BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Cruz Castillo (University of Illinois Urbana-Champaign) DTSTART:20230925T180000Z DTEND:20230925T190000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/2 DESCRIPTION:Title: Sign changes of the error term in the Piltz divisor prob lem\nby Cruz Castillo (University of Illinois Urbana-Champaign) as par t of CRG Weekly Seminars\n\n\nAbstract\nFor an integer $k\\geq 3$\; $\\Del ta_k(x):=\\sum_{n\\leq x} d_k(n)-\\Res_{s=1} \\Big(\\frac{\\zeta^k(s)x^s}{ s}\\Big)$\, 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 larg e enough $X$\; $\\Delta_k(x)$ changes sign at least once in the interval $ [X\, X + C_kX^{1-1/k}]$ for some positive constant $C_k$. For a large para meter $X$\, we show that if the Lindelöf hypothesis is true\, then there exist many disjoint subintervals of $[X\, 2X]$\, each of length $X^{1-1/k- \\epsilon}$ such that $\\Delta_k(x)$ does not change sign in any of these subintervals. 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 r esults may be viewed as higher-degree analogues of a theorem of Heath-Brow n and Tsang\, who studied the case $k = 2$. This is joint work with Siegfr ed Baluyot.\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Neea Palojärvi (University of Helsinki) DTSTART:20231016T180000Z DTEND:20231016T190000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/3 DESCRIPTION:Title: Conditional estimates for logarithms and logarithmic der ivatives in the Selberg class\nby Neea Palojärvi (University of Helsi nki) as part of CRG Weekly Seminars\n\n\nAbstract\nThe Selberg class consi sts of functions sharing similar properties to the Riemann zeta function. The Riemann zeta function is one example of the functions in this class. T he estimates for logarithms of Selberg class functions and their logarithm ic derivatives are connected to\, for example\, primes in arithmetic progr essions.\n\nIn this talk\, I will discuss about effective and explicit est imates for logarithms and logarithmic derivatives of the Selberg class fun ctions when $\\Re(s) \\geq \\frac12+ \\delta$ where $\\delta >0$. All res ults are under the Generalized Riemann hypothesis and some of them are als o under assumption of a polynomial Euler product representation or the str ong $\\lambda$-conjecture. The talk is based on a joint work with Aleksand er Simonič (University of New South Wales Canberra).\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Sebastian Zuniga Alterman (University of Turku) DTSTART:20231204T190000Z DTEND:20231204T200000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/4 DESCRIPTION:Title: Möbius function\, an identity factory with applications \nby Sebastian Zuniga Alterman (University of Turku) as part of CRG We ekly Seminars\n\n\nAbstract\nBy using an identity relating a sum to an int egral\, we obtain a family of identities for the averages $\\displaystyle M(X)= \\sum_{n\\leq X} \\mu(n)$ and $\\displaystyle m(X)= \\sum_{n\\leq X } \\frac{\\mu(n)}{n}$. Further\, by choosing some specific families\, we s tudy two summatory functions related to the Möbius function\, $\\mu(n)$\, namely $\\displaystyle \\sum_{n\\leq X} \\frac{\\mu(n)}{n^s}$ and $\\disp laystyle \\sum_{n\\leq X} \\frac{\\mu(n) }{n^s}\\log(X/n) $\, where $s$ is a complex number and $\\Re s >0$. We also explore some applications and e xamples when $s$ is real. (joint work with O. Ramaré)\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Michaela Cully-Hugill (University of New South Wales Canberra) DTSTART:20231003T230000Z DTEND:20231004T000000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/5 DESCRIPTION:Title: An explicit estimate on the mean value of the error in t he prime number theorem in intervals\nby Michaela Cully-Hugill (Univer sity of New South Wales Canberra) as part of CRG Weekly Seminars\n\n\nAbst ract\nThe prime number theorem (PNT) gives us the density of primes amongs t the natural numbers. We can extend this idea to consider whether we have the asymptotic number of primes predicted by the PNT in a given interval. Currently\, this has only been proven for sufficiently large intervals. W e can also consider whether the PNT holds for sufficiently large intervals ‘on average’. This requires estimating the mean-value of the error in the PNT in intervals. A new explicit estimate for this will be given base d on the work of Selberg in 1943\, along with two applications: one for pr imes in intervals\, and one for Goldbach numbers in intervals.\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Lucile Devin (Université du Littoral Côte d'Opale) DTSTART:20231023T180000Z DTEND:20231023T190000Z DTSTAMP:20260404T095030Z 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 pa rt of CRG Weekly Seminars\n\n\nAbstract\nGeneralizing the original Chebysh ev bias can go in many directions: one can adapt the setting to virtually any equidistribution result encoded by a finite number of $L$-functions. I n this talk\, we will discuss what happens when one needs an infinite numb er of $L$-functions. This will be illustrated by the following question: g iven a prime that can be written as a sum of two squares $p = a^²+4b^²$\ , how does the congruence class of $a>0$ distribute?\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Shivani Goel (Indraprastha Institute of Information Technology\, D elhi) DTSTART:20231030T180000Z DTEND:20231030T190000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/8 DESCRIPTION:Title: On the Hardy Littlewood $3$-tuple prime conjecture and c onvolutions of Ramanujan sums\nby Shivani Goel (Indraprastha Institute of Information Technology\, Delhi) as part of CRG Weekly Seminars\n\n\nAb stract\nThe Hardy and Littlewood $k$-tuple prime conjecture is one of the most enduring unsolved problems in mathematics. In 1999\, Gadiyar and Padm a presented a heuristic derivation of the $2$-tuples conjecture by employi ng the orthogonality principle of Ramanujan sums. Building upon their work \, we explore triple convolution Ramanujan sums and use this approach to p rovide a heuristic derivation of the Hardy-Littlewood conjecture concernin g prime $3$-tuples. Furthermore\, we estimate the triple convolution of th e Jordan totient function using Ramanujan sums.\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Vorrapan Chandee (Kansas State University) DTSTART:20231106T190000Z DTEND:20231106T200000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/9 DESCRIPTION:Title: The eighth moment of $\\Gamma_1(q)$ $L$-functions\nb y Vorrapan Chandee (Kansas State University) as part of CRG Weekly Seminar s\n\n\nAbstract\nIn this talk\, I will discuss my on-going joint work with Xiannan Li on an unconditional asymptotic formula for the eighth moment o f $\\Gamma_1(q)$ $L$-functions\, which are associated with eigenforms for the congruence subgroups $\\Gamma_1(q)$. Similar to a large family of Diri chlet $L$-functions\, the family of $\\Gamma_1(q)$ $L$-functions has a siz e around $q^2$ while the conductor is of size $q$. An asymptotic large sie ve of the family is available by the work of Iwaniec and Xiaoqing Li\, and they observed that this family of harmonics is not perfectly orthogonal. This introduces certain subtleties in our work.\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrew Pearce-Crump (University of York) DTSTART:20231120T190000Z DTEND:20231120T200000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/11 DESCRIPTION:Title: Characteristic polynomials\, the Hybrid model\, and the Ratios Conjecture\nby Andrew Pearce-Crump (University of York) as par t of CRG Weekly Seminars\n\n\nAbstract\nIn the 1960s Shanks conjectured th at the $\\zeta'(\\rho)$\, where $\\rho$ is a non-trivial zero of zeta\, is both real and positive in the mean. Conjecturing and proving this result has a rich history\, but efforts to generalise it to higher moments have s o far failed. Building on the work of Keating and Snaith using characteris tic polynomials from Random Matrix Theory\, the Hybrid model of Gonek\, Hu ghes and Keating\, and the Ratios Conjecture of Conrey\, Farmer\, and Zirn bauer\, we have been able to produce new conjectures for the full asymptot ics of higher moments of the derivatives of zeta. This is joint work with Chris Hughes.\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Siegfred Baluyot (American Institute of Mathematics) DTSTART:20231127T190000Z DTEND:20231127T200000Z DTSTAMP:20260404T095030Z UID:crgseminarfall2023/12 DESCRIPTION:Title: Twisted moments of characteristic polynomials of random matrices\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 con jectural 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 predictions for all integral moments. In 2015\, Conr ey and Keating revisited the approach of using divisor sum heuristics to p redict asymptotic formulas for moments of zeta. Essentially\, their method estimates moments of zeta using lower twisted moments. In this talk\, I w ill discuss a rigorous random matrix theory analogue of the Conrey-Keating heuristic. This is ongoing joint work with Brian Conrey.\n LOCATION:https://stable.researchseminars.org/talk/crgseminarfall2023/12/ END:VEVENT END:VCALENDAR