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BEGIN:VEVENT
SUMMARY:Chiara Bellotti (University of New South Wales\, Canberra)
DTSTART:20240117T000000Z
DTEND:20240117T010000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/1/">Explicit bounds for $\\zeta$ and a new zero-free region<
 /a>\nby Chiara Bellotti (University of New South Wales\, Canberra) as part
  of CRG Weekly Seminars\n\n\nAbstract\nIn this talk we prove that $|\\zeta
 (\\sigma+it)|\\le 70.7 |t|^{4.438(1-\\sigma)^{3/2}}\\log^{2/3}|t|$ for $1/
 2\\le\\sigma\\le 1$ and $|t|\\ge 3$\, combining new explicit bounds for th
 e Vinogradov integral with exponential sums estimates. As a consequence\, 
 we improve the explicit zero-free region for $\\zeta(s)$\, showing that $\
 \zeta(\\sigma+it)$ has no zeros in the region $\\sigma \\geq 1-1 /\\left(5
 3.989(\\log |t|)^{2 / 3}(\\log \\log |t|)^{1 / 3}\\right)$ for $|t| \\geq 
 3$.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winston Heap (Norwegian University of Science and Technology)
DTSTART:20240122T190000Z
DTEND:20240122T200000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/2/">Mean values of long Dirichlet polynomials</a>\nby Winsto
 n Heap (Norwegian University of Science and Technology) as part of CRG Wee
 kly Seminars\n\n\nAbstract\nWe discuss the role of long Dirichlet polynomi
 als in number theory. We first survey some applications of mean values of 
 long Dirichlet polynomials over primes in the theory of the Riemann zeta f
 unction which includes central limit theorems and pair correlation of zero
 s. We then give some examples showing how\, on assuming the Riemann Hypoth
 esis\, one can compute asymptotics for such mean values without using the 
 Hardy-Littlewood conjectures for additive correlations of the von-Mangoldt
  functions.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emily Quesada-Herrera (Graz University of Technology)
DTSTART:20240129T190000Z
DTEND:20240129T200000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/3/">Fourier optimization and the least quadratic non-residue
 </a>\nby Emily Quesada-Herrera (Graz University of Technology) as part of 
 CRG Weekly Seminars\n\n\nAbstract\nWe will explore how a Fourier optimizat
 ion framework may be used to study two classical problems in number theory
  involving Dirichlet characters: The problem of estimating the least chara
 cter non-residue\; and the problem of estimating the least prime in an ari
 thmetic progression. In particular\, we show how this Fourier framework le
 ads to subtle\, but conceptually interesting\, improvements on the best cu
 rrent asymptotic bounds under the Generalized Riemann Hypothesis\, given b
 y Lamzouri\, Li\, and Soundararajan. Based on joint work with Emanuel Carn
 eiro\, Micah Milinovich\, and Antonio Ramos.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivian Kuperberg (ETH Zurich)
DTSTART:20240212T190000Z
DTEND:20240212T200000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/4/">Consecutive sums of two squares in arithmetic progressio
 ns</a>\nby Vivian Kuperberg (ETH Zurich) as part of CRG Weekly Seminars\n\
 n\nAbstract\nIn 2000\, Shiu proved that there are infinitely many primes w
 hose last digit is 1 such that the next prime also ends in a 1. However\, 
 it is an open problem to show that there are infinitely many primes ending
  in 1 such that the next prime ends in 3. In this talk\, we'll instead con
 sider the sequence of sums of two squares in increasing order. In particul
 ar\, we'll show that there are infinitely many sums of two squares ending 
 in 1 such that the next sum of two squares ends in 3. We'll show further t
 hat all patterns of length 3 occur infinitely often: for any modulus q\, e
 very sequence (a mod q\, b mod q\, c mod q) appears infinitely often among
  consecutive sums of two squares. We'll discuss some of the proof techniqu
 es\, and explain why they fail for primes. Joint work with Noam Kimmel.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bittu (Indraprastha Institute of Information Technology\, Delhi)
DTSTART:20240226T190000Z
DTEND:20240226T200000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/5
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/5/">Spacing statistics of the Farey sequence</a>\nby Bittu (
 Indraprastha Institute of Information Technology\, Delhi) as part of CRG W
 eekly Seminars\n\n\nAbstract\nThe Farey sequence $\\mathcal{F}_Q$ of order
  $Q$ is an ascending sequence of fractions $a/b$ in the unit interval $(0\
 ,1]$ such that $(a\,b)=1$ and $0< a \\leq b \\leq Q $. The study of the Fa
 rey fractions is of major interest because of their role in problems relat
 ed to the Diophantine approximation. Also\, there is a connection between 
 the distribution of Farey fractions and the Riemann hypothesis\, which mot
 ivates their study. In this talk\, we will discuss the distribution of Far
 ey fractions with some divisibility constraints on denominators by studyin
 g their pair correlation measure. This is based on the joint work with Sne
 ha Chaubey.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Quanli Shen (Shandong University\, Weihai)
DTSTART:20240318T180000Z
DTEND:20240318T190000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/6/">The fourth moment of quadratic Dirichlet $L$-functions</
 a>\nby Quanli Shen (Shandong University\, Weihai) as part of CRG Weekly Se
 minars\n\n\nAbstract\nI will discuss the fourth moment of quadratic Dirich
 let $L$-functions where we prove an asymptotic formula with four main term
 s unconditionally. Previously the asymptotic formula was established with 
 the leading main term under generalized Riemann hypothesis. This work is b
 ased on Li's recent work on the second moment of quadratic twists of modul
 ar $L$-functions. It is joint work with Joshua Stucky.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Greg Knapp (University of Calgary)
DTSTART:20240410T180000Z
DTEND:20240410T190000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/7/">Bounds on the Number of Solutions to Thue Equations</a>\
 nby Greg Knapp (University of Calgary) as part of CRG Weekly Seminars\n\n\
 nAbstract\nIn 1909\, Thue proved that when $F(x\,y)$ is an irreducible\, h
 omogeneous\, polynomial with integer coefficients and degree at least $3$\
 , the inequality $|F(x\,y)| \\leq h$ has finitely many integer-pair soluti
 ons 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 T
 hue’s Inequality\, with Mueller and Schmidt initiating the modern approa
 ch to this problem in the 1980s.  In this talk\, I will describe different
  techniques used by Akhtari and Bengoechea\; Baker\; Mueller and Schmidt\;
  Saradha and Sharma\; and Thomas to make progress on this general problem.
   After that\, I will discuss some improvements that can be made to a coun
 ting technique used in association with “the gap principle” and how th
 ose improvements lead to better bounds on the number of solutions to Thue
 ’s Inequality.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Stadlmann (University of Oxford)
DTSTART:20240304T190000Z
DTEND:20240304T200000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/9
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/9/">Primes in arithmetic progressions to smooth moduli</a>\n
 by Julia Stadlmann (University of Oxford) as part of CRG Weekly Seminars\n
 \n\nAbstract\nThe twin prime conjecture asserts that there are infinitely 
 many primes p for which p+2 is also prime. This conjecture appears far out
  of reach of current mathematical techniques. However\, in 2013 Zhang achi
 eved a breakthrough\, showing that there exists some positive integer h fo
 r which p and p+h are both prime infinitely often. Equidistribution estima
 tes for primes in arithmetic progressions to smooth moduli were a key ingr
 edient of his work. In this talk\, I will sketch what role these estimates
  play in proofs of bounded gaps between primes. I will also show how a ref
 inement of the q-van der Corput method can be used to improve on equidistr
 ibution estimates of the Polymath project for primes in APs to smooth modu
 li.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:(Talk cancelled) Sneha Chaubey (Indraprastha Institute of Informat
 ion Technology\, Delhi)
DTSTART:20240311T180000Z
DTEND:20240311T190000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/10
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/10/">(Talk cancelled) Distribution of spacings of real-value
 d sequences</a>\nby (Talk cancelled) Sneha Chaubey (Indraprastha Institute
  of Information Technology\, Delhi) as part of CRG Weekly Seminars\n\n\nAb
 stract\nThe topic on the distribution of sequences saw its light with the 
 seminal paper of Weyl. While the classical notion of equidistribution modu
 lo one addresses the “global” behaviour of the fractional parts of a s
 equence\, quantities such as $k$-point correlations and nearest neighbour 
 gap distributions are useful in investigating the sequence on finer scales
 . In this talk\, we discuss these fine-scale statistics for real-valued ar
 ithmetic sequences\, and show that the limiting distribution of the neares
 t neighbour gaps of real-valued lacunary sequences is Poissonian. We also 
 prove the Poissonian behavior of the $2$-point correlation function for ce
 rtain classes of real-valued vector sequences. This is achieved by extrapo
 lating conditions on the number of solutions of Diophantine inequalities u
 sing twisted moments of the Riemann zeta function.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jérémy Dousselin (Institut Élie Cartan de Lorraine\, Nancy)
DTSTART:20240325T180000Z
DTEND:20240325T190000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/11
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/11/">Zeros of linear combinations of Dirichlet $L$-functions
  on the critical line</a>\nby Jérémy Dousselin (Institut Élie Cartan de
  Lorraine\, Nancy) as part of CRG Weekly Seminars\n\n\nAbstract\nFix $N\\g
 eq 1$ and let $L_1$\, $L_2$\, ...\, $L_N$ be Dirichlet $L$-functions with 
 distinct\, primitive and even Dirichlet characters. We assume that these f
 unctions satisfy the same functional equation. Let\n\\[F(s):=\\sum_{j=1}^N
  c_jL_j(s)\\]\nbe a linear combination of these functions ($c_j\\in\\mathb
 b{R}^*$ are distinct).$F$ is known to have two kinds of zeros: trivial one
 s\, and non-trivial zeros which are confined in a vertical strip. We denot
 e the number of non-trivial zeros $\\rho$ with $\\Im(\\rho)\\leq T$ by $N(
 T)$\, and we let $N_0(T)$ be the number of these zeros that are on the cri
 tical line.At the end of the 90s\, Selberg proved that this linear combina
 tion had 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 \\fr
 ac c{N^2}\\]\nfor some $c>0$.Our goal is to provide an explicit value for 
 $c$\, and also to improve the lower bound above by showing that\n\\[\\kapp
 a_F\\geq \\frac{2.16\\times 10^{-6}}{N\\log N}\,\\]\nfor any large enough 
 $N$.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugenia Rosu (Mathematical Institute\, Universiteit Leiden)
DTSTART:20240205T190000Z
DTEND:20240205T200000Z
DTSTAMP:20260404T111443Z
UID:crgseminarwinter24/13
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/crgse
 minarwinter24/13/">A higher degree Weierstrass function</a>\nby Eugenia Ro
 su (Mathematical Institute\, Universiteit Leiden) as part of CRG Weekly Se
 minars\n\n\nAbstract\nThe Weierstrass $\\wp$ function plays a great role i
 n the classic theory of complex elliptic curves. A related function\, the 
 Weierstrass zeta-function\, is used by Guerzhoy to construct preimages und
 er the $\\xi$-operator of newforms of weight 2\, corresponding to elliptic
  curves. In this talk\, I will discuss a generalization of the Weierstrass
  zeta-function and an application to harmonic Maass forms. More precisely\
 , I will describe a construction of a preimage of the $\\xi$-operator of a
  newform of weight k for k>2. This is based on joint work with C. Alfes-Ne
 umann\, J. Funke and M. Mertens.\n
LOCATION:https://stable.researchseminars.org/talk/crgseminarwinter24/13/
END:VEVENT
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