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SUMMARY:Youness Lamzouri (Institut Élie Cartan de Lorraine\, Nancy)
DTSTART;VALUE=DATE-TIME:20230111T200000Z
DTEND;VALUE=DATE-TIME:20230111T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/1
DESCRIPTION:Title: Zeros of linear combinations of $L$-functions near the critical
line\nby Youness Lamzouri (Institut Élie Cartan de Lorraine\, Nancy)
as part of CRG Weekly Seminars\n\n\nAbstract\nIn this talk\, I will prese
nt 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 cr
itical line. More precisely\, we let $L_1\, \\dots\, L_J$ be distinct prim
itive $L$-functions belonging to a large class (which conjecturally contai
ns all $L$-functions arising from automorphic representations on $\\text{G
L}(n)$)\, 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 $\\sigma\\geq 1/2+1/G(T)$ and $t\\i
n [T\, 2T]$\, uniformly in the range $\\log \\log T \\leq G(T)\\leq (\\log
T)^{\\nu}$\, where $\\nu\\asymp 1/J$. This establishes a general form of
a conjecture of Hejhal in this range. The strategy of the proof relies on
comparing the distribution of $F(\\sigma+it)$ to that of an associated pro
babilistic random model.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/1/
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BEGIN:VEVENT
SUMMARY:Enrique Treviño (Lake Forest College)
DTSTART;VALUE=DATE-TIME:20230118T200000Z
DTEND;VALUE=DATE-TIME:20230118T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/2
DESCRIPTION:Title: Least quadratic non-residue and related problems\nby Enriqu
e Treviño (Lake Forest College) as part of CRG Weekly Seminars\n\nAbstrac
t: TBA\n\nIn this talk we will talk about explicit estimates for character
sums which have allowed us to find explicit estimates for the least quadr
atic non-residue and other related problems.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel R. Johnston (UNSW Canberra)
DTSTART;VALUE=DATE-TIME:20230125T200000Z
DTEND;VALUE=DATE-TIME:20230125T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/3
DESCRIPTION:Title: An explicit error term in the prime number theorem for large $x
$\nby Daniel R. Johnston (UNSW Canberra) as part of CRG Weekly Seminar
s\n\n\nAbstract\nIn 1896\, the prime number theorem was established\, show
ing that $\\pi(x)\\sim \\textrm{li}(x)$. Perhaps the most widely used esti
mates in explicit analytic number theory are bounds on $|\\pi(x)-\\textrm{
li}(x)|$ or the related error term $|\\theta(x)-x|$. In this talk we discu
ss methods one can use to obtain good bounds on these error terms when $x$
is large. Moreover\, we will explore the many ways in which these bounds
could be improved in the future.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alisa A. Sedunova (St. Petersburg State University)
DTSTART;VALUE=DATE-TIME:20230208T200000Z
DTEND;VALUE=DATE-TIME:20230208T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/4
DESCRIPTION:Title: A logarithmic improvement in the Bombieri-Vinogradov theorem\nby Alisa A. Sedunova (St. Petersburg State University) as part of CRG W
eekly Seminars\n\n\nAbstract\nWe improve the best known to date result 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 smooth trun
cation inside the procedure\, and an estimate due to Barban-Vehov and Grah
am related to Selberg's sieve. We give effective and non-effective version
s of the result.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asif Zaman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20230215T200000Z
DTEND;VALUE=DATE-TIME:20230215T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/5
DESCRIPTION:Title: A uniform prime number theorem for arithmetic progressions\
nby Asif Zaman (University of Toronto) as part of CRG Weekly Seminars\n\n\
nAbstract\nI will describe a version of the prime number theorem for arith
metic progressions that is uniform enough to deduce the Siegel-Walfisz the
orem\, Hoheisel's asymptotic for short intervals\, a Brun-Titchmarsh bound
and Linnik's bound for the least prime in an arithmetic progression. The
proof combines Vinogradov-Korobov's zero-free region\, a log-free zero den
sity estimate and the Deuring-Heilbronn zero repulsion phenomenon. This is
joint work with Jesse Thorner.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ghaith Hiary (Ohio State University)
DTSTART;VALUE=DATE-TIME:20230301T200000Z
DTEND;VALUE=DATE-TIME:20230301T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/6
DESCRIPTION:Title: A new explicit bound for the Riemann zeta function\nby Ghai
th Hiary (Ohio State University) as part of CRG Weekly Seminars\n\n\nAbstr
act\nI give a new explicit bound for the Riemann zeta function on the crit
ical line. This is joint work with Dhir Patel and Andrew Yang. The context
of this work highlights the importance of reliability and reproducibility
of explicit bounds in analytic number theory.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wanlin Li (Washington University in St. Louis)
DTSTART;VALUE=DATE-TIME:20230201T200000Z
DTEND;VALUE=DATE-TIME:20230201T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/7
DESCRIPTION:Title: The central value of Dirichlet $L$-functions over function fiel
ds and related topics\nby Wanlin Li (Washington University in St. Loui
s) as part of CRG Weekly Seminars\n\n\nAbstract\nA Dirichlet character ove
r $\\mathbb{F}_q(t)$ corresponds to a curve over $\\mathbb{F}_q$. Using th
is connection to geometry\, we construct families of characters whose $L$-
functions vanish (resp. does not vanish) at the central point. The existen
ce of infinitely many vanishing $L$-functions is in contrast with the situ
ation over the rational numbers\, where a conjecture of Chowla predicts th
ere should be no such. Towards Chowla's conjecture\, for each fixed $q$\,
we present an explicit upper bound on the number of such quadratic charact
ers 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 interesting qu
estions related to this problem. Some results in this talk are from projec
ts joint with Ravi Donepudi\, Jordan Ellenberg and Mark Shusterman.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anne-Maria Ernvall-Hytönen (University of Helsinki)
DTSTART;VALUE=DATE-TIME:20230308T200000Z
DTEND;VALUE=DATE-TIME:20230308T210000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/8
DESCRIPTION:Title: Euler's divergent series and primes in arithmetic progressions<
/a>\nby Anne-Maria Ernvall-Hytönen (University of Helsinki) as part of CR
G Weekly Seminars\n\n\nAbstract\nEuler's divergent series $\\sum_{n = 0}^\
\infty n! z^n$ which converges only for $z = 0$ becomes an interesting obj
ect when evaluated with respect to a $p$-adic norm (which will be introduc
ed in the talk). Very little is known about the values of the series. For
example\, it is an open question whether the value at one is irrational (o
r even non-zero). As individual values are difficult to reach\, it makes s
ense to try to say something about collections of values over sufficiently
large sets of primes. This leads to looking at primes in arithmetic progr
essions\, which is in turn raises a need for an explicit bound for the num
ber of primes in an arithmetic progression under the generalized Riemann h
ypothesis.\n\nDuring the talk\, I will speak about both sides of the story
: why we needed good explicit bounds for the number of primes in arithmeti
c progressions while working with questions about irrationality\, and how
we then proved such a bound.\n\nThe talk is joint work with Tapani Matala-
aho\, Neea Palojärvi and Louna Seppälä. (Questions about irrationality
with T. M. and L. S. and primes in arithmetic progressions with N. P.)\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jyothsnaa Sivaraman (Chennai Mathematical Institute)
DTSTART;VALUE=DATE-TIME:20230315T190000Z
DTEND;VALUE=DATE-TIME:20230315T200000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/9
DESCRIPTION:Title: Products of primes in ray classes\nby Jyothsnaa Sivaraman (
Chennai Mathematical Institute) as part of CRG Weekly Seminars\n\n\nAbstra
ct\nIn 1944\, Linnik showed that the least prime in an arithmetic progress
ion given by $a \\bmod q$ for $(a\,q)=1$ is at most $cq^L$ for some absolu
tely computable constants $c$ and $L$.\nA lot of work has gone in computin
g explicit bounds for $c$ and $L$. The best known bound is due to Xylouris
(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 Walker gave
a completely explicit result if one prime is replaced by a product of pri
mes. They showed that each co-prime class modulo $q$ contains a product of
three primes each less than $q^{16/3}$. This was improved by Ramar$\\acut
e{\\text{e}}$\, 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 Ramar$\\
acute{\\text{e}}$.\n
LOCATION:https://researchseminars.org/talk/crgseminarwinter23/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olga Balkanova (Steklov Mathematical Institute)
DTSTART;VALUE=DATE-TIME:20230405T190000Z
DTEND;VALUE=DATE-TIME:20230405T200000Z
DTSTAMP;VALUE=DATE-TIME:20240624T072454Z
UID:crgseminarwinter23/10
DESCRIPTION:Title: The second moment of symmetric square $L$-functions over Gauss
ian integers\nby Olga Balkanova (Steklov Mathematical Institute) as pa
rt of CRG Weekly Seminars\n\n\nAbstract\nWe prove an explicit formula for
the first moment of Maass form symmetric square $L$-functions defined 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://researchseminars.org/talk/crgseminarwinter23/10/
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