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SUMMARY:Akshaa Vatwani (Indian Institute of Technology Gandhinagar)
DTSTART:20220915T170000Z
DTEND:20220915T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/1
 /">Joint extreme values of $L$-functions</a>\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:20220922T170000Z
DTEND:20220922T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/2
 /">Joint value distribution of $L$-functions</a>\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:20220929T170000Z
DTEND:20220929T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/3
 /">Races of irreducible monic polynomials in function fields</a>\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:20221006T170000Z
DTEND:20221006T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/4
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/4
 /">Multiplicative functions in short intervals</a>\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:20221020T170000Z
DTEND:20221020T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/5
 /">Moments and Periods for $GL(3)$</a>\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:20221027T170000Z
DTEND:20221027T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/6
 /">Uniform distribution and geometric incidence theory</a>\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:20221103T170000Z
DTEND:20221103T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/7
 /">Quadratic twists of modular $L$-functions</a>\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:20221117T150000Z
DTEND:20221117T160000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/9
 /">Vorono$\\ddot{\\textrm{\\i}}$ summation formula for the generalized div
 isor function $\\sigma_z^{(k)}(n)$</a>\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:20221124T180000Z
DTEND:20221124T190000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/1
 0/">On non-Archimedean analogue of a question of Atkin and Serre</a>\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:20221201T180000Z
DTEND:20221201T190000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/1
 1/">The value distribution of the Hurwitz zeta function with an irrational
  shift</a>\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:20221013T170000Z
DTEND:20221013T180000Z
DTSTAMP:20260422T213016Z
UID:crgseminar/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/crgseminar/1
 2/">Extreme values of the Riemann zeta and Dirichlet $L$-functions at crit
 ical points</a>\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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