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SUMMARY:Cruz Castillo (University of Illinois Urbana-Champaign)
DTSTART:20230925T180000Z
DTEND:20230925T190000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/2
DESCRIPTION:Title: Sign changes of the error term in the Piltz divisor problem
\nby Cruz Castillo (University of Illinois Urbana-Champaign) as part of CR
G Weekly Seminars\n\n\nAbstract\nFor an integer $k\\geq 3$\; $\\Delta_k(x)
:=\\sum_{n\\leq x} d_k(n)-\\Res_{s=1} \\Big(\\frac{\\zeta^k(s)x^s}{s}\\Big
)$\, where $d_k(n)$ is the $k$-fold divisor function\, and $\\zeta(s)$ is
the Riemann zeta-function. In the 1950's\, Tong showed for all large enoug
h $X$\; $\\Delta_k(x)$ changes sign at least once in the interval $[X\, X
+ C_kX^{1-1/k}]$ for some positive constant $C_k$. For a large parameter $
X$\, we show that if the Lindelöf hypothesis is true\, then there exist m
any disjoint subintervals of $[X\, 2X]$\, each of length $X^{1-1/k-\\epsil
on}$ such that $\\Delta_k(x)$ does not change sign in any of these subinte
rvals. If the Riemann hypothesis is true\, then we can improve the length
of the subintervals to $\\ll X^{1-1/k} (\\log X)^{-k^2-2}$. These results
may be viewed as higher-degree analogues of a theorem of Heath-Brown and T
sang\, who studied the case $k = 2$. This is joint work with Siegfred Balu
yot.\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/2/
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BEGIN:VEVENT
SUMMARY:Neea Palojärvi (University of Helsinki)
DTSTART:20231016T180000Z
DTEND:20231016T190000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/3
DESCRIPTION:Title: Conditional estimates for logarithms and logarithmic derivative
s in the Selberg class\nby Neea Palojärvi (University of Helsinki) as
part of CRG Weekly Seminars\n\n\nAbstract\nThe Selberg class consists of
functions sharing similar properties to the Riemann zeta function. The Rie
mann zeta function is one example of the functions in this class. The esti
mates for logarithms of Selberg class functions and their logarithmic deri
vatives are connected to\, for example\, primes in arithmetic progressions
.\n\nIn this talk\, I will discuss about effective and explicit estimates
for logarithms and logarithmic derivatives of the Selberg class functions
when $\\Re(s) \\geq \\frac12+ \\delta$ where $\\delta >0$. All results ar
e under the Generalized Riemann hypothesis and some of them are also under
assumption of a polynomial Euler product representation or the strong $\\
lambda$-conjecture. The talk is based on a joint work with Aleksander Simo
nič (University of New South Wales Canberra).\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/3/
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BEGIN:VEVENT
SUMMARY:Sebastian Zuniga Alterman (University of Turku)
DTSTART:20231204T190000Z
DTEND:20231204T200000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/4
DESCRIPTION:Title: Möbius function\, an identity factory with applications\nb
y Sebastian Zuniga Alterman (University of Turku) as part of CRG Weekly Se
minars\n\n\nAbstract\nBy using an identity relating a sum to an integral\,
we obtain a family of identities for the averages $\\displaystyle M(X)= \
\sum_{n\\leq X} \\mu(n)$ and $\\displaystyle m(X)= \\sum_{n\\leq X} \\fra
c{\\mu(n)}{n}$. Further\, by choosing some specific families\, we study tw
o summatory functions related to the Möbius function\, $\\mu(n)$\, namely
$\\displaystyle \\sum_{n\\leq X} \\frac{\\mu(n)}{n^s}$ and $\\displaystyl
e \\sum_{n\\leq X} \\frac{\\mu(n) }{n^s}\\log(X/n) $\, where $s$ is a comp
lex number and $\\Re s >0$. We also explore some applications and examples
when $s$ is real. (joint work with O. Ramaré)\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/4/
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BEGIN:VEVENT
SUMMARY:Michaela Cully-Hugill (University of New South Wales Canberra)
DTSTART:20231003T230000Z
DTEND:20231004T000000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/5
DESCRIPTION:Title: An explicit estimate on the mean value of the error in the prim
e number theorem in intervals\nby Michaela Cully-Hugill (University of
New South Wales Canberra) as part of CRG Weekly Seminars\n\n\nAbstract\nT
he prime number theorem (PNT) gives us the density of primes amongst the n
atural numbers. We can extend this idea to consider whether we have the as
ymptotic number of primes predicted by the PNT in a given interval. Curren
tly\, this has only been proven for sufficiently large intervals. We can a
lso consider whether the PNT holds for sufficiently large intervals ‘on
average’. This requires estimating the mean-value of the error in the PN
T in intervals. A new explicit estimate for this will be given based on th
e work of Selberg in 1943\, along with two applications: one for primes in
intervals\, and one for Goldbach numbers in intervals.\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucile Devin (Université du Littoral Côte d'Opale)
DTSTART:20231023T180000Z
DTEND:20231023T190000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/7
DESCRIPTION:Title: Biases in the distribution of Gaussian primes and other stories
\nby Lucile Devin (Université du Littoral Côte d'Opale) as part of C
RG Weekly Seminars\n\n\nAbstract\nGeneralizing the original Chebyshev bias
can go in many directions: one can adapt the setting to virtually any equ
idistribution result encoded by a finite number of $L$-functions. In this
talk\, we will discuss what happens when one needs an infinite number of $
L$-functions. This will be illustrated by the following question: given a
prime that can be written as a sum of two squares $p = a^²+4b^²$\, how d
oes the congruence class of $a>0$ distribute?\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/7/
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BEGIN:VEVENT
SUMMARY:Shivani Goel (Indraprastha Institute of Information Technology\, D
elhi)
DTSTART:20231030T180000Z
DTEND:20231030T190000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/8
DESCRIPTION:Title: On the Hardy Littlewood $3$-tuple prime conjecture and convolut
ions of Ramanujan sums\nby Shivani Goel (Indraprastha Institute of Inf
ormation Technology\, Delhi) as part of CRG Weekly Seminars\n\n\nAbstract\
nThe Hardy and Littlewood $k$-tuple prime conjecture is one of the most en
during unsolved problems in mathematics. In 1999\, Gadiyar and Padma prese
nted a heuristic derivation of the $2$-tuples conjecture by employing the
orthogonality principle of Ramanujan sums. Building upon their work\, we e
xplore triple convolution Ramanujan sums and use this approach to provide
a heuristic derivation of the Hardy-Littlewood conjecture concerning prime
$3$-tuples. Furthermore\, we estimate the triple convolution of the Jorda
n totient function using Ramanujan sums.\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vorrapan Chandee (Kansas State University)
DTSTART:20231106T190000Z
DTEND:20231106T200000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/9
DESCRIPTION:Title: The eighth moment of $\\Gamma_1(q)$ $L$-functions\nby Vorra
pan Chandee (Kansas State University) as part of CRG Weekly Seminars\n\n\n
Abstract\nIn this talk\, I will discuss my on-going joint work with Xianna
n Li on an unconditional asymptotic formula for the eighth moment of $\\Ga
mma_1(q)$ $L$-functions\, which are associated with eigenforms for the con
gruence subgroups $\\Gamma_1(q)$. Similar to a large family of Dirichlet $
L$-functions\, the family of $\\Gamma_1(q)$ $L$-functions has a size aroun
d $q^2$ while the conductor is of size $q$. An asymptotic large sieve of t
he family is available by the work of Iwaniec and Xiaoqing Li\, and they o
bserved that this family of harmonics is not perfectly orthogonal. This in
troduces certain subtleties in our work.\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Pearce-Crump (University of York)
DTSTART:20231120T190000Z
DTEND:20231120T200000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/11
DESCRIPTION:Title: Characteristic polynomials\, the Hybrid model\, and the Ratios
Conjecture\nby Andrew Pearce-Crump (University of York) as part of CR
G Weekly Seminars\n\n\nAbstract\nIn the 1960s Shanks conjectured that the
$\\zeta'(\\rho)$\, where $\\rho$ is a non-trivial zero of zeta\, is both r
eal and positive in the mean. Conjecturing and proving this result has a r
ich history\, but efforts to generalise it to higher moments have so far f
ailed. Building on the work of Keating and Snaith using characteristic pol
ynomials from Random Matrix Theory\, the Hybrid model of Gonek\, Hughes an
d Keating\, and the Ratios Conjecture of Conrey\, Farmer\, and Zirnbauer\,
we have been able to produce new conjectures for the full asymptotics of
higher moments of the derivatives of zeta. This is joint work with Chris H
ughes.\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siegfred Baluyot (American Institute of Mathematics)
DTSTART:20231127T190000Z
DTEND:20231127T200000Z
DTSTAMP:20260422T213010Z
UID:crgseminarfall2023/12
DESCRIPTION:Title: Twisted moments of characteristic polynomials of random matric
es\nby Siegfred Baluyot (American Institute of Mathematics) as part of
CRG Weekly Seminars\n\n\nAbstract\nIn the late 90's\, Keating and Snaith
used random matrix theory to predict the exact leading terms of conjectura
l asymptotic formulas for all integral moments of the Riemann zeta-functio
n. Prior to their work\, no number-theoretic argument or heuristic has led
to such exact predictions for all integral moments. In 2015\, Conrey and
Keating revisited the approach of using divisor sum heuristics to predict
asymptotic formulas for moments of zeta. Essentially\, their method estima
tes moments of zeta using lower twisted moments. In this talk\, I will dis
cuss a rigorous random matrix theory analogue of the Conrey-Keating heuris
tic. This is ongoing joint work with Brian Conrey.\n
LOCATION:https://researchseminars.org/talk/crgseminarfall2023/12/
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