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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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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:20240614T063124Z
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
BEGIN:VEVENT
SUMMARY:Mayank Pandey (Princeton University)
DTSTART;VALUE=DATE-TIME:20230918T160000Z
DTEND;VALUE=DATE-TIME:20230918T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/35
DESCRIPTION:Title: $L^1$ means of exponential sums with multiplicative coefficients\nby Mayank Pandey (Princeton University) as part of FRG Grad Seminar (A
verages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nWe
discuss some recent results on the $L^1$ norm of exponential sums with mu
ltiplicative functions\, with specific results for the Mobius and Liouvill
e functions. Joint work with Maksym Radziwill.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henryk Iwaniec (Rutgers University)
DTSTART;VALUE=DATE-TIME:20230925T160000Z
DTEND;VALUE=DATE-TIME:20230925T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/36
DESCRIPTION:Title: Integer Parts Mutually Coprime\nby Henryk Iwaniec (Rutgers Uni
versity) as part of FRG Grad Seminar (Averages of of L-functions and Arith
metic Stratification)\n\n\nAbstract\nOne of the problems which is the subj
ect of this talk concerns the density of integers $n$ for which the intege
r parts $[(n+i)^c]$ are pairwise coprime with $i=1\,...\,k$. Here $c$ is a
ny constant\, $1Short Second Moment Bound for GL(2) L-functions in the Level Aspec
t\nby Agniva Dasgupta (Texas A&M University) as part of FRG Grad Semin
ar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstrac
t\nWe will discuss my recent work on moments of $L$-functions at the centr
al point. Early results in this area were concerned with 'full' moments\,
by studying expressions like $\\int_{T}^{2T} \\left \\vert L(f\,\\frac12+i
t) \\right \\vert^k dt$\, or $\\sum_{\\chi (\\text{mod }q)} \\left \\vert
L(f\\otimes \\chi\, \\frac12) \\right \\vert^k$. A 1978 paper of Iwaniec p
roved a Lindelöf-on-average upper bound on a 'short' fourth moment\, by s
howing $\\int_{T}^{T+T^{\\frac23}} \\left \\vert{\\zeta(\\frac12+it)} \\ri
ght \\vert ^4 \\ll T^{\\frac23 + \\varepsilon}$. Good (1982) proved a simi
lar upper bound for a short (second) moment for level 1 cusp forms. We pr
ove a level-aspect analogue to Good's result. We assume $q=p^3$ for an od
d prime $p$\, and for the short second moment\, we consider the twists of
a level 1 cusp form along a coset of subgroup of the characters modulo $q^
{\\frac23}$.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winston Heap (Norwegian University of Science and Technology (NTNU
))
DTSTART;VALUE=DATE-TIME:20231023T160000Z
DTEND;VALUE=DATE-TIME:20231023T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/38
DESCRIPTION:Title: Simultaneous extreme values of zeta and L-functions\nby Winsto
n Heap (Norwegian University of Science and Technology (NTNU)) as part of
FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification
)\n\n\nAbstract\nWe use a modification of the resonance method to demonstr
ate simultaneous large values of L-functions on the critical line. The met
hod extends to other families and can be used to show both simultaneous la
rge and small values. Joint work with Junxian Li.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Curran (University of Oxford)
DTSTART;VALUE=DATE-TIME:20231106T170000Z
DTEND;VALUE=DATE-TIME:20231106T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/39
DESCRIPTION:Title: Correlations of the Riemann zeta function\nby Michael Curran (
University of Oxford) as part of FRG Grad Seminar (Averages of of L-functi
ons and Arithmetic Stratification)\n\n\nAbstract\nShifted moments of the R
iemann zeta function\, introduced by Chandee\, are natural generalizations
of the moments of zeta. While the moments of zeta capture large values of
zeta\, the shifted moments capture how the values of zeta are correlated
along the half line. I will describe recent and forthcoming work giving sh
arp bounds for shifted moments assuming the Riemann hypothesis\, improving
previous work of Chandee and Ng\, Shen\, and Wong.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Stadlmann (University of Oxford)
DTSTART;VALUE=DATE-TIME:20231113T170000Z
DTEND;VALUE=DATE-TIME:20231113T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/40
DESCRIPTION:Title: Primes in arithmetic progressions to smooth moduli\nby Julia S
tadlmann (University of Oxford) as part of FRG Grad Seminar (Averages of o
f L-functions and Arithmetic Stratification)\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 mat
hematical techniques. However\, in 2013\, Zhang achieved a breakthrough\,
showing that there exists some positive integer $h$ for which $p$ and $p+h
$ are both prime infinitely often. Equidistribution estimates for primes i
n arithmetic progressions to smooth moduli were a key ingredient of his wo
rk.\n\nIn this talk\, I will sketch what role these estimates play in proo
fs of bounded gaps between primes. I will also show how a refinement of th
e q-van der Corput method can be used to improve on equidistribution estim
ates of the Polymath project for primes in APs to smooth moduli.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Lester (King's College London)
DTSTART;VALUE=DATE-TIME:20231120T170000Z
DTEND;VALUE=DATE-TIME:20231120T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/41
DESCRIPTION:Title: Around the Gauss circle problem\nby Steve Lester (King's Colle
ge London) as part of FRG Grad Seminar (Averages of of L-functions and Ari
thmetic Stratification)\n\n\nAbstract\nHardy conjectured that the error te
rm arising from approximating the number of lattice points lying in a radi
us-$R$ disc by its area is $O(R^{1/2+o(1)})$. One source of support for th
is conjecture is a folklore heuristic that uses i.i.d. random variables to
model the lattice points lying near the boundary and square-root cancella
tion of sums of these random variables. In this talk I will examine this h
euristic and discuss how lattice points near the circle interact with one
another. This is joint work with Igor Wigman.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Yiasemides (London School of Economics)
DTSTART;VALUE=DATE-TIME:20231127T170000Z
DTEND;VALUE=DATE-TIME:20231127T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/42
DESCRIPTION:Title: Lattice Points in Function Fields\, and Hankel Matrices.\nby M
ichael Yiasemides (London School of Economics) as part of FRG Grad Seminar
(Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\
nIn this talk we discuss the function field analogue of lattice points in
thin elliptic annuli. We will begin with a general introduction to lattice
points in the classical setting\, including briefly highlighting connecti
ons to physics and various areas of number theory\; before introducing the
function field analogue and stating our results on the mean and variance
of lattice points in elliptic annuli. Our approach is unique to the functi
on field setting\, and it translates the problem to one involving Hankel m
atrices over finite fields. We will summarise this approach\, before finis
hing by highlighting some interesting connections between Hankel matrices
and number theory in function fields.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrés Chirre (Pontificia Universidad Católica del Perú)
DTSTART;VALUE=DATE-TIME:20231211T170000Z
DTEND;VALUE=DATE-TIME:20231211T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/44
DESCRIPTION:Title: Remarks on a formula of Ramanujan\nby Andrés Chirre (Pontific
ia Universidad Católica del Perú) as part of FRG Grad Seminar (Averages
of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn this ta
lk\, we will discuss a well-known formula of Ramanujan and its relationshi
p with the partial sums of the Möbius function. Under some conjectures\,
we analyze a finer structure of the involved terms. It is a joint work wit
h Steven M. Gonek.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Gerspach (KTH Royal Institute of Technology)
DTSTART;VALUE=DATE-TIME:20231218T170000Z
DTEND;VALUE=DATE-TIME:20231218T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/45
DESCRIPTION:Title: Heuristics and random models for quadratic character sums\nby
Maxim Gerspach (KTH Royal Institute of Technology) as part of FRG Grad Sem
inar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstr
act\nIn this talk\, I will discuss heuristics for low moments of quadratic
character sums\, i.e. for low powers (between 0 and 2) of quadratic chara
cter sums averaged over the conductor. I will begin by talking about the r
ational setting and then go over to the function field setting. These heur
istics are backed up by rigorous results in a random model that I will des
cribe. Moreover\, I will touch upon the extent to which these heuristics c
an be made rigorous in the deterministic setting.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sacha Mangerel (Durham University)
DTSTART;VALUE=DATE-TIME:20231030T160000Z
DTEND;VALUE=DATE-TIME:20231030T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/48
DESCRIPTION:Title: Large order Dirichlet characters and an analogue of a conjecture o
f Vinogradov\nby Sacha Mangerel (Durham University) as part of FRG Gra
d Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\n
Abstract\nLet $q$ be a large prime. According to an old and famous conject
ure of I.M. Vinogradov\, for any $c > 0$ and $q$ sufficiently large\, the
least quadratic non-residue $n$ modulo $q$ should satisfy $n < q^c$. This
statement would be implied by non-trivial upper bounds for averages of the
Legendre symbol $(n/q)$ with $n < q^c$. Currently the best such results\,
due essentially to Burgess\, are satisfactory only when $c > 1/4$\, due t
o the potential obstruction\, difficult to rule out\, that $(n/q) = +1$ fo
r "many" initial integers $n$. \n\nIn this talk I will discuss a generalis
ation of Vinogradov's conjecture to other primitive Dirichlet characters $
\\chi$ modulo $q$\, seeking the least $n$ for which $\\chi(n)$ is not $1$.
I will explain some recent work of mine that shows\, using techniques fro
m elementary additive combinatorics\, that when the order $d$ of $\\chi$ i
s a prime that grows with $q$:\n\ni) the aforementioned obstruction does n
ot occur\, \nii) the analogue of Vinogradov's conjecture for $\\chi$ does
hold\, and moreover \niii) for each $c > 0$ and $q$ large enough with resp
ect to $c$\, $\\chi(n) = 1$ occurs rarely when $n < q^c$. \n\nThese result
s are connected with averaged cancellation in short sums of $\\chi$ over $
n < q^c$ for arbitrarily small $c > 0$\, going beyond Burgess' estimate.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivian Kuperberg (ETH Zürich)
DTSTART;VALUE=DATE-TIME:20240226T170000Z
DTEND;VALUE=DATE-TIME:20240226T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/49
DESCRIPTION:Title: Sums of odd-ly many fractions\nby Vivian Kuperberg (ETH Züric
h) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic
Stratification)\n\n\nAbstract\nIn this talk\, I will discuss new bounds on
constrained sets of fractions. Specifically\, I will discuss the answer t
o the following question\, which arises in multiple areas of number theory
: for an integer $k \\ge 2$\, consider the set of $k$-tuples of reduced fr
actions $\\frac{a_1}{q_1}\, \\dots\, \\frac{a_k}{q_k} \\in I$\, where $I$
is an interval around $0$.\nHow many $k$-tuples are there with $\\sum_i \\
frac{a_i}{q_i} \\in \\mathbb Z$?\n\nWhen $k$ is even\, the answer is well-
known: the main contribution to the number of solutions comes from "diagon
al'' terms\, where the fractions $\\frac{a_i}{q_i}$ cancel in pairs. When
$k$ is odd\, the answer is much more mysterious! In work with Bloom\, we p
rove a near-optimal upper bound on this problem when $k$ is odd. I will al
so discuss applications of this problem to estimating moments of the distr
ibutions of primes and reduced residues.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandre de Faveri (Stanford University)
DTSTART;VALUE=DATE-TIME:20240212T170000Z
DTEND;VALUE=DATE-TIME:20240212T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/51
DESCRIPTION:Title: An inequality for GL(3) Fourier coefficients\nby Alexandre de
Faveri (Stanford University) as part of FRG Grad Seminar (Averages of of L
-functions and Arithmetic Stratification)\n\n\nAbstract\nWe prove a certai
n comparison inequality for partial sums of Fourier coefficients of Hecke-
Maass cuspforms in GL(3). This is a higher rank generalization of a result
of Soundararajan\, and has applications to distribution of mass in GL(3).
Joint work with Zvi Shem-Tov.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jackie Voros (University of Bristol)
DTSTART;VALUE=DATE-TIME:20240219T170000Z
DTEND;VALUE=DATE-TIME:20240219T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/52
DESCRIPTION:Title: On the average least negative Hecke eigenvalue\nby Jackie Voro
s (University of Bristol) as part of FRG Grad Seminar (Averages of of L-fu
nctions and Arithmetic Stratification)\n\n\nAbstract\nIn this talk we disc
uss the first sign change of Fourier coefficients of newforms\, or equival
ently Hecke eigenvalues. We will see this to be an analogue of the least q
uadratic non-residue problem\, of which the average was investigated by Er
dős in 1961. In fact\, we will see that the average least negative prime
Hecke eigenvalue holds the same (finite) value as the average least quadra
tic non-residue\, under GRH. This is mainly due to the fact that Hecke eig
envalues at primes are equidistributed with respect to the Sato-Tate measu
re\, a consequence of the Sato-Tate conjecture that was proven in 2011. We
further explore the so-called vertical Sato-Tate conjecture to show the a
verage least Hecke eigenvalue has a finite value unconditionally.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prahlad Sharma (Max Planck Institute for Mathematics\, Bonn)
DTSTART;VALUE=DATE-TIME:20240318T160000Z
DTEND;VALUE=DATE-TIME:20240318T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/53
DESCRIPTION:Title: Counting special points on quadratic surfaces\nby Prahlad Shar
ma (Max Planck Institute for Mathematics\, Bonn) as part of FRG Grad Semin
ar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstrac
t\nWe show how the modern versions of the circle method can be combined wi
th the equidistribution of quadratic roots\, allowing us to count special
points on quadratic surfaces. For example\, we will obtain asymptotic for
integer points on general quadratic surfaces with prime coordinates and in
short intervals.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trevor Wooley (Purdue University)
DTSTART;VALUE=DATE-TIME:20240129T170000Z
DTEND;VALUE=DATE-TIME:20240129T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/54
DESCRIPTION:Title: Primes as sums of k-th powers\, and Freiman's theorem\nby Trev
or Wooley (Purdue University) as part of FRG Grad Seminar (Averages of of
L-functions and Arithmetic Stratification)\n\n\nAbstract\nSuppose that one
seeks to apply the circle method to the problem of representing a large i
nteger n as the sum of a prime number and a number of k-th powers. The Wey
l sum over the prime is small on a set of minor arcs\, but the complementa
ry set of major arcs is incompatible with conventional technology for hand
ling the corresponding Weyl sums over the k-th powers. In this talk we exp
lain progress on this problem that delivers conclusions with only slightly
more than 2k of these k-th powers. The key idea is to obtain partial info
rmation concerning moments on minor arcs of large height well beyond the c
onventional range accessible to Poisson summation. Similar ideas yield pro
gress on such problems as that of Freiman concerning sums of mixed powers.
This is work joint with Joerg Bruedern.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seth Hardy (University of Warwick)
DTSTART;VALUE=DATE-TIME:20240304T170000Z
DTEND;VALUE=DATE-TIME:20240304T180000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/55
DESCRIPTION:Title: Bounds for exponential sums with random multiplicative coefficient
s\nby Seth Hardy (University of Warwick) as part of FRG Grad Seminar (
Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nT
he study of exponential sums with multiplicative coefficients is classical
in analytic number theory\, yet our understanding of them is far from com
plete. This is unsurprising\, seeing as multiplicative functions alone are
often difficult objects to grasp. However\, in recent years\, our underst
anding of random multiplicative functions has flourished\, and pioneering
work has been conducted by Benatar\, Nishry\, and Rodgers to uncover how e
xponential sums behave when their coefficients are given by random multipl
icative functions. In this talk\, we will introduce random multiplicative
functions\, discuss some of the literature surrounding them\, and outline
recent work on conjecturally sharp lower bounds for the maximum size of ex
ponential sums involving them.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Quanli Shen (Shandong University\, Weihai)
DTSTART;VALUE=DATE-TIME:20240325T160000Z
DTEND;VALUE=DATE-TIME:20240325T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/56
DESCRIPTION:Title: The fourth moment of quadratic Dirichlet L-functions II\nby Qu
anli Shen (Shandong University\, Weihai) as part of FRG Grad Seminar (Aver
ages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nI wil
l discuss the fourth moment of quadratic Dirichlet L-functions where we pr
ove an asymptotic formula with four main terms unconditionally. Previously
the asymptotic formula was established with the leading main term under t
he generalized Riemann hypothesis. This work is based on Li's recent work
on the second moment of quadratic twists of modular L-functions. It is jo
int work with Joshua Stucky.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Gonek (University of Rochester)
DTSTART;VALUE=DATE-TIME:20240409T160000Z
DTEND;VALUE=DATE-TIME:20240409T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/57
DESCRIPTION:Title: Universality of $L$-Functions over finite function fields\nby
Steve Gonek (University of Rochester) as part of FRG Grad Seminar (Average
s of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nWe prove
that Dirichlet $L$-functions corresponding to Dirichlet characters for $
\\mathbb{F}_{q}[x]$ with $q$ odd are universal in the following sense. Let
$\\mathscr Q$ denote either the set of all prime polynomials $Q$ in $
\\mathbb F_q[x]$\, or the set of all polynomials $Q$ that are products of
a fixed set of prime polynomials $Q_1\, Q_2\, \\ldots\, Q_r \\in \\math
bb F_q[x]$. Let $U $ be the open rectangle with vertices $s_1+ia\, s_2+ia\
, s_2+ib\, s_1+ib\,$ where $\\frac12Twisted moments of characteristic polynomials of random matrices\nby Siegfred Baluyot (American Institute of Mathematics) as part of FRG
Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n
\n\nAbstract\nIn the late 90's\, Keating and Snaith used random matrix the
ory to predict the exact leading terms of conjectural asymptotic formulas
for all integral moments of the Riemann zeta-function. Prior to their work
\, no number-theoretic argument or heuristic has led to such exact predict
ions for all integral moments. In 2015\, Conrey and Keating revisited the
approach of using divisor sum heuristics to predict asymptotic formulas fo
r moments of zeta. Essentially\, their method estimates moments of zeta us
ing lower twisted moments. In this talk\, I will discuss a rigorous random
matrix theory analogue of the Conrey-Keating heuristic. This is ongoing j
oint work with Brian Conrey.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ertan Elma (University of Lethbridge)
DTSTART;VALUE=DATE-TIME:20240311T160000Z
DTEND;VALUE=DATE-TIME:20240311T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/59
DESCRIPTION:by Ertan Elma (University of Lethbridge) as part of FRG Grad S
eminar (Averages of of L-functions and Arithmetic Stratification)\n\nAbstr
act: TBA\n
LOCATION:https://researchseminars.org/talk/frggradseminar/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Keating (University of Oxford)
DTSTART;VALUE=DATE-TIME:20240416T160000Z
DTEND;VALUE=DATE-TIME:20240416T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/60
DESCRIPTION:Title: Joint Moments\nby Jonathan Keating (University of Oxford) as p
art of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratif
ication)\n\n\nAbstract\nI will discuss the evaluation of the joint moments
of the characteristic polynomials of random unitary matrices and their de
rivatives\, and in this context the joint moments of the Riemann zeta-func
tion and its derivates\, on the critical line.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ertan Elma (University of Lethbridge)
DTSTART;VALUE=DATE-TIME:20240401T160000Z
DTEND;VALUE=DATE-TIME:20240401T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/61
DESCRIPTION:Title: A discrete mean value of the Riemann zeta function and its derivat
ives\nby Ertan Elma (University of Lethbridge) as part of FRG Grad Sem
inar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstr
act\nIn this talk\, we will discuss an estimate for a discrete mean value
of the Riemann zeta function and its derivatives multiplied by Dirichlet p
olynomials. Assuming the Riemann Hypothesis\, we obtain a lower bound for
the 2kth moment of all the derivatives of the Riemann zeta function evalua
ted at its nontrivial zeros. This is based on a joint work with Kübra Ben
li and Nathan Ng.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Conrey (American Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20240429T160000Z
DTEND;VALUE=DATE-TIME:20240429T170000Z
DTSTAMP;VALUE=DATE-TIME:20240614T063124Z
UID:frggradseminar/62
DESCRIPTION:Title: Averages of L-functions and arithmetic stratification: a report on
the FRG\nby Brian Conrey (American Institute of Mathematics) as part
of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratificat
ion)\n\n\nAbstract\nI will give an update on some of the work that has bee
n done since the FRG started.\n
LOCATION:https://researchseminars.org/talk/frggradseminar/62/
END:VEVENT
END:VCALENDAR