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BEGIN:VEVENT
SUMMARY:Caroline Turnage-Butterbaugh (Carleton College)
DTSTART;VALUE=DATE-TIME:20220922T160000Z
DTEND;VALUE=DATE-TIME:20220922T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/1
DESCRIPTION:Title: Moments of Dirichlet L-functions\nby Caroline Turnage-Butterbau
gh (Carleton College) as part of FRG Grad Seminar (Averages of of L-functi
ons and Arithmetic Stratification)\n\n\nAbstract\nIn recent decades there
has been much interest and measured progress in the study of moments of L-
functions. Despite a great deal of effort spanning over a century\, asympt
otic formulas for moments of L-functions remain stubbornly out of reach in
all but a few cases. I will begin this talk by reviewing what is known fo
r moments of the Riemann zeta-function on the critical line\, and we will
then consider the problem for the family of all Dirichlet L-functions of e
ven primitive characters of bounded conductor. A heuristic of Conrey\, Far
mer\, Keating\, Rubenstein\, and Snaith gives a precise prediction for the
asymptotic formula for the general 2kth moment of this family. I will out
line how to harness the asymptotic large sieve to prove an asymptotic form
ula for the general 2kth moment of approximations of this family. The resu
lt\, which assumes the generalized Lindelöf hypothesis for large values o
f k\, agrees with the prediction of CFKRS. Moreover\, it provides the firs
t rigorous evidence beyond the so-called “diagonal terms” in their con
jectured asymptotic formula for this family of L-functions. This is joint
work with Siegfred Baluyot.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hua Lin (UC Irvine)
DTSTART;VALUE=DATE-TIME:20220926T160000Z
DTEND;VALUE=DATE-TIME:20220926T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/3
DESCRIPTION:Title: One-level density of zeros of Dirichlet L-function over function fi
elds\nby Hua Lin (UC Irvine) as part of FRG Grad Seminar (Averages of
of L-functions and Arithmetic Stratification)\n\n\nAbstract\nFor this talk
\, we compute the one-level density of zeros of cubic and quartic Dirichle
t $L$-functions over function fields $\\mathbb{F}_q[t]$ in the Kummer sett
ing ($q\\equiv1\\pmod{\\ell}$) and for order $\\ell=3\,4\,6$ in the non-Ku
mmer setting ($q\\not\\equiv1\\pmod{\\ell}$). In each case\, we obtain a m
ain term predicted by Random Matrix Theory (RMT) and a lower order term no
t predicted by RMT. We also confirm the symmetry type of the family is uni
tary\, supporting the Katz and Sarnak philosophy. I will first talk about
some history and background on the subject\, make the analogy and describe
the primitive characters over function fields in each setting\, and then
show the computation in more detail.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louis Gaudet (Rutgers University)
DTSTART;VALUE=DATE-TIME:20221003T160000Z
DTEND;VALUE=DATE-TIME:20221003T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/4
DESCRIPTION:Title: The least Euler prime via a sieve approach\nby Louis Gaudet (Ru
tgers University) as part of FRG Grad Seminar (Averages of of L-functions
and Arithmetic Stratification)\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lasse Grimmelt (University of Oxford)
DTSTART;VALUE=DATE-TIME:20221010T160000Z
DTEND;VALUE=DATE-TIME:20221010T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/5
DESCRIPTION:Title: Primes in large arithmetic progressions and applications to additiv
e problems\nby Lasse Grimmelt (University of Oxford) as part of FRG Gr
ad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\
nAbstract\nResults on the distribution of primes up to $X$ in an arithmeti
c progression with modulus $q$ fall\, depending on the relative size of $q
$ and $X$\, roughly speaking into three categories. For small $q$ (say up
to a power of $\\log X$)\, multiplicative analytic methods in the form of
Dirichlet L-functions are used\, in the medium range ($q < N^{1/2-\\epsilo
n}$) the large sieve gives us the Bombieri-Vinogradov Theorem\, and finall
y one can handle slightly larger $q$ by bounds for sums of Kloosterman sum
s. In this talk I will give a background about these results and highlight
some recent progress in the third category. I will also explain how this
progress can be applied to additive problems involving (subsets of) the pr
imes.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Goldston (San José State University)
DTSTART;VALUE=DATE-TIME:20221017T160000Z
DTEND;VALUE=DATE-TIME:20221017T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/6
DESCRIPTION:Title: Small Gaps and Spacings between Riemann zeta-function zeros\nby
Dan Goldston (San José State University) as part of FRG Grad Seminar (Av
erages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nI w
ill discuss joint work with Hung Bui\, Micah Milinovich\, and Hugh Montgom
ery on differences between consecutive zeros of the Riemann zeta-function
that are smaller than the average spacing between zeros. We assume the Rie
mann Hypothesis. One result is that by using the pair correlation method o
ne can prove there is a positive proportion of consecutive zeros closer th
an 0.6039 times the average spacing. One limitation of this method is that
these close pairs of zeros could all be multiple zeros\, and thus the met
hod may not be finding any small gaps between zeros at all - here we requi
re a gap between two numbers to have non-zero length because that is what
a gap is. We refer to differences between consecutive zeros including diff
erences equal to zero as “spacings”. There are three methods known to
deal with close zeros\, and all three actually produce small spacings betw
een zeros rather than small gaps. (One method is unconditional\, the other
two assume RH.) For small gaps\, or differences between distinct zeros\,
the three methods only produce gaps larger than the average spacing. Our s
econd result is based on a new fourth method that on RH proves there are s
mall gaps between zeros closer than 0.991 times the average spacing betwee
n zeros. The method however does not produce a positive proportion of such
gaps\, and I believe proving this on RH for a positive proportion is a di
fficult problem.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:George Dickinson (University of Manchester)
DTSTART;VALUE=DATE-TIME:20221219T170000Z
DTEND;VALUE=DATE-TIME:20221219T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/7
DESCRIPTION:Title: Second moments of Dirichlet L-functions\nby George Dickinson (U
niversity of Manchester) as part of FRG Grad Seminar (Averages of of L-fun
ctions and Arithmetic Stratification)\n\n\nAbstract\nThe asymptotic formul
ae for moments of L-functions are well studied objects in analytic number
theory as they are useful tools when investigating the L-functions themsel
ves. Often especially useful are the moments that have been twisted by a D
irichlet polynomial\, and the longer the twist the better. However\, findi
ng formulae gets more difficult as the length increases. In this talk\, we
will compare methods for finding different types of twisted second moment
s of Dirichlet L-functions\, as well as looking at some of their applicati
ons.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emma Bailey (CUNY)
DTSTART;VALUE=DATE-TIME:20221031T160000Z
DTEND;VALUE=DATE-TIME:20221031T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/8
DESCRIPTION:Title: Large values of $\\zeta$ on the critical line\nby Emma Bailey (
CUNY) as part of FRG Grad Seminar (Averages of of L-functions and Arithmet
ic Stratification)\n\n\nAbstract\nSelberg’s central limit theorem tells
us that typically $|\\zeta(1/2 + it)|$ is of size $\\exp(\\sqrt{\\log \\lo
g T})$ for $t\\in [T\, 2T]$. One can ask about /atypical/ values\, or abou
t large deviations to Selberg’s central limit theorem. By exploring a co
nnection between $\\zeta$ and branching random walks\, we are able to show
that the Gaussian tail extends to the right\, on the scale of the varianc
e. In this talk I will focus on the connection to branching random walks a
nd show how this probabilistic interpretation allows us to understand larg
e values of zeta. This is based on joint work with Louis-Pierre Arguin.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daodao Yang (TU Graz)
DTSTART;VALUE=DATE-TIME:20221107T170000Z
DTEND;VALUE=DATE-TIME:20221107T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/9
DESCRIPTION:Title: Large values of derivatives of the Riemann zeta function and relate
d problems\nby Daodao Yang (TU Graz) as part of FRG Grad Seminar (Aver
ages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nLarge
values of the Riemann zeta function and L-functions are classical topics
in analytic number theory\, which can be dated back to a result of Bohr an
d Landau in 1910. Resonance methods are modern tools to produce large valu
es of zeta and L-functions. GCD sums are one of important ingredients\, wh
ich naturally appears in a Diophantine approximation problem considered by
Hardy and Littlewood in 1922. I will talk on producing large values of de
rivatives of zeta and L-functions via resonance methods. On the other hand
\, I will talk on conditional upper bounds and asymptotic formulas when as
suming RH (GRH) and a conjecture of Granville-Soundararajan on character s
ums. If time permits\, the log-type GCD sums and related spectral norms wi
ll be discussed.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jakob Streipel (University of Maine)
DTSTART;VALUE=DATE-TIME:20221114T160000Z
DTEND;VALUE=DATE-TIME:20221114T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/10
DESCRIPTION:Title: Using second moments to count zeros\nby Jakob Streipel (Univer
sity of Maine) as part of FRG Grad Seminar (Averages of of L-functions and
Arithmetic Stratification)\n\n\nAbstract\nUsing Selberg's somewhat strang
e looking version of the argument principle\, it is possible to count zero
s of families of L-functions using upper bounds on second moments. We will
explore this argument principle\, how one uses it\, and some applications
of it to various zero counting problems\, old and new.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Dunn (Caltech)
DTSTART;VALUE=DATE-TIME:20221121T170000Z
DTEND;VALUE=DATE-TIME:20221121T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/11
DESCRIPTION:Title: Bias in cubic Gauss sums: Patterson's conjecture\nby Alexander
Dunn (Caltech) as part of FRG Grad Seminar (Averages of of L-functions an
d Arithmetic Stratification)\n\n\nAbstract\nWe prove\, in this joint work
with Maksym Radziwill\, a 1978 conjecture of S. Patterson (conditional on
the Generalised Riemann hypothesis) concerning the bias of cubic Gauss su
ms. This explains a well-known numerical bias in the distribution of cubic
Gauss sums first observed by Kummer in 1846.\n\nOne important byproduct
of our proof is that we show Heath-Brown's cubic large sieve is sharp unde
r GRH. This disproves the popular belief that the cubic large sieve can b
e improved.\n\nAn important ingredient in our proof is a dispersion estima
te for cubic Gauss sums. It can be interpreted as a cubic large sieve wit
h correction by a non-trivial asymptotic main term.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aled Walker (King's College\, London)
DTSTART;VALUE=DATE-TIME:20221128T170000Z
DTEND;VALUE=DATE-TIME:20221128T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/12
DESCRIPTION:Title: Correlations of sieve weights and distributions of zeros\nby A
led Walker (King's College\, London) as part of FRG Grad Seminar (Averages
of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn this t
alk\, we will briefly review Montgomery's pair correlation conjecture for
the zeros of the Riemann zeta function\, before discussing a (conditional
) partial lower bound on the Fourier transform of this pair correlation fu
nction: the so-called 'form factor' $F_T(x)$. The methods\, based in part
on ideas of Goldston and Gonek\, utilise some new correlation estimates fo
r Selberg sieve weights.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asif Zaman (University of Toronto)
DTSTART;VALUE=DATE-TIME:20221205T170000Z
DTEND;VALUE=DATE-TIME:20221205T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/13
DESCRIPTION:Title: Random multiplicative functions and a simplified model\nby Asi
f Zaman (University of Toronto) as part of FRG Grad Seminar (Averages of o
f L-functions and Arithmetic Stratification)\n\n\nAbstract\nOver the past
few years\, there has been a lot of interest in random multiplicative func
tions and their partial sums. This subject has many intriguing questions a
nd connections to other areas of number theory and probability. In joint w
ork with Soundararajan\, we have introduced a simplified model of partial
sums of random multiplicative functions and established a result parallel
to Harper’s breakthrough on better-than-squareroot cancellation. In this
expository talk\, I will review some of the history of random multiplicat
ive functions\, and illustrate how random multiplicative functions connect
to our simplified model.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Dobner (University of Michigan)
DTSTART;VALUE=DATE-TIME:20221212T170000Z
DTEND;VALUE=DATE-TIME:20221212T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/14
DESCRIPTION:Title: Optimization and moment methods in number theory\nby Alexander
Dobner (University of Michigan) as part of FRG Grad Seminar (Averages of
of L-functions and Arithmetic Stratification)\n\n\nAbstract\nA common tech
nique in analytic number theory is to turn a number theoretic problem into
some sort of optimization problem which is hopefully more tractable. A we
ll known example is the Selberg sieve method which turns classical sieving
problems into a quadratic optimization problem. This technique also appe
ars in conjunction with the so-called moment method from probability theor
y. In this talk I'll summarize several instances of this including finding
primes in bounded intervals\, finding small/large gaps between zeta zeros
\, and finding large values of Dirichlet series.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Ng (University of Lethbridge)
DTSTART;VALUE=DATE-TIME:20230123T170000Z
DTEND;VALUE=DATE-TIME:20230123T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/15
DESCRIPTION:Title: The eighth moment of the Riemann zeta function\nby Nathan Ng (
University of Lethbridge) as part of FRG Grad Seminar (Averages of of L-fu
nctions and Arithmetic Stratification)\n\n\nAbstract\nIn recent work (http
s://arxiv.org/abs/2204.13891)\, Quanli Shen\, Peng-Jie Wong\, and I have s
hown that the Riemann hypothesis and a conjecture for quaternary additive
divisor sums implies the conjectured asymptotic for the eighth moment of t
he Riemann zeta function. This builds on earlier work on the sixth moment
of the Riemann zeta function (Ng\, Discrete Analysis\, 2021). One key dif
ference is that sharp bounds for shifted moments of the zeta function on t
he critical line are required. In this talk\, I will discuss some of the
ideas that go into the proof.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Farmer (American Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20230130T170000Z
DTEND;VALUE=DATE-TIME:20230130T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/16
DESCRIPTION:Title: The zeta function when it is particularly large\nby David Farm
er (American Institute of Mathematics) as part of FRG Grad Seminar (Averag
es of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nWhat do
es the zeta function look like in a neighborhood of its largest values? N
obody knows for sure\, because particularly large values have never been c
omputed. We will give a plausible answer by combining theorems from analy
tic number theory\, first principles reasoning\, and examples of random ch
aracteristic polynomials.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matilde Lalín (Université de Montréal)
DTSTART;VALUE=DATE-TIME:20230417T160000Z
DTEND;VALUE=DATE-TIME:20230417T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/17
DESCRIPTION:Title: The distribution of values of cubic $L$-functions at $s=1$\nby
Matilde Lalín (Université de Montréal) as part of FRG Grad Seminar (Av
erages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nWe
investigate the distribution of values of cubic Dirichlet $L$-functions at
$s=1$. Following ideas of Granville and Soundararajan\, and Dahl and Lamz
ouri for quadratic $L$-functions\, we model values of $L(1\,\\chi)$ with t
he distribution of random Euler products $L(1\,\\mathbb{X})$ for certain f
amily of random variables $\\mathbb{X}(p)$ attached to each prime. We obta
in a description of the proportion of $|L(1\,\\chi)|$ that are larger or t
hat are smaller than a given bound\, and yield more light into the Littlew
ood bounds. Unlike the quadratic case\, there is a clear asymmetry between
lower and upper bounds for the cubic case.\n\nThis is joint work with Pra
nendu Darbar\, Chantal David\, and Allysa Lumley.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keshav Aggarwal (Alfréd Rényi Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20230206T170000Z
DTEND;VALUE=DATE-TIME:20230206T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/19
DESCRIPTION:Title: Bound for the existence of prime gap graphs\nby Keshav Aggarwa
l (Alfréd Rényi Institute of Mathematics) as part of FRG Grad Seminar (A
verages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nGi
ven a sequence $\\mathbf{D}$ of non-negative integers\, it is interesting
to know whether there exists a graph with vertices of degrees equaling the
integers in $\\mathbf{D}$. If that happens\, we say $\\mathbf{D}$ is grap
hic. Clearly\, if the sequence is graphic\, then the sum of its members mu
st be even. However\, it is not self-evident whether a given sequence is g
raphic. There are exponentially many different realizations for almost eve
ry graphic degree sequence. At the same time\, the number of all graphic d
egree sequences is infinitesimal compared to the number of integer partiti
ons of the sum of the degrees. Therefore it is incredibly hard to come up
with an interesting (or non-trivial) graphic degree sequence.\n\nLet us ca
ll a simple graph on $n>2$ vertices a prime gap graph if its vertex degree
s are $1$ and the first $n-1$ prime gaps. Recently\, Erdős-Harcos-Kharel-
Maga-Mezei-Toroczkai showed that the prime gap\nsequence is graphic for la
rge enough $n$. In a joint work with Robin Frot\, we make their work effec
tive.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rizwanur Khan (University of Mississippi)
DTSTART;VALUE=DATE-TIME:20230213T170000Z
DTEND;VALUE=DATE-TIME:20230213T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/20
DESCRIPTION:Title: The fourth moment of Dirichlet L-functions and related problems\nby Rizwanur Khan (University of Mississippi) as part of FRG Grad Semina
r (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract
\nI will discuss asymptotics for the fourth moment of Dirichlet L-function
s and related problems\, especially with regards to simplifying existing a
pproaches and sharpening the error terms in these asymptotics. This is joi
nt work with Zeyuan Zhang.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katy Woo (Princeton University)
DTSTART;VALUE=DATE-TIME:20230220T170000Z
DTEND;VALUE=DATE-TIME:20230220T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/21
DESCRIPTION:Title: Small scale distribution of primes in four-term arithmetic progres
sions\nby Katy Woo (Princeton University) as part of FRG Grad Seminar
(Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\n
In 1985\, Maier demonstrated that there are short intervals with exception
ally large or small numbers of primes. In this talk\, I will discuss adapt
ing Maier's matrix method to look at the small scale distribution of prime
s in three-term and four-term arithmetic progressions. I aim to highlight
the similarities and differences in the proofs for the two cases\; the for
mer uses the classical circle method\, whereas the latter requires tools f
rom ergodic theory. This is based on joint work with Mayank Pandey.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Whitehead (Swarthmore College)
DTSTART;VALUE=DATE-TIME:20230313T160000Z
DTEND;VALUE=DATE-TIME:20230313T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/22
DESCRIPTION:Title: Multiple Dirichlet Series and Moments of L-functions\nby Ian W
hitehead (Swarthmore College) as part of FRG Grad Seminar (Averages of of
L-functions and Arithmetic Stratification)\n\n\nAbstract\nWeyl group multi
ple Dirichlet series are multivariable analogues of Dirichlet L-functions.
Their meromorphic continuation leads to asymptotics for moments in famili
es of L-functions\, most notably the family of quadratic Dirichlet L-funct
ions. In this talk I will present work of Diaconu-Goldfeld-Hoffstein and C
hinta-Gunnells which constructs multiple Dirichlet series associated with
various moments of quadratic L-functions. There is an important distinctio
n between series with finite groups of functional equations\, where meromo
rphic continuation is proven\, and series with infinite groups of function
al equations\, where it is an open question. If time permits\, I will disc
uss work of Diaconu-Pașol\, Sawin\, and myself which takes an axiomatic a
pproach to defining these series.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Zenz (Brown University)
DTSTART;VALUE=DATE-TIME:20230320T160000Z
DTEND;VALUE=DATE-TIME:20230320T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/23
DESCRIPTION:Title: On the Distribution of Holomorphic Cusp Forms and Applications
\nby Peter Zenz (Brown University) as part of FRG Grad Seminar (Averages o
f of L-functions and Arithmetic Stratification)\n\n\nAbstract\nArithmetic
Quantum Chaos (AQC) is an active area of research at the intersection of n
umber theory and physics. One major goal in AQC is to study the mass distr
ibution of Hecke Maass cusp forms on hyperbolic surfaces as the Laplace ei
genvalue tends to infinity. In this talk we will focus on analogous questi
ons for holomorphic Hecke cusp forms. We review solved and open conjecture
s in this direction\, like the Quantum Unique Ergodicity Conjecture and th
e Random Wave Conjecture. We then divert our attention to similar question
s\, when restricted to certain subsets of the fundamental domain. Finally\
, we elaborate on how to use some of the mentioned distribution results to
detect real zeros of holomorphic cusp forms low in the fundamental domain
.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksander Simonič (UNSW Canberra)
DTSTART;VALUE=DATE-TIME:20230410T160000Z
DTEND;VALUE=DATE-TIME:20230410T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/24
DESCRIPTION:Title: Some conditional estimates for functions in the Selberg class\
nby Aleksander Simonič (UNSW Canberra) as part of FRG Grad Seminar (Avera
ges of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn thi
s talk I will present recent progress in obtaining conditional (GRH) estim
ates for $(L'/L)(s)$ and $\\log{L(s)}$\, when $L$ is an element of the Sel
berg class of functions and $s$ is not too close to the critical line. We
are able to obtain effective results while assuming the strong $\\lambda$-
conjecture and a polynomial Euler product representation for $L$. If time
permits\, I will also briefly touch on similar results for $s$ being close
to the critical line. This is a joint work with N. Palojärvi.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eun Hye Lee (Stony Brook University)
DTSTART;VALUE=DATE-TIME:20230424T160000Z
DTEND;VALUE=DATE-TIME:20230424T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/25
DESCRIPTION:Title: The Shintani Zeta Functions\nby Eun Hye Lee (Stony Brook Unive
rsity) as part of FRG Grad Seminar (Averages of of L-functions and Arithme
tic Stratification)\n\n\nAbstract\nCounting number fields is a central int
erest in number theory. In this talk\, I will introduce Shintani zeta func
tions\, the counting functions for the number of cubic fields\, and survey
some of the results on them. Time permitting\, I will also discuss some k
ey points of some of the proofs.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiannan Li (Kansas State University)
DTSTART;VALUE=DATE-TIME:20230227T170000Z
DTEND;VALUE=DATE-TIME:20230227T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/26
DESCRIPTION:Title: Quadratic Twists of Modular L-functions\nby Xiannan Li (Kansas
State University) as part of FRG Grad Seminar (Averages of of L-functions
and Arithmetic Stratification)\n\n\nAbstract\nThe behavior of quadratic t
wists of modular L-functions is at the critical point is related both to c
oefficients of half integer weight modular forms and data on elliptic curv
es. Here we describe a proof of an asymptotic for the second moment of thi
s family of L-functions\, previously available conditionally on the Genera
lized 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/frggradseminar/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiseok Yeon (Purdue University)
DTSTART;VALUE=DATE-TIME:20230501T160000Z
DTEND;VALUE=DATE-TIME:20230501T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/27
DESCRIPTION:Title: The Hasse principle for random projective hypersurfaces via the ci
rcle method\nby Kiseok Yeon (Purdue University) as part of FRG Grad Se
minar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbst
ract\nIn this talk\, we introduce a framework via the circle method in ord
er to confirm the Hasse principle for random projective hypersurfaces over
$\\mathbb{Q}$. In particular\, we mainly give a motivation for developing
this framework by providing the overall history of the problems of confir
ming the Hasse principle for projective hypersurfaces over $\\mathbb{Q}$.
Next\, we provide a sketch of the proof of our main result and show a part
of the estimates used in the proof. Furthermore\, if time allows\, we int
roduce an auxiliary mean value theorem which plays a crucial role in our a
rgument and may be of independent interest.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaime Hernandez Palacios (University of Mississippi)
DTSTART;VALUE=DATE-TIME:20230529T160000Z
DTEND;VALUE=DATE-TIME:20230529T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/28
DESCRIPTION:Title: Gaps between zeros of zeta and L-functions of high degree\nby
Jaime Hernandez Palacios (University of Mississippi) as part of FRG Grad S
eminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbs
tract\nThere is a great deal of evidence\, both theoretical and experiment
al\, that the distribution of zeros of zeta and L-functions can be modeled
using statistics of eigenvalues of random matrices from classical compact
groups. In particular\, we expect that there are arbitrarily large and sm
all normalized gaps between the ordinates of (high) zeros zeta and L-funct
ions. Previous results are known for zeta and L-functions of degrees 1 and
2. We discuss some new results for higher degrees\, including Dedekind ze
ta-functions associated to Galois extensions of the rational numbers and p
rincipal automorphic L-functions.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro Fazzari (American Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20230508T160000Z
DTEND;VALUE=DATE-TIME:20230508T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/29
DESCRIPTION:Title: Averages of long Dirichlet polynomials with modular coefficients\nby Alessandro Fazzari (American Institute of Mathematics) as part of F
RG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)
\n\n\nAbstract\nWe study the moments of L-functions associated with primit
ive cusp forms\, in the weight aspect. In particular\, we present recent j
oint work with Brian Conrey\, where we obtain an asymptotic formula for th
e twisted $r$th moment of a long Dirichlet polynomial approximation of suc
h L-functions. This result\, which is conditional on the Generalized Linde
löf Hypothesis\, agrees with the prediction of the recipe by Conrey\, Far
mer\, Keating\, Rubinstein and Snaith.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ofir Gorodetsky (University of Oxford)
DTSTART;VALUE=DATE-TIME:20230310T170000Z
DTEND;VALUE=DATE-TIME:20230310T180000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/30
DESCRIPTION:Title: How many smooth numbers and smooth polynomials are there?\nby
Ofir Gorodetsky (University of Oxford) as part of FRG Grad Seminar (Averag
es of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nSmooth
numbers are integers whose prime factors are all small (smaller than some
threshold $y$). In the 80s they became important outside of pure math\, be
cause Pomerance's quadratic sieve algorithm for factoring integers relied
on them and on their distribution.\n\nThe density of smooth numbers below
x can be approximated -- in some range -- using a peculiar function $\\rho
$ called Dickman's function\, which is defined using a delay-differential
equation. \nAll of the above is also true for smooth polynomials\, which a
re defined similarly and have practical applications.\n\nWe'll survey thes
e topics and discuss recent results whose proofs rely on relating the numb
er of smooth numbers to the Riemann zeta function and its zeros.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emily Quesada-Herrera (Technische Universität Graz)
DTSTART;VALUE=DATE-TIME:20230605T160000Z
DTEND;VALUE=DATE-TIME:20230605T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/32
DESCRIPTION:Title: On the vertical distribution of the zeros of the Riemann zeta-func
tion\nby Emily Quesada-Herrera (Technische Universität Graz) as part
of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratificat
ion)\n\n\nAbstract\nIn 1973\, assuming the Riemann hypothesis (RH)\, Montg
omery studied the vertical distribution of zeta zeros\, and conjectured th
at they behave like the eigenvalues of some random matrices. We will discu
ss some models for zeta zeros – starting from the random matrix model bu
t going beyond it – and related questions\, conjectures and results on s
tatistical information on the zeros. In particular\, assuming RH and a con
jecture of Chan for how often gaps between zeros can be close to a fixed n
on-zero value\, we will discuss our proof of a conjecture of Berry (1988)
for the number variance of zeta zeros\, in a regime where random matrix mo
dels alone do not accurately predict the actual behavior (based on joint w
ork with Meghann Moriah Lugar and Micah B. Milinovich).\n
LOCATION:https://researchseminars.org/talk/frggradseminar/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vorappan (Fai) Chandee (Kansas State University)
DTSTART;VALUE=DATE-TIME:20230515T160000Z
DTEND;VALUE=DATE-TIME:20230515T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/33
DESCRIPTION:Title: The eighth moment of $\\Gamma_1(q)$ L-functions\nby Vorappan (
Fai) Chandee (Kansas State University) as part of FRG Grad Seminar (Averag
es of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn this
talk\, I will discuss my joint work with Xiannan Li on an unconditional a
symptotic formula for the eighth moment of $\\Gamma_1(q)$ L-functions\, wh
ich are associated with eigenforms for the congruence subgroups $\\Gamma_1
(q)$. Similar to a large family of Dirichlet L-functions\, the family of $
\\Gamma_1(q)$ L-functions has a size around $q^2$ while the conductor is o
f size $q$. An asymptotic large sieve of the family is available by the w
ork of Iwaniec and Xiaoqing Li\, and they observed that this family of har
monics is not perfectly orthogonal. This introduces certain subtleties in
our work.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Shparlinski (UNSW\, Sydney)
DTSTART;VALUE=DATE-TIME:20230522T160000Z
DTEND;VALUE=DATE-TIME:20230522T170000Z
DTSTAMP;VALUE=DATE-TIME:20230610T044807Z
UID:frggradseminar/34
DESCRIPTION:Title: Bilinear forms with Kloosterman and Salie Sums and Moments of L-fu
nctions\nby Igor Shparlinski (UNSW\, Sydney) as part of FRG Grad Semin
ar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstrac
t\nWe present some recent results on bilinear forms with complete and inco
mplete Kloosterman and Salie sums. These results are of independent intere
st and also play a major role in bounding error terms in asymptotic formul
as for moments of various L-functions. We then describe several results ab
out non-correlation of Kloosterman and Salie sums between themselves and a
lso with some classical number-theoretic functions such as the Mobius func
tion\, the divisor function and the sum of binary digits\, etc. Some open
problems will be outlined as well.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/34/
END:VEVENT
END:VCALENDAR