BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Florent Jouve (Institut de Mathématiques de Bordeaux\, France) DTSTART:20231005T140000Z DTEND:20231005T145000Z DTSTAMP:20260415T034322Z UID:Frobenius4/1 DESCRIPTION:Title: Moments in the Chebotarev Density Theorem\nby Florent Jouve (Instit ut de Mathématiques de Bordeaux\, France) as part of Around Frobenius Dis tributions and Related Topics IV\n\n\nAbstract\nI will report on joint wor k with Régis de La Bretèche and Daniel Fiorilli in which we consider wei ghted moments for the distribution of Frobenius elements in conjugacy clas ses of Galois groups of normal number field extensions. The question is in spired by results of Hooley and recent progress due to de La Bretèche--Fi orilli concerning moments for the distribution of primes in arithmetic pro gressions . As in the latter case\, our results are conditional on GRH and confirm that the moments considered should be Gaussian. Time permitting w e will mention another notion of moments for which particular Galois group structures exclude a Gaussian behaviour.\n LOCATION:https://researchseminars.org/talk/Frobenius4/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Lilian Matthiessen (KTH Royal Institute of Technology\, Sweden) DTSTART:20231005T150000Z DTEND:20231005T155000Z DTSTAMP:20260415T034322Z UID:Frobenius4/2 DESCRIPTION:Title: Distributional properties of smooth numbers: Smooth numbers are orthogo nal to nilsequences\nby Lilian Matthiessen (KTH Royal Institute of Tec hnology\, Sweden) as part of Around Frobenius Distributions and Related To pics IV\n\n\nAbstract\nAn integer is called y-smooth if all of its prime f actors are of size at most y. The y-smooth numbers below x form a subset o f the integers below x which is\, in general\, sparse but is known to enjo y good equidistribution properties in progressions and short intervals. Di stributional properties of y-smooth numbers found striking applications in \, for instance\, integer factorisation algorithms or in work of Vaughan a nd Wooley on improving bounds in Waring's problem. In this talk I will dis cuss joint work with Mengdi Wang which considers some finer aspects of the distribution of y-smooth numbers. More precisely\, we show for a very lar ge range of the parameter y that y-smooth number are (in a certain sense) discorrelated with "nilsequences". Through work of Green\, Tao and Ziegler \, our result is closely related to the Diophantine problem of studying so lutions to certain systems of linear equations in the set of y-smooth numb ers.\n LOCATION:https://researchseminars.org/talk/Frobenius4/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Lemke Oliver (Tufts University\, USA) DTSTART:20231005T170000Z DTEND:20231005T175000Z DTSTAMP:20260415T034322Z UID:Frobenius4/3 DESCRIPTION:Title: Uniform exponent bounds on the number of primitive extensions of number fields\nby Robert Lemke Oliver (Tufts University\, USA) as part of Ar ound Frobenius Distributions and Related Topics IV\n\n\nAbstract\nA folklo re conjecture asserts the existence of a constant $c_n > 0$ such that $N_n (X) \\sim c_n X$ as $X\\to \\infty$\, where $N_n(X)$ is the number of degr ee $n$ extensions $K/\\mathbb{Q}$ with discriminant bounded by $X$. This conjecture is known if $n \\leq 5$\, but even the weaker conjecture that t here exists an absolute constant $C\\geq 1$ such that $N_n(X) \\ll_n X^C$ remains unknown and apparently out of reach.\n\nHere\, we make progress on this weaker conjecture (which we term the "uniform exponent conjecture") in two ways. First\, we reduce the general problem to that of studying re lative extensions of number fields whose Galois group is an almost simple group in its smallest degree permutation representation. Second\, for alm ost all such groups\, we prove the strongest known upper bound on the numb er of such extensions. These bounds have the effect of resolving the unif orm exponent conjecture for solvable groups\, sporadic groups\, exceptiona l groups\, and classical groups of bounded rank. This is forthcoming work that grew out of conversations with M. Bhargava.\n LOCATION:https://researchseminars.org/talk/Frobenius4/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Vorrapan Chandee (Kansas State University\, USA) DTSTART:20231005T180000Z DTEND:20231005T185000Z DTSTAMP:20260415T034322Z UID:Frobenius4/4 DESCRIPTION:Title: On Benford's law for multiplicative functions\nby Vorrapan Chandee (Kansas State University\, USA) as part of Around Frobenius Distributions and Related Topics IV\n\n\nAbstract\nBenford's law is a phenomenon about t he first digits of the numbers in data sets. In particular\, the leading d igits does not exhibit uniform distribution as might be naively expected\, but rather\, the digit appears the most\, followed by \, and so on until . In this talk\, I will discuss my recent joint work with Xiannan Li\, Pau l Pollack and Akash Sigha Roy on a criterion to determine whether a real m ultiplicative function is a Benford sequence. The criterion implies that t he divisor functions and Hecke eigenvalues of newforms\, such as Ramanujan tau function\, are Benford. In contrast to earlier work\, our approach is based on Halasz's Theorem.\n LOCATION:https://researchseminars.org/talk/Frobenius4/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Anders Södergren (Chalmers University of Technology\, Sweden) DTSTART:20231006T140000Z DTEND:20231006T145000Z DTSTAMP:20260415T034322Z UID:Frobenius4/5 DESCRIPTION:Title: Non-vanishing at the central point of the Dedekind zeta functions of no n-Galois cubic fields\nby Anders Södergren (Chalmers University of Te chnology\, Sweden) as part of Around Frobenius Distributions and Related T opics IV\n\n\nAbstract\nIt is believed that for every $S_n$-number field\, i.e. every degree extension of the rationals whose normal closure has Gal ois group $S_n$\, the Dedekind zeta function is non-vanishing at the centr al point. In the case $n=2$ Soundararajan established\, in spectacular wor k improving on earlier work of Jutila\, the non-vanishing of the Dedekind zeta function for at least 87.5% of the fields in certain families of quad ratic fields. In this talk\, I will present joint work with Arul Shankar a nd Nicolas Templier\, in which we study the case $n=3$. In particular\, I will discuss some of the main ideas in our proof that the Dedekind zeta fu nctions of infinitely many $S_3$-fields have non-vanishing central value.\ n LOCATION:https://researchseminars.org/talk/Frobenius4/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Nina Zubrilina (Princeton University\, USA) DTSTART:20231006T150000Z DTEND:20231006T155000Z DTSTAMP:20260415T034322Z UID:Frobenius4/6 DESCRIPTION:Title: Root Number Correlation Bias of Fourier Coefficients of Modular Forms\nby Nina Zubrilina (Princeton University\, USA) as part of Around Frobe nius Distributions and Related Topics IV\n\n\nAbstract\nIn a recent machin e learning-based study\, He\, Lee\, Oliver\, and Pozdnyakov observed a str iking oscillating pattern in the average value of the $P$-th Frobenius tra ce of elliptic curves of prescribed rank and conductor in an interval rang e. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root numbe r. In my talk\, I will discuss this root number correlation bias when the average is taken over all weight 2 modular newforms. I will point to a sou rce of this phenomenon in this case and compute the correlation function e xactly.\n LOCATION:https://researchseminars.org/talk/Frobenius4/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Arul Shankar (University of Toronto\, Canada) DTSTART:20231006T170000Z DTEND:20231006T175000Z DTSTAMP:20260415T034322Z UID:Frobenius4/7 DESCRIPTION:Title: Frobenius equidistribution in families of number fields\nby Arul Sh ankar (University of Toronto\, Canada) as part of Around Frobenius Distrib utions and Related Topics IV\n\n\nAbstract\nI will describe the notion of Frobenius equidistribution (also called Sato--Tate equidistribution) in fa milies of $S_n$-number fields. In the fundamental case of families of -num ber fields\, this equidistribution is only known in the case $n=3$\, due t o Davenport--Heilbronn\, as well as $n=4$ and $n=5$\, due to Bhargava. Mor eover\, assuming this equidistribution\, Bhargava uses the Serre mass form ula to develop heuristics for the asymptotics of $S_n$-number fields\, whe n they are ordered by discriminant.\nI will then discuss results in two di fferent directions. In the first direction\, we consider the following que stion: what do we expect to happen when we order fields by natural invaria nts other than the discriminant? To shed some light on this\, I will descr ibe joint work with Frank Thorne in which we give a complete answer in the case of cubic fields. Second\, I will describe joint work with Jacob Tsim erman\, in which we develop heuristics which give evidence for Frobenius e quidistribution in families of all number fields.\n LOCATION:https://researchseminars.org/talk/Frobenius4/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexandre de Faveri (Institute for Advanced Study\, USA) DTSTART:20231006T180000Z DTEND:20231006T185000Z DTSTAMP:20260415T034322Z UID:Frobenius4/8 DESCRIPTION:Title: An inequality for GL(3) Fourier coefficients\nby Alexandre de Faver i (Institute for Advanced Study\, USA) as part of Around Frobenius Distrib utions and Related Topics IV\n\n\nAbstract\nWe prove a certain comparison inequality for partial sums of Fourier coefficients of Hecke-Maass cuspfor ms in GL(3). This is a higher rank generalization of a result of Soundarar ajan\, and has applications to distribution of mass in GL(3). Joint work w ith Zvi Shem-Tov.\n LOCATION:https://researchseminars.org/talk/Frobenius4/8/ END:VEVENT END:VCALENDAR