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SUMMARY:Chiara Bellotti (University of New South Wales\, Canberra)
DTSTART;VALUE=DATE-TIME:20240117T000000Z
DTEND;VALUE=DATE-TIME:20240117T010000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/1
DESCRIPTION:Title: Explicit bounds for $\\zeta$ and a new zero-free region\nby
Chiara Bellotti (University of New South Wales\, Canberra) as part of CRG
Weekly Seminars\n\n\nAbstract\nIn this talk we prove that $|\\zeta(\\sigm
a+it)|\\le 70.7 |t|^{4.438(1-\\sigma)^{3/2}}\\log^{2/3}|t|$ for $1/2\\le\\
sigma\\le 1$ and $|t|\\ge 3$\, combining new explicit bounds for the Vinog
radov integral with exponential sums estimates. As a consequence\, we impr
ove the explicit zero-free region for $\\zeta(s)$\, showing that $\\zeta(\
\sigma+it)$ has no zeros in the region $\\sigma \\geq 1-1 /\\left(53.989(\
\log |t|)^{2 / 3}(\\log \\log |t|)^{1 / 3}\\right)$ for $|t| \\geq 3$.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winston Heap (Norwegian University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20240122T190000Z
DTEND;VALUE=DATE-TIME:20240122T200000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/2
DESCRIPTION:Title: Mean values of long Dirichlet polynomials\nby Winston Heap
(Norwegian University of Science and Technology) as part of CRG Weekly Sem
inars\n\n\nAbstract\nWe discuss the role of long Dirichlet polynomials in
number theory. We first survey some applications of mean values of long Di
richlet polynomials over primes in the theory of the Riemann zeta function
which includes central limit theorems and pair correlation of zeros. We t
hen give some examples showing how\, on assuming the Riemann Hypothesis\,
one can compute asymptotics for such mean values without using the Hardy-L
ittlewood conjectures for additive correlations of the von-Mangoldt functi
ons.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emily Quesada-Herrera (Graz University of Technology)
DTSTART;VALUE=DATE-TIME:20240129T190000Z
DTEND;VALUE=DATE-TIME:20240129T200000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/3
DESCRIPTION:Title: Fourier optimization and the least quadratic non-residue\nb
y Emily Quesada-Herrera (Graz University of Technology) as part of CRG Wee
kly Seminars\n\n\nAbstract\nWe will explore how a Fourier optimization fra
mework may be used to study two classical problems in number theory involv
ing Dirichlet characters: The problem of estimating the least character no
n-residue\; and the problem of estimating the least prime in an arithmetic
progression. In particular\, we show how this Fourier framework leads to
subtle\, but conceptually interesting\, improvements on the best current a
symptotic bounds under the Generalized Riemann Hypothesis\, given by Lamzo
uri\, Li\, and Soundararajan. Based on joint work with Emanuel Carneiro\,
Micah Milinovich\, and Antonio Ramos.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivian Kuperberg (ETH Zurich)
DTSTART;VALUE=DATE-TIME:20240212T190000Z
DTEND;VALUE=DATE-TIME:20240212T200000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/4
DESCRIPTION:Title: Consecutive sums of two squares in arithmetic progressions\
nby Vivian Kuperberg (ETH Zurich) as part of CRG Weekly Seminars\n\n\nAbst
ract\nIn 2000\, Shiu proved that there are infinitely many primes whose la
st digit is 1 such that the next prime also ends in a 1. However\, it is a
n open problem to show that there are infinitely many primes ending in 1 s
uch that the next prime ends in 3. In this talk\, we'll instead consider t
he sequence of sums of two squares in increasing order. In particular\, we
'll show that there are infinitely many sums of two squares ending in 1 su
ch that the next sum of two squares ends in 3. We'll show further that all
patterns of length 3 occur infinitely often: for any modulus q\, every se
quence (a mod q\, b mod q\, c mod q) appears infinitely often among consec
utive sums of two squares. We'll discuss some of the proof techniques\, an
d explain why they fail for primes. Joint work with Noam Kimmel.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bittu (Indraprastha Institute of Information Technology\, Delhi)
DTSTART;VALUE=DATE-TIME:20240226T190000Z
DTEND;VALUE=DATE-TIME:20240226T200000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/5
DESCRIPTION:Title: Spacing statistics of the Farey sequence\nby Bittu (Indrapr
astha Institute of Information Technology\, Delhi) as part of CRG Weekly S
eminars\n\n\nAbstract\nThe Farey sequence $\\mathcal{F}_Q$ of order $Q$ is
an ascending sequence of fractions $a/b$ in the unit interval $(0\,1]$ su
ch that $(a\,b)=1$ and $0< a \\leq b \\leq Q $. The study of the Farey fra
ctions is of major interest because of their role in problems related to t
he Diophantine approximation. Also\, there is a connection between the dis
tribution of Farey fractions and the Riemann hypothesis\, which motivates
their study. In this talk\, we will discuss the distribution of Farey frac
tions with some divisibility constraints on denominators by studying their
pair correlation measure. This is based on the joint work with Sneha Chau
bey.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Quanli Shen (Shandong University\, Weihai)
DTSTART;VALUE=DATE-TIME:20240318T180000Z
DTEND;VALUE=DATE-TIME:20240318T190000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/6
DESCRIPTION:Title: The fourth moment of quadratic Dirichlet $L$-functions\nby
Quanli Shen (Shandong University\, Weihai) as part of CRG Weekly Seminars\
n\n\nAbstract\nI will discuss the fourth moment of quadratic Dirichlet $L$
-functions where we prove an asymptotic formula with four main terms uncon
ditionally. Previously the asymptotic formula was established with the lea
ding main term under generalized Riemann hypothesis. This work is based on
Li's recent work on the second moment of quadratic twists of modular $L$-
functions. It is joint work with Joshua Stucky.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Greg Knapp (University of Calgary)
DTSTART;VALUE=DATE-TIME:20240410T180000Z
DTEND;VALUE=DATE-TIME:20240410T190000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/7
DESCRIPTION:Title: Bounds on the Number of Solutions to Thue Equations\nby Gre
g Knapp (University of Calgary) as part of CRG Weekly Seminars\n\n\nAbstra
ct\nIn 1909\, Thue proved that when $F(x\,y)$ is an irreducible\, homogene
ous\, polynomial with integer coefficients and degree at least $3$\, the i
nequality $|F(x\,y)| \\leq h$ has finitely many integer-pair solutions for
any positive $h$. Because of this result\, the inequality $| F(x\,y) | \
\leq h$ is known as Thue’s Inequality. Much work has been done to find
sharp bounds on the size and number of integer-pair solutions to Thue’s
Inequality\, with Mueller and Schmidt initiating the modern approach to t
his problem in the 1980s. In this talk\, I will describe different techni
ques used by Akhtari and Bengoechea\; Baker\; Mueller and Schmidt\; Saradh
a and Sharma\; and Thomas to make progress on this general problem. After
that\, I will discuss some improvements that can be made to a counting te
chnique used in association with “the gap principle” and how those imp
rovements lead to better bounds on the number of solutions to Thue’s Ine
quality.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Stadlmann (University of Oxford)
DTSTART;VALUE=DATE-TIME:20240304T190000Z
DTEND;VALUE=DATE-TIME:20240304T200000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/9
DESCRIPTION:Title: Primes in arithmetic progressions to smooth moduli\nby Juli
a Stadlmann (University of Oxford) as part of CRG Weekly Seminars\n\n\nAbs
tract\nThe twin prime conjecture asserts that there are infinitely many pr
imes p for which p+2 is also prime. This conjecture appears far out of rea
ch of current mathematical techniques. However\, in 2013 Zhang achieved a
breakthrough\, showing that there exists some positive integer h for which
p and p+h are both prime infinitely often. Equidistribution estimates for
primes in arithmetic progressions to smooth moduli were a key ingredient
of his work. In this talk\, I will sketch what role these estimates play i
n proofs of bounded gaps between primes. I will also show how a refinement
of the q-van der Corput method can be used to improve on equidistribution
estimates of the Polymath project for primes in APs to smooth moduli.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:(Talk cancelled) Sneha Chaubey (Indraprastha Institute of Informat
ion Technology\, Delhi)
DTSTART;VALUE=DATE-TIME:20240311T180000Z
DTEND;VALUE=DATE-TIME:20240311T190000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/10
DESCRIPTION:Title: (Talk cancelled) Distribution of spacings of real-valued seque
nces\nby (Talk cancelled) Sneha Chaubey (Indraprastha Institute of Inf
ormation Technology\, Delhi) as part of CRG Weekly Seminars\n\n\nAbstract\
nThe topic on the distribution of sequences saw its light with the seminal
paper of Weyl. While the classical notion of equidistribution modulo one
addresses the “global” behaviour of the fractional parts of a sequence
\, quantities such as $k$-point correlations and nearest neighbour gap dis
tributions are useful in investigating the sequence on finer scales. In th
is talk\, we discuss these fine-scale statistics for real-valued arithmeti
c sequences\, and show that the limiting distribution of the nearest neigh
bour gaps of real-valued lacunary sequences is Poissonian. We also prove t
he Poissonian behavior of the $2$-point correlation function for certain c
lasses of real-valued vector sequences. This is achieved by extrapolating
conditions on the number of solutions of Diophantine inequalities using tw
isted moments of the Riemann zeta function.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jérémy Dousselin (Institut Élie Cartan de Lorraine\, Nancy)
DTSTART;VALUE=DATE-TIME:20240325T180000Z
DTEND;VALUE=DATE-TIME:20240325T190000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/11
DESCRIPTION:Title: Zeros of linear combinations of Dirichlet $L$-functions on the
critical line\nby Jérémy Dousselin (Institut Élie Cartan de Lorrai
ne\, Nancy) as part of CRG Weekly Seminars\n\n\nAbstract\nFix $N\\geq 1$ a
nd let $L_1$\, $L_2$\, ...\, $L_N$ be Dirichlet $L$-functions with distinc
t\, primitive and even Dirichlet characters. We assume that these function
s satisfy the same functional equation. Let\n\\[F(s):=\\sum_{j=1}^N c_jL_j
(s)\\]\nbe a linear combination of these functions ($c_j\\in\\mathbb{R}^*$
are distinct).$F$ is known to have two kinds of zeros: trivial ones\, and
non-trivial zeros which are confined in a vertical strip. We denote the n
umber of non-trivial zeros $\\rho$ with $\\Im(\\rho)\\leq T$ by $N(T)$\, a
nd we let $N_0(T)$ be the number of these zeros that are on the critical l
ine.At the end of the 90s\, Selberg proved that this linear combination ha
d a positive proportion of zeros on the critical line\, by showing that\n\
\[\\kappa_F:=\\liminf_T\\frac{N_0(2T)-N_0(T)}{N(2T)-N(T)}\\geq \\frac c{N^
2}\\]\nfor some $c>0$.Our goal is to provide an explicit value for $c$\, a
nd also to improve the lower bound above by showing that\n\\[\\kappa_F\\ge
q \\frac{2.16\\times 10^{-6}}{N\\log N}\,\\]\nfor any large enough $N$.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugenia Rosu (Mathematical Institute\, Universiteit Leiden)
DTSTART;VALUE=DATE-TIME:20240205T190000Z
DTEND;VALUE=DATE-TIME:20240205T200000Z
DTSTAMP;VALUE=DATE-TIME:20240804T055505Z
UID:crgseminarwinter24/13
DESCRIPTION:Title: A higher degree Weierstrass function\nby Eugenia Rosu (Mat
hematical Institute\, Universiteit Leiden) as part of CRG Weekly Seminars\
n\n\nAbstract\nThe Weierstrass $\\wp$ function plays a great role in the c
lassic theory of complex elliptic curves. A related function\, the Weierst
rass zeta-function\, is used by Guerzhoy to construct preimages under the
$\\xi$-operator of newforms of weight 2\, corresponding to elliptic curves
. In this talk\, I will discuss a generalization of the Weierstrass zeta-f
unction and an application to harmonic Maass forms. More precisely\, I wil
l describe a construction of a preimage of the $\\xi$-operator of a newfor
m of weight k for k>2. This is based on joint work with C. Alfes-Neumann\,
J. Funke and M. Mertens.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter24/13/
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