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SUMMARY:Youness Lamzouri (Institut Élie Cartan de Lorraine\, Nancy)
DTSTART:20230111T200000Z
DTEND:20230111T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/1
DESCRIPTION:Title: Zeros of linear combinations of $L$-functions near the c
ritical line\nby Youness Lamzouri (Institut Élie Cartan de Lorraine\,
Nancy) as part of CRG Weekly Seminars\n\n\nAbstract\nIn this talk\, I wil
l present a recent joint work with Yoonbok Lee\, where we investigate the
number of zeros of linear combinations of $L$-functions in the vicinity of
the critical line. More precisely\, we let $L_1\, \\dots\, L_J$ be distin
ct primitive $L$-functions belonging to a large class (which conjecturally
contains all $L$-functions arising from automorphic representations on $\
\text{GL}(n)$)\, and $b_1\, \\dots\, b_J$ be real numbers. Our main result
is an asymptotic formula for the number of zeros of $F(\\sigma+it)=\\sum_
{j\\leq J} b_j L_j(\\sigma+it)$ in the region $\\sigma\\geq 1/2+1/G(T)$ an
d $t\\in [T\, 2T]$\, uniformly in the range $\\log \\log T \\leq G(T)\\leq
(\\log T)^{\\nu}$\, where $\\nu\\asymp 1/J$. This establishes a general f
orm of a conjecture of Hejhal in this range. The strategy of the proof rel
ies on comparing the distribution of $F(\\sigma+it)$ to that of an associa
ted probabilistic random model.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enrique Treviño (Lake Forest College)
DTSTART:20230118T200000Z
DTEND:20230118T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/2
DESCRIPTION:Title: Least quadratic non-residue and related problems\nby
Enrique Treviño (Lake Forest College) as part of CRG Weekly Seminars\n\n
Abstract: TBA\n\nIn this talk we will talk about explicit estimates for ch
aracter sums which have allowed us to find explicit estimates for the leas
t quadratic non-residue and other related problems.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel R. Johnston (UNSW Canberra)
DTSTART:20230125T200000Z
DTEND:20230125T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/3
DESCRIPTION:Title: An explicit error term in the prime number theorem for l
arge $x$\nby Daniel R. Johnston (UNSW Canberra) as part of CRG Weekly
Seminars\n\n\nAbstract\nIn 1896\, the prime number theorem was established
\, showing that $\\pi(x)\\sim \\textrm{li}(x)$. Perhaps the most widely us
ed estimates in explicit analytic number theory are bounds on $|\\pi(x)-\\
textrm{li}(x)|$ or the related error term $|\\theta(x)-x|$. In this talk w
e discuss methods one can use to obtain good bounds on these error terms w
hen $x$ is large. Moreover\, we will explore the many ways in which these
bounds could be improved in the future.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alisa A. Sedunova (St. Petersburg State University)
DTSTART:20230208T200000Z
DTEND:20230208T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/4
DESCRIPTION:Title: A logarithmic improvement in the Bombieri-Vinogradov the
orem\nby Alisa A. Sedunova (St. Petersburg State University) as part o
f CRG Weekly Seminars\n\n\nAbstract\nWe improve the best known to date res
ult of Dress-Iwaniec-Tenenbaum\, getting $(\\log{x})^2$ instead of $(\\log
x)^{5/2}$. We use a weighted form of Vaughan's identity\, allowing a smoo
th truncation inside the procedure\, and an estimate due to Barban-Vehov a
nd Graham related to Selberg's sieve. We give effective and non-effective
versions of the result.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asif Zaman (University of Toronto)
DTSTART:20230215T200000Z
DTEND:20230215T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/5
DESCRIPTION:Title: A uniform prime number theorem for arithmetic progressio
ns\nby Asif Zaman (University of Toronto) as part of CRG Weekly Semina
rs\n\n\nAbstract\nI will describe a version of the prime number theorem fo
r arithmetic progressions that is uniform enough to deduce the Siegel-Walf
isz theorem\, Hoheisel's asymptotic for short intervals\, a Brun-Titchmars
h bound and Linnik's bound for the least prime in an arithmetic progressio
n. The proof combines Vinogradov-Korobov's zero-free region\, a log-free z
ero density estimate and the Deuring-Heilbronn zero repulsion phenomenon.
This is joint work with Jesse Thorner.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ghaith Hiary (Ohio State University)
DTSTART:20230301T200000Z
DTEND:20230301T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/6
DESCRIPTION:Title: A new explicit bound for the Riemann zeta function\n
by Ghaith Hiary (Ohio State University) as part of CRG Weekly Seminars\n\n
\nAbstract\nI give a new explicit bound for the Riemann zeta function on t
he critical line. This is joint work with Dhir Patel and Andrew Yang. The
context of this work highlights the importance of reliability and reproduc
ibility of explicit bounds in analytic number theory.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wanlin Li (Washington University in St. Louis)
DTSTART:20230201T200000Z
DTEND:20230201T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/7
DESCRIPTION:Title: The central value of Dirichlet $L$-functions over functi
on fields and related topics\nby Wanlin Li (Washington University in S
t. Louis) as part of CRG Weekly Seminars\n\n\nAbstract\nA Dirichlet charac
ter over $\\mathbb{F}_q(t)$ corresponds to a curve over $\\mathbb{F}_q$. U
sing this connection to geometry\, we construct families of characters who
se $L$-functions vanish (resp. does not vanish) at the central point. The
existence of infinitely many vanishing $L$-functions is in contrast with t
he situation over the rational numbers\, where a conjecture of Chowla pred
icts there should be no such. Towards Chowla's conjecture\, for each fixed
$q$\, we present an explicit upper bound on the number of such quadratic
characters which decreases as $q$ grows and it goes to $0$ percent as $q$
goes to infinity. In this talk\, I will also discuss phenomena and interes
ting questions related to this problem. Some results in this talk are from
projects joint with Ravi Donepudi\, Jordan Ellenberg and Mark Shusterman.
\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne-Maria Ernvall-Hytönen (University of Helsinki)
DTSTART:20230308T200000Z
DTEND:20230308T210000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/8
DESCRIPTION:Title: Euler's divergent series and primes in arithmetic progre
ssions\nby Anne-Maria Ernvall-Hytönen (University of Helsinki) as par
t of CRG Weekly Seminars\n\n\nAbstract\nEuler's divergent series $\\sum_{n
= 0}^\\infty n! z^n$ which converges only for $z = 0$ becomes an interest
ing object when evaluated with respect to a $p$-adic norm (which will be i
ntroduced in the talk). Very little is known about the values of the serie
s. For example\, it is an open question whether the value at one is irrati
onal (or even non-zero). As individual values are difficult to reach\, it
makes sense to try to say something about collections of values over suffi
ciently large sets of primes. This leads to looking at primes in arithmeti
c progressions\, which is in turn raises a need for an explicit bound for
the number of primes in an arithmetic progression under the generalized Ri
emann hypothesis.\n\nDuring the talk\, I will speak about both sides of th
e story: why we needed good explicit bounds for the number of primes in ar
ithmetic progressions while working with questions about irrationality\, a
nd how we then proved such a bound.\n\nThe talk is joint work with Tapani
Matala-aho\, Neea Palojärvi and Louna Seppälä. (Questions about irratio
nality with T. M. and L. S. and primes in arithmetic progressions with N.
P.)\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jyothsnaa Sivaraman (Chennai Mathematical Institute)
DTSTART:20230315T190000Z
DTEND:20230315T200000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/9
DESCRIPTION:Title: Products of primes in ray classes\nby Jyothsnaa Siva
raman (Chennai Mathematical Institute) as part of CRG Weekly Seminars\n\n\
nAbstract\nIn 1944\, Linnik showed that the least prime in an arithmetic p
rogression given by $a \\bmod q$ for $(a\,q)=1$ is at most $cq^L$ for some
absolutely computable constants $c$ and $L$.\nA lot of work has gone in c
omputing explicit bounds for $c$ and $L$. The best known bound is due to X
ylouris (2011) who showed that $c$ can be taken to be $1$ and $L$ to be $5
$ for $q$ sufficiently large. In 2018\, Ramar$\\acute{\\text{e}}$ and Walk
er gave a completely explicit result if one prime is replaced by a product
of primes. They showed that each co-prime class modulo $q$ contains a pro
duct of three primes each less than $q^{16/3}$. This was improved by Ramar
$\\acute{\\text{e}}$\, Srivastava and Serra to $650 q^3$ in 2020. In this
talk we will introduce analogous results in the set up of narrow ray class
fields of number fields. This is joint work with Deshouillers\, Gun and R
amar$\\acute{\\text{e}}$.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olga Balkanova (Steklov Mathematical Institute)
DTSTART:20230405T190000Z
DTEND:20230405T200000Z
DTSTAMP:20260404T111444Z
UID:crgseminarwinter23/10
DESCRIPTION:Title: The second moment of symmetric square $L$-functions ove
r Gaussian integers\nby Olga Balkanova (Steklov Mathematical Institute
) as part of CRG Weekly Seminars\n\n\nAbstract\nWe prove an explicit formu
la for the first moment of Maass form symmetric square $L$-functions defin
ed over Gaussian integers. As a consequence\, we derive a new upper bound
for the second moment. This is joint work with Dmitry Frolenkov.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter23/10/
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