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SUMMARY:Akshaa Vatwani (Indian Institute of Technology Gandhinagar)
DTSTART;VALUE=DATE-TIME:20220915T170000Z
DTEND;VALUE=DATE-TIME:20220915T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/1
DESCRIPTION:Title: Joint extreme values of $L$-functions\nby Akshaa Vatwani (Indian In
stitute of Technology Gandhinagar) as part of CRG Weekly Seminars\n\n\nAbs
tract\nWe consider $L$-functions $L_1\,\\ldots\,L_k$ from the Selberg cla
ss having polynomial Euler product and satisfying Selberg's orthonormality
condition. We show that on every vertical line $s=\\sigma+it$ in the comp
lex plane with $\\sigma \\in(1/2\,1)$\, these $L$-functions simultaneously
take "large" values inside a small neighborhood. \nOur method extends to
$\\sigma=1$ unconditionally\, and to $\\sigma =1/2$ on the generalized Rie
mann hypothesis. We also obtain similar joint omega results for arguments
of the given $L$-functions. \nThis is joint work with Kamalakshya Mahatab
and Łukasz Pańkowski.\n
LOCATION:https://researchseminars.org/talk/crgseminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Junxian Li (Mathematisches Institut der Universität Bonn)
DTSTART;VALUE=DATE-TIME:20220922T170000Z
DTEND;VALUE=DATE-TIME:20220922T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/2
DESCRIPTION:Title: Joint value distribution of $L$-functions\nby Junxian Li (Mathemati
sches Institut der Universität Bonn) as part of CRG Weekly Seminars\n\n\n
Abstract\nIt is believed that distinct primitive $L$-functions are “stat
istically independent”. The independence can be interpreted in many diff
erent ways. We are interested in the joint value distributions and their a
pplications in moments and extreme values for distinct $L$-functions. We d
iscuss some large deviation estimates in Selberg and Bombieri-Hejhal’s c
entral limit theorem for values of several $L$-functions. On the critical
line\, values of distinct primitive $L$-functions behave independently in
a strong sense. However\, away from the critical line\, values of distinct
Dirichlet $L$-functions begin to exhibit some correlations.\n\nThis is ba
sed on joint works with Shota Inoue.\n
LOCATION:https://researchseminars.org/talk/crgseminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Youssef Sedrati (Institut Élie Cartan de Lorraine\, Nancy)
DTSTART;VALUE=DATE-TIME:20220929T170000Z
DTEND;VALUE=DATE-TIME:20220929T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/3
DESCRIPTION:Title: Races of irreducible monic polynomials in function fields\nby Youss
ef Sedrati (Institut Élie Cartan de Lorraine\, Nancy) as part of CRG Week
ly Seminars\n\n\nAbstract\nChebyshev noticed in 1853 that there is a predo
minance\, for “most” real numbers $x ≥ 2$\, of the number of primes
$≤ x$ and congruent to $3$ modulo $4$ over primes $≤ x$ and congruent
to $1$ modulo $4$. Since then\, several generalizations of this phenomenon
have been studied\, notably in the case of prime number races with three
or more competitors by Y. Lamzouri. In this talk\, I will present results
related to the generalization of Y. Lamzouri’s work in the context of po
lynomial rings over finite fields. I will also discuss results concerning
races of irreducible monic polynomials involving two competitors. In parti
cular\, I will give examples where the races in the function field setting
behave differently than in the number field setting.\n
LOCATION:https://researchseminars.org/talk/crgseminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pranendu Darbar (The Norwegian University of Science and Technolog
y)
DTSTART;VALUE=DATE-TIME:20221006T170000Z
DTEND;VALUE=DATE-TIME:20221006T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/4
DESCRIPTION:Title: Multiplicative functions in short intervals\nby Pranendu Darbar (Th
e Norwegian University of Science and Technology) as part of CRG Weekly Se
minars\n\n\nAbstract\nIn this talk\, we are interested in a general class
of multiplicative functions. For a function that belongs to this class\, w
e will relate \nits “short average” to its “long average”. More pr
ecisely\, we will compute the variance of such a function over short inter
vals by using Fourier analysis and by counting rational points on certain
binary forms.\n\nThe discussion is applicable to some interesting multipli
cative functions such as \n\\[\n\\mu_k(n)\, \\\, \\\, \\frac{\\phi(n)}{n}
\, \\\, \\\, \\frac{n}{\\phi(n)}\, \\\, \\\, \\mu^2(n)\\frac{\\phi(n)}{n}
\, \\\,\\\, \\sigma_{\\alpha}(n)\, \\\,\\\,\n (-1)^{\\#\\{p\\\,: \\\, p^k|
n\\}}(n)\,\n\\]\nand many others and it provides various new results and i
mprovements to the previous result in the literature. This is a joint work
with Mithun Kumar Das.\n
LOCATION:https://researchseminars.org/talk/crgseminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chung-Hang (Kevin) Kwan (University College London)
DTSTART;VALUE=DATE-TIME:20221020T170000Z
DTEND;VALUE=DATE-TIME:20221020T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/5
DESCRIPTION:Title: Moments and Periods for $GL(3)$\nby Chung-Hang (Kevin) Kwan (Univer
sity College London) as part of CRG Weekly Seminars\n\n\nAbstract\nIn the
past century\, the studies of moments of $L$-functions have been important
in number\ntheory and are well-motivated by a variety of arithmetic appli
cations.\n\nThis talk will begin with two problems in elementary number th
eory\, followed by a survey of\ntechniques in the past and the present. We
will slowly move towards the perspectives of period\nintegrals which will
be used to illustrate the interesting structures behind moments. In parti
cular\,\nwe shall focus on the “Motohashi phenomena”.\n
LOCATION:https://researchseminars.org/talk/crgseminar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayla Gafni (University of Mississippi)
DTSTART;VALUE=DATE-TIME:20221027T170000Z
DTEND;VALUE=DATE-TIME:20221027T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/6
DESCRIPTION:Title: Uniform distribution and geometric incidence theory\nby Ayla Gafni
(University of Mississippi) as part of CRG Weekly Seminars\n\n\nAbstract\n
The Szemeredi-Trotter Incidence Theorem\, a central result in geometric co
mbinatorics\, bounds the number of incidences between n points and m lines
in the Euclidean plane. Replacing lines with circles leads to the unit di
stance problem\, which asks how many pairs of points in a planar set of n
points can be at a unit distance. The unit distance problem breaks down in
dimensions $4$ and higher due to degenerate configurations that attain th
e trivial bound. However\, nontrivial results are possible under certain s
tructural assumptions about the point set. In this talk\, we will give an
overview of the history of these and other incidence results. Then we will
introduce a quantitative notion of uniform distribution and use that prop
erty to obtain nontrivial bounds on unit distances and point-hyperplane in
cidences in higher-dimensional Euclidean space. This is based on joint wor
k with Alex Iosevich and Emmett Wyman.\n
LOCATION:https://researchseminars.org/talk/crgseminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiannan Li (Kansas State University)
DTSTART;VALUE=DATE-TIME:20221103T170000Z
DTEND;VALUE=DATE-TIME:20221103T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/7
DESCRIPTION:Title: Quadratic twists of modular $L$-functions\nby Xiannan Li (Kansas St
ate University) as part of CRG Weekly Seminars\n\n\nAbstract\nThe behavior
of quadratic twists of modular $L$-functions at the critical point is rel
ated both to coefficients of half integer weight modular forms and data on
elliptic curves. Here we describe a proof of an asymptotic for the second
moment of this family of $L$-functions\, previously available conditional
ly on the Generalized Riemann Hypothesis by the work of Soundararajan and
Young. Our proof depends on deriving an optimal large sieve type bound.\n
LOCATION:https://researchseminars.org/talk/crgseminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Atul Dixit (Indian Institute of Technology Gandhinagar)
DTSTART;VALUE=DATE-TIME:20221117T150000Z
DTEND;VALUE=DATE-TIME:20221117T160000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/9
DESCRIPTION:Title: Vorono$\\ddot{\\textrm{\\i}}$ summation formula for the generalized div
isor function $\\sigma_z^{(k)}(n)$\nby Atul Dixit (Indian Institute of
Technology Gandhinagar) as part of CRG Weekly Seminars\n\n\nAbstract\nFor
a fixed $z\\in \\mathbb C$ and a fixed $k\\in \\mathbb N$\, let $\\sigma_
z^{(k)}(n)$ denote the sum of $z$-th powers of those divisors $d$ of $n$ w
hose $k$-th powers also divide $n$. This arithmetic function is a simultan
eous generalization of the well-known divisor function $\\sigma_z(n)$ as w
ell as a divisor function $d^{(k)}(n)$ first studied by Wigert. A Vorono$\
\ddot{\\textrm{\\i}}$ summation formula is obtained for $\\sigma_z^{(k)}(n
)$. An interesting thing to note here is that this arithmetic function doe
s not fall under the purview of the setting of the Hecke functional functi
on with multiple gamma factors studied by Chandrasekharan and Narasimhan.
Some applications of the Vorono$\\ddot{\\textrm{\\i}}$ summation formula
will be given. This is joint work with Bibekananda Maji and Akshaa Vatwani
.\n
LOCATION:https://researchseminars.org/talk/crgseminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanoli Gun (The Institute of Mathematical Sciences)
DTSTART;VALUE=DATE-TIME:20221124T180000Z
DTEND;VALUE=DATE-TIME:20221124T190000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/10
DESCRIPTION:Title: On non-Archimedean analogue of a question of Atkin and Serre\nby S
anoli Gun (The Institute of Mathematical Sciences) as part of CRG Weekly S
eminars\n\n\nAbstract\nLet $\\tau$ be the Ramanujan tau function.\nIt is a
well known question of Atkin and Serre that for any\n$\\epsilon > 0$\, th
ere exists a constant $c(\\epsilon) >0$\nsuch that $|\\tau(p)| \\ge c(\\ep
silon) p^{(k-3)/2 - \\epsilon}$.\nIn this talk\, we will address a non-Arc
himedean\nanalogue of this question which improves the recent\nbound of Be
nnett\, Gherga\, Patel and Siksek.\nThis is a report on a joint work with
Yuri Bilu and Sunil Naik.\n
LOCATION:https://researchseminars.org/talk/crgseminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anurag Sahay (University of Rochester)
DTSTART;VALUE=DATE-TIME:20221201T180000Z
DTEND;VALUE=DATE-TIME:20221201T190000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/11
DESCRIPTION:Title: The value distribution of the Hurwitz zeta function with an irrational
shift\nby Anurag Sahay (University of Rochester) as part of CRG Weekl
y Seminars\n\n\nAbstract\nThe Hurwitz zeta function $\\zeta(s\,\\alpha)$ i
s a shifted integer analogue of the Riemann zeta function which shares man
y of its properties\, but is not an "$L$-function" under any reasonable de
finition of the word. We will first review the basics of the value distrib
ution of the Riemann zeta function in the critical strip (moments\, Bohr--
Jessen theory...) and then contrast it with the value distribution of the
Hurwitz zeta function.\n\nOur focus will be on shift parameters $\\alpha \
\notin \\mathbb{Q}$\, i.e.\, algebraic irrational or transcendental. We wi
ll present a new result (joint with Winston Heap) on moments of these obje
cts on the critical line.\n
LOCATION:https://researchseminars.org/talk/crgseminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shashank Chorge (University of Rochester)
DTSTART;VALUE=DATE-TIME:20221013T170000Z
DTEND;VALUE=DATE-TIME:20221013T180000Z
DTSTAMP;VALUE=DATE-TIME:20230921T164450Z
UID:crgseminar/12
DESCRIPTION:Title: Extreme values of the Riemann zeta and Dirichlet $L$-functions at crit
ical points\nby Shashank Chorge (University of Rochester) as part of C
RG Weekly Seminars\n\n\nAbstract\nWe compute extreme values of the Riemann
zeta function at the critical\npoints of the zeta function in the critica
l strip. i.e. the points where $\\zeta'(s) = 0$ and $\\Re s < 1$. We show
that the values taken by the zeta function at these points\nare very simil
ar to the extreme values taken without any restrictions. We will\nshow geo
metric significance of such points.\n\nWe also compute extreme values of D
irichlet $L$-functions at the critical points of the zeta function to the
right of $\\Re s = 1$. It shows statistical independence of $L$-functions
and zeta function in a certain way as these values are very similar to the
values taken by $L$-functions without any restriction.\n
LOCATION:https://researchseminars.org/talk/crgseminar/12/
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