Munsch in whi
ch we pursue this line of research.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary Soroker (Tel Aviv)
DTSTART:20201112T120000Z
DTEND:20201112T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/13
DESCRIPTION:Title: Random Polynomials\, Probabilistic Galois Theory\, and Fin
ite Field Arithmetic\nby Lior Bary Soroker (Tel Aviv) as part of Tel A
viv number theory seminar\n\n\nAbstract\nWe will discuss recent advances o
n the following two question: \nLet $A(X) = \\sum \\pm X^i$ be a random po
lynomial of degree n with coefficients taking \nthe values -1\,1 independe
ntly each with probability 1/2.\n\nQ1: What is the probability that A is i
rreducible as the degree goes to infinity\n\nQ2: What is the typical Galoi
s of A?\n\nOne believes that the answers are YES and THE FULL SYMMETRIC GR
OUP\, respectively.\nThese questions were studied extensively in recent ye
ars\, and we will survey \nthe tools developed to attack these problem and
partial results.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoav Gath (Technion)
DTSTART:20201119T120000Z
DTEND:20201119T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/14
DESCRIPTION:Title: The lattice point counting problem on Heisenberg groups\nby Yoav Gath (Technion) as part of Tel Aviv number theory seminar\n\n\n
Abstract\nEuclidean lattice point counting problems\, the classical exampl
e of which is the Gauss circle problem\, are an important topic in classic
al analysis and have been the driving force behind much of the development
s in the area of analytic number theory in the 20th century. While it is w
ell known that homogeneous groups provide a natural setting to generalize
many questions of Euclidean harmonic analysis\, it was only recently that
analogues of the Euclidean lattice point counting problem were considered
for a certain family of 2-step nilpotent homogeneous groups. I will presen
t the lattice point counting problem for Cygan-Koranyi norm balls on the H
eisenberg groups\, which is the analogue of the lattice point counting pro
blem for Euclidean balls. I will describe recently obtained results relati
ng to the distribution and moments of the error term on the Heisenberg gro
ups\, and discuss the similarities (and stark differences) between the Euc
lidean and Heisenberg case.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nadav Yesha (Haifa)
DTSTART:20201210T120000Z
DTEND:20201210T130000Z
DTSTAMP:20260404T095030Z
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 n
atural way to test for the randomness of a sequence of points in $\\mathbb
R/\\mathbb Z$ is to consider its local statistics such as the $k$-level c
orrelations and the nearest-neighbour gap distribution\, and compare them
to those of a sequence of uniform independent random points (Poisson stati
stics). \nIn this talk I will describe recent results concerning two impor
tant examples of such sequences:\n\n- The sequence $\\{x^n\\}$\, where in
a joint work with Aistleitner\, Baker and Technau we showed that for almos
t all $x>1$\, all the correlations and hence the normalized gaps have a Po
issonian limit distribution.\n\n- The sequence $\\{n^x\\}$\, where in a jo
int work with Technau we showed Poissonian $k$-level correlations for almo
st all $x$ sufficiently large (depending on $k$).\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Erez Nesharim (Hebrew University)
DTSTART:20201217T120000Z
DTEND:20201217T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/16
DESCRIPTION:Title: Diophantine approximation over function fields and the t-a
dic Littlewood conjecture\nby Erez Nesharim (Hebrew University) as par
t of Tel Aviv number theory seminar\n\n\nAbstract\nAbstract: The Littlewoo
d conjecture in simultaneous approximation and the p-adic Littlewood conje
cture are famous open problems in the intersection of number theory and dy
namics. In a joint work with Faustin Adiceam and Fred Lunnon we show that
an analogue of the p-adic Littlewood conjecture over $\\mathbb F_3((1/t))$
is false. The counterexample is given by the Laurent series whose coeffic
ients are the regular paper folding sequence\, and the method of proof is
by reduction to the non vanishing of certain Hankel determinants. The proo
f is computer assisted and it uses substitution tilings of $\\mathbb Z^2$
and a generalisation of Dodgson's condensation algorithm for computing the
determinant of any Hankel matrix.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noam Kimmel (TAU)
DTSTART:20210107T120000Z
DTEND:20210107T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/17
DESCRIPTION:Title: On covariance of eigenvalue counts and lattice point probl
ems\nby Noam Kimmel (TAU) as part of Tel Aviv number theory seminar\n\
n\nAbstract\nWe explore the covariance of error terms coming from Weyl's c
onjecture regarding the number of Dirichlet eigenvalues up to size $X$.\nW
e also consider this problem in short intervals\, i.e. the error term of t
he number of eigenvalues in the window $[X\, X+S]$ for some $S(X)$.\nWe lo
ok at these error terms for planar domains where the Dirichlet eigenvalues
can be explicitly calculated.\nIn these cases\, the error term is closely
related to the error term from the classical lattice points counting prob
lem of expanding planar domains.\nWe give a formula for the covariance of
such error terms\, for general planar domains.\nWe also give a formula for
the covariance of error terms in short intervals\, for sufficiently large
intervals.\nGoing back to the Dirichlet eigenvalue problem\, we give resu
lts regarding the covariance of the error terms in short intervals of 'gen
eric' rectangles.\nWe also explore a specific example\, namely we compute
the covariance between the error terms of an equilateral triangle and vari
ous rectangles.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ezra Waxman (TU Dresden)
DTSTART:20210304T120000Z
DTEND:20210304T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/18
DESCRIPTION:Title: Artin Twin Primes and Poisson Binomial Distributions\n
by Ezra Waxman (TU Dresden) as part of Tel Aviv number theory seminar\n\n\
nAbstract\nWe say that a prime $p$ is an Artin prime for $g$ if $g$ is a p
rimitive root mod $p$. For appropriately chosen integers $g$ and $d$\, we
present a conjecture for the asymptotic number of prime pairs $(p\,p+d)$
such that both $p$ and $p+d$ are Artin primes for $g$. Our model suggests
that the distribution of Artin prime pairs\, amongst the ordinary prime p
airs\, is largely governed by a Poisson binomial distribution. Time permi
tting\, we moreover present a conjecture for the variance of Artin primes
across short intervals of ordinary primes\, obtained via similar heuristic
methods (Joint work with Magdaléna Tinková and Mikuláš Zindulka).\n
LOCATION:https://stable.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:20260404T095030Z
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 importan
t objects in spectral geometry is the counting function for the eigenvalue
s $\\lambda_j$ for the Dirichlet Laplacian associated with planar domains.
The simplest domains are squares\, disks and ellipses. \nIt is well-known
that for each of these domains its eigenvalue counting function\n $\\#\\{
\\lambda_j\\le\\mu^2 \\}$ \n has an asymptotic containing two main term
s $a \\mu^2 -b \\mu$ \nand a remainder of size $o(\\mu)$. To improve the
estimate of the remainder term had been one of the most attractive problem
s in spectral geometry for decades.\n\n\nI will introduce background brief
ly and explain how to transfer the above problem into problems of counting
lattice points\, to which tools from analysis and analytic number theory
can be applied. I will mention our progresses for disks\, annuli and balls
in high dimensions\, joint with Wolfgang Mueller\, Weiwei Wang and Zuoqin
Wang.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalie Evans (Queen Mary)
DTSTART:20210408T110000Z
DTEND:20210408T120000Z
DTSTAMP:20260404T095030Z
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-
Littlewood 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-ze
ro even integer $h$. While this conjecture remains wide open\, Matomäki\,
Radziwill and Tao proved that it holds on average over $h$\, improving on
a previous result of Mikawa. In this talk we will discuss an almost prime
analogue of the Hardy-Littlewood conjecture for which we can go beyond wh
at is known for primes. We will describe some recent work in which we prov
e an asymptotic formula for the number of almost primes $n=p_1p_2 \\le X$
such that $n+h$ has exactly two prime factors which holds for a very short
average over $h$.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Etai Leumi (Tel Aviv)
DTSTART:20210422T110000Z
DTEND:20210422T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/21
DESCRIPTION:Title: The LCM problem for function fields\nby Etai Leumi (Te
l Aviv) as part of Tel Aviv number theory seminar\n\n\nAbstract\nCilleruel
o conjectured that for any irreducible polynomial $f$ with integer coeffic
ients\, and degree greater than one\, the least common multiple of the val
ues 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. \nH
e proved this only for $\\deg(f)=2$. No example in higher degree is known.
We study the analogue of this conjecture for function fields\, \nwhere we
replace the integers by the ring of polynomials over a finite field. In t
hat setting we are able to establish some instances of the conjecture for
higher degrees. \nThe examples are all "special" polynomials $f(X)$\, wh
ich have the property that the bivariate polynomial $f(X)-f(Y)$ factors in
to linear terms in the base field.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Misha Sodin (Tel Aviv)
DTSTART:20210429T110000Z
DTEND:20210429T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/22
DESCRIPTION:Title: Equidistribution of zeroes of power series and binary corr
elations of coefficients.\nby Misha Sodin (Tel Aviv) as part of Tel Av
iv number theory seminar\n\n\nAbstract\nAbstract:\nWe will discuss global
and local 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 appr
oach to several examples of the sequence r of very different\norigin\, in
particular various sequences of arithmetic origin such as the Möbius func
tion where we see connections to Chowla’s conjecture\, random multiplica
tive\nfunctions\, and the function $e(xn^2)$ where the Diophantine nature
of x plays a role.\n\nThe talk will be based on joint work with Alexander
Borichev and Jacques Benatar\n(arXiv:1908.09161\, arXiv:2104.04812)\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Leuthi (Tel Aviv)
DTSTART:20210506T110000Z
DTEND:20210506T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/23
DESCRIPTION:Title: Random walks on homogeneous spaces\, Spectral Gaps\, and K
hinchine's theorem on fractals\nby Manuel Leuthi (Tel Aviv) as part of
Tel Aviv number theory seminar\n\n\nAbstract\nKhinchine's classical theor
em in Euclidean space gives a zero one law describing the approximability
of typical points by rational points. In 1984\, Mahler asked how well poin
ts on the middle third Cantor set can be approximated by rational numbers\
, both from within and from outside Cantor's set. His question fits into a
n attempt to determine conditions under which subsets of Euclidean space i
nherit the Diophantine properties of the ambient space. \nFor certain frac
tals significant progress has been made regarding the Diophantine properti
es of typical points\, albeit\, almost all known results have been of "con
vergence type". In this talk\, we will discuss the first instances where a
complete analogue of Khinchine’s theorem for fractal measures is obtain
ed. Our results hold for fractals generated by rational similarities of Eu
clidean space that have sufficiently small Hausdorff co-dimension. The mai
n new ingredient is an effective equidistribution theorem for associated f
ractal measures on the space of unimodular lattices. The latter is establi
shed using a spectral gap property of a type of Markov operators associate
d with the generating similarities. This is joint work with Osama Khalil.\
n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jens Marklof (Bristol)
DTSTART:20211028T110000Z
DTEND:20211028T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/24
DESCRIPTION:Title: How random are the roots of quadratic congruences?\nby
Jens Marklof (Bristol) as part of Tel Aviv number theory seminar\n\n\nAbs
tract\nIn 1963 Christopher Hooley showed that the roots of a quadratic con
gruence mod m\, appropriately normalized and averaged\, are uniformly dist
ributed mod 1. In this lecture\, which is based joint work with Matthew We
lsh (Bristol)\, we will study pseudo-randomness properties of the roots on
finer scales and prove for instance that the pair correlation density con
verges to an intriguing limit. A key step in our approach is to translate
the problem to convergence of certain geodesic random line processes in th
e hyperbolic plane.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura Monk (MPIM)
DTSTART:20211125T120000Z
DTEND:20211125T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/25
DESCRIPTION:Title: Geometry and spectrum of random hyperbolic surfaces\nb
y Laura Monk (MPIM) as part of Tel Aviv number theory seminar\n\n\nAbstrac
t\nThe aim of this talk is to describe typical compact hyperbolic surfaces
: results will be stated for most surfaces rather than every single one of
them. In order to motivate this idea\, I will first present examples intr
oduced in literature as limiting cases of famous theorems\, and argue that
they might be seen as "atypical". This will allow us to appreciate the co
ntrast with a fast-growing family of new results in both geometry and spec
tral theory\,\nwhich are established with probability close to one in vari
ous settings\, while being false for these atypical surfaces. In particula
r\, I will discuss results on the distribution of eigenvalues and the geom
etry of long geodesics\, as well as ongoing research on spectral gaps.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Par Kurlberg (KTH)
DTSTART:20211104T120000Z
DTEND:20211104T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/26
DESCRIPTION:Title: Cancellation in exponential sums over small multiplicative
subgroups of Z/pZ\nby Par Kurlberg (KTH) as part of Tel Aviv number t
heory seminar\n\n\nAbstract\nWe will sketch the proof of a breakthrough re
sult (from 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" multiplic
ative subgroup $H$ of the finite field $Z/pZ$. The result was discussed in
the first talk of the semester\, for showing that the digits of $1/p$ are
uniformly distributed if the period is not too small. The proof uses idea
s from additive combinatorics\, in particular the "sum-product theorem" an
d the Balog-Gowers-Szemeredi theorem (roughly\, subsets of $Z/pZ$ with "la
rge additive energy" must contain "large" subsets $S$ with property that
the sumset $S+S$ is "small").\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sa'ar Zehavi (TAU)
DTSTART:20211021T110000Z
DTEND:20211021T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/27
DESCRIPTION:Title: Sectorial equidistribution of the roots of $x^2=-1 \\mod p
$\nby Sa'ar Zehavi (TAU) as part of Tel Aviv number theory seminar\n\n
Lecture 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 F
ermat says 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 equidistr
ibuted is a famous theorem of Duke\, Friedlander and Iwaniec from 1995. We
examine what happens to the distribution when one adds a restriction on t
he primes which has to do with the angle in the plane formed by their corr
esponding representation as a sum of squares. This simple arithmetic quest
ion has a solution which involves multiple disciplines of number theory\,
but the talk does not assume any previous background.\n\nThe talk will be
delivered in person\, no online access.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Sartori (Tel Aviv)
DTSTART:20211111T120000Z
DTEND:20211111T130000Z
DTSTAMP:20260404T095030Z
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 he
ld in Schreiber 309\, TAU.\n\nAbstract\nGiven a prime $p$\, the generators
of the multiplicative group of the integers modulo $p$ are called primiti
ve roots. In 1930 Vinogradov conjectured that the smallest generator\, the
least primitive root\, is smaller than any power of $p$. This talk will b
e a general introduction to the subject. I will discuss the classic result
s of Vinogradov 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://stable.researchseminars.org/talk/TAUNumbertheory/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ofir Gorodetsky (Oxford)
DTSTART:20211209T120000Z
DTEND:20211209T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/29
DESCRIPTION:Title: Sums of two squares are strongly biased towards quadratic
residues\nby Ofir Gorodetsky (Oxford) as part of Tel Aviv number theor
y seminar\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 p
rimes of the form 1 mod 4. This was confirmed theoretically much later by
Rubinstein and Sarnak in a logarithmic density sense. Our understanding of
this is conditional on the generalized Riemann Hypothesis as well as Line
ar Independence of the zeros of L-functions.\n\nWe investigate similar que
stions for sums of two squares in arithmetic progressions. We find a signi
ficantly stronger bias than in primes\, which happens for almost all integ
ers in a natural density sense. Because the bias is more pronounced\, we d
o not need to assume Linear Independence of zeros\, only a Chowla-type Con
jecture on non-vanishing of L-functions at 1/2.\nThe bias is stronger beca
use it arises from a multiplicative contribution of squares as opposed to
additive contribution (as in the case of primes). \n\nTo illustrate\, we h
ave under GRH that the number of sums of two squares up to x that are 1 mo
d 3 is greater than those that are 2 mod 3 for all but o(x) integers.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Lester (King's College London)
DTSTART:20211216T120000Z
DTEND:20211216T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/30
DESCRIPTION:Title: Spacing statistics for lattice points on circles\nby S
teve Lester (King's College London) as part of Tel Aviv number theory semi
nar\n\n\nAbstract\nIn this talk I will describe the distribution of lattic
e points lying on circles. A striking result of Kátai and Környei shows
that along a density one subsequence of admissible radii the angles of lat
tice points lying on circles are uniformly distributed in the limit as the
radius tends to infinity. Their result goes further\, proving that unifor
m distribution persists even at very small scales\, meaning that the angle
s are uniformly 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 number of gaps between consecutive angles of size less than $1/N$. I
will discuss some recent joint work with Pär Kurlberg in which we comput
e the nearest neighbor spacing of the angles along a density one subsequen
ce of admissible radii.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Verónica Becher (Universidad de Buenos Aires)
DTSTART:20220224T120000Z
DTEND:20220224T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/31
DESCRIPTION:Title: Poisson generic real numbers.\nby Verónica Becher (Un
iversidad de Buenos Aires) as part of Tel Aviv number theory seminar\n\n\n
Abstract\nYears ago\, Zeev Rudnick defined the Poisson generic real number
s as those where the number of occurrences of the long strings \nin the i
nitial segments of their fractional expansions in some base follow the Poi
sson distribution. \nPeres and Weiss proved that almost all real numbers\,
with respect to Lebesgue measure\, are Poisson generic. \nThey also showe
d that Poisson genericity implies Borel normality\, but the two notions do
not coincide\, witnessed by the famous \nChampernowne constant. I will
discuss these results and present a construction of a Poisson generic rea
l for a fixed parameter lambda.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xin Zhang (Hong Kong University)
DTSTART:20220310T120000Z
DTEND:20220310T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/32
DESCRIPTION:Title: Traces of elements in thin subgroups of the modular group<
/a>\nby Xin Zhang (Hong Kong University) as part of Tel Aviv number theory
seminar\n\n\nAbstract\nFor a subgroup of the modular group\, we ask which
integers occur as the trace of an element of the subgroup? For the modula
r group itself\, every integer occurs. The question is particularly intere
sting for "thin" groups\, which are certain subgroups of infinite index. W
e use circle method to prove a local-global theorem on the set of traces\,
when the subgroup contains a parabolic element. This yields fine informat
ion on the length spectrum (the set of lengths of closed geodesics) of the
hyperbolic surface associated to this group. This is joint work with Alex
Kontorovich.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lior Bary Soroker (Tel Aviv)
DTSTART:20220303T120000Z
DTEND:20220303T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/33
DESCRIPTION:Title: Distribution of rational points on elliptic curves\nby
Lior Bary Soroker (Tel Aviv) as part of Tel Aviv number theory seminar\n\
nLecture held in Schreiber 309\, TAU.\n\nAbstract\nHilbert's irreducibilit
y theorem may be formulated as the statement the rational points on the li
ne cannot be covered by rational points coming from finitely many covers.\
nWhen one wants to replace the line\, by more complicated varieties\, such
as elliptic curves or algebraic groups\, such a naive statement fails\, d
ue to the existences of isogenies. It turns out the correct generalization
is for ramified covers. \n\nIn the talk\, we will discuss some recent pro
gress on these problems.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ezra Wazman (University of Haifa)
DTSTART:20220324T120000Z
DTEND:20220324T130000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/34
DESCRIPTION:Title: The Correction Factor in Artin's Primitive Root Conjecture
: Classically 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
density of primes $p$ for which $g$ is a primitive root mod $p$. By ob
serving numerical deviations from Artin’s originally predicted asymptoti
c\, Derrick and Emma Lehmer (1957) identified the need for an additional c
orrection factor\; leading to a modified conjecture which was eventually p
roved to be correct by Hooley (1967) under the assumption of GRH. An appr
opriate analogue of Artin's primitive root conjecture may also be formulat
ed over a global function field\, where Bilharz provided a proof that is c
orrect under the assumption that $g$ is a "geometric" element. In this t
alk we discuss the correction factor that emerges when one removes the ass
umption that $g$ is geometric\; thereby completing the proof of Artin's p
rimitive root conjecture for arbitrary function fields in one variable ove
r a finite field.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Wigman (King's College London)
DTSTART:20220407T110000Z
DTEND:20220407T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/35
DESCRIPTION:Title: The Robin problem for rectangles\nby Igor Wigman (King
's College London) as part of Tel Aviv number theory seminar\n\nLecture he
ld in Schreiber 309\, TAU.\n\nAbstract\nRobin boundary conditions are used
in heat conductance theory to interpolate between a perfectly insulating
boundary\, described by Neumann boundary conditions\, and a temperature fi
xing boundary\, described by Dirichlet boundary conditions. The Neumann an
d Dirichlet spectrum of a square are just sums of two squares\, hence have
a direct arithmetic significance. The Robin spectrum is more mysterious a
nd until now the number theory behind it was not explored. We study the st
atistics and the arithmetic properties of the Robin spectrum of a rectangl
e. Based on a joint work with Zeev Rudnick.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacques Benatar (Tel Aviv)
DTSTART:20220428T110000Z
DTEND:20220428T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/36
DESCRIPTION:Title: Extremal bounds for Dirichlet polynomials with multiplicat
ive coefficients\nby Jacques Benatar (Tel Aviv) as part of Tel Aviv nu
mber theory seminar\n\n\nAbstract\nI will discuss some recent work\, joint
with Alon Nishry and Brad Rodgers\, concerning the distribution of Dirich
let and trigonometric polynomials generated by multiplicative coefficients
$f(n)$.\n\nIn the first part of the talk we will explore some old and new
results for deterministic sequences $f(n)$ (Möbius\, Legendre symbol\,..
.)\, stopping along our journey to marvel at a variety of wild and thorny
conjectures. The second half of the talk will be devoted to Steinhaus rand
om multiplicative coefficients $f(n)=X(n)$.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noam Kimmel (Tel Aviv)
DTSTART:20220512T110000Z
DTEND:20220512T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/37
DESCRIPTION:Title: The Least Common Multiple of Polynomial sequences\nby
Noam Kimmel (Tel Aviv) as part of Tel Aviv number theory seminar\n\n\nAbs
tract\nThe prime number theorem can be stated as saying that the logarithm
of the least common multiple (LCM) of the first $N$ integers is asymptoti
cally equal to $N$\, as was known to Chebyshev. \n\nMotivated by this form
ulation\, we look at a generalization - the least common multiple of polyn
omial sequences. The case of a polynomial in one variable was first studie
d by Cilleruelo in 2011\, who determined the asymptotics of the quadratic
case\, and has since been explored by various other researchers. The concl
usion is that for an irreducible polynomial $F(x)$ of degree $d\\geq 2$\,
\n$\\log LCM (F(1)\,\\dots F(N))$ grows roughly as $N \\log N$\, though
we still do not know the asymptotics\, conjectured to be $(d-1)N \\log(N)$
. \n\nIn this talk we consider polynomials in two variables. We discover
that already in the quadratic case\, there is a range of asymptotic behavi
ours. We 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://stable.researchseminars.org/talk/TAUNumbertheory/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zeev Rudnick (Tel Aviv)
DTSTART:20220519T110000Z
DTEND:20220519T120000Z
DTSTAMP:20260404T095030Z
UID:TAUNumbertheory/38
DESCRIPTION:Title: eigenvalue statistics for hyperbolic surfaces of large gen
us\nby Zeev Rudnick (Tel Aviv) as part of Tel Aviv number theory semin
ar\n\n\nAbstract\nAn outstanding conjecture in quantum chaos is that the s
tatistics of the energy levels of "generic" chaotic systems with time reve
rsal symmetry are described by those of the Gaussian Orthogonal Ensemble (
GOE) in Random Matrix Theory. Conjectural examples are the eigenvalues of
the Laplacian on a "generic" hyperbolic surface. This conjecture has prove
d to be extremely difficult\, with no single case being proved\, the close
st case being some results for the Riemann zeros which seem to have simila
r statistics\, those of the Gaussian Unitary Ensemble. It has long been de
sired to improve the situation by averaging over a suitable ensemble of ch
aotic systems. I will describe a version of such ensemble averaging on the
moduli space of compact hyperbolic surfaces\, equipped with the Weil-Pete
rsson measure\, using the pioneering work of Maryam Mirzakhani. For a suit
able quantity\, we obtain a small confirmation of GOE statistics.\n
LOCATION:https://stable.researchseminars.org/talk/TAUNumbertheory/38/
END:VEVENT
END:VCALENDAR