BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Caroline Turnage-Butterbaugh (Carleton College) DTSTART:20220922T160000Z DTEND:20220922T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/1 DESCRIPTION:Title: Moments of Dirichlet L-functions\nby Caroline Turnage-Bu tterbaugh (Carleton College) as part of FRG Grad Seminar (Averages of of L -functions and Arithmetic Stratification)\n\n\nAbstract\nIn recent decades there has been much interest and measured progress in the study of moment s of L-functions. Despite a great deal of effort spanning over a century\, asymptotic formulas for moments of L-functions remain stubbornly out of r each in all but a few cases. I will begin this talk by reviewing what is k nown for moments of the Riemann zeta-function on the critical line\, and w e will then consider the problem for the family of all Dirichlet L-functio ns of even primitive characters of bounded conductor. A heuristic of Conre y\, Farmer\, Keating\, Rubenstein\, and Snaith gives a precise prediction for the asymptotic formula for the general 2kth moment of this family. I w ill outline how to harness the asymptotic large sieve to prove an asymptot ic formula for the general 2kth moment of approximations of this family. T he result\, which assumes the generalized Lindelöf hypothesis for large v alues of k\, agrees with the prediction of CFKRS. Moreover\, it provides t he first rigorous evidence beyond the so-called “diagonal terms” in th eir conjectured asymptotic formula for this family of L-functions. This is joint work with Siegfred Baluyot.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Hua Lin (UC Irvine) DTSTART:20220926T160000Z DTEND:20220926T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/3 DESCRIPTION:Title: One-level density of zeros of Dirichlet L-function over func tion fields\nby Hua Lin (UC Irvine) as part of FRG Grad Seminar (Avera ges of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nFor th is talk\, we compute the one-level density of zeros of cubic and quartic D irichlet $L$-functions over function fields $\\mathbb{F}_q[t]$ in the Kumm er setting ($q\\equiv1\\pmod{\\ell}$) and for order $\\ell=3\,4\,6$ in the non-Kummer setting ($q\\not\\equiv1\\pmod{\\ell}$). In each case\, we obt ain a main term predicted by Random Matrix Theory (RMT) and a lower order term not predicted by RMT. We also confirm the symmetry type of the family is unitary\, supporting the Katz and Sarnak philosophy. I will first talk about some history and background on the subject\, make the analogy and d escribe the primitive characters over function fields in each setting\, an d then show the computation in more detail.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Louis Gaudet (Rutgers University) DTSTART:20221003T160000Z DTEND:20221003T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/4 DESCRIPTION:Title: The least Euler prime via a sieve approach\nby Louis Gau det (Rutgers University) as part of FRG Grad Seminar (Averages of of L-fun ctions and Arithmetic Stratification)\n\nAbstract: TBA\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Lasse Grimmelt (University of Oxford) DTSTART:20221010T160000Z DTEND:20221010T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/5 DESCRIPTION:Title: Primes in large arithmetic progressions and applications to additive problems\nby Lasse Grimmelt (University of Oxford) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratificatio n)\n\n\nAbstract\nResults on the distribution of primes up to $X$ in an ar ithmetic progression with modulus $q$ fall\, depending on the relative siz e 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 f orm of Dirichlet L-functions are used\, in the medium range ($q < N^{1/2-\ \epsilon}$) the large sieve gives us the Bombieri-Vinogradov Theorem\, and finally one can handle slightly larger $q$ by bounds for sums of Klooster man sums. In this talk I will give a background about these results and hi ghlight some recent progress in the third category. I will also explain ho w this progress can be applied to additive problems involving (subsets of) the primes.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Dan Goldston (San José State University) DTSTART:20221017T160000Z DTEND:20221017T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/6 DESCRIPTION:Title: Small Gaps and Spacings between Riemann zeta-function zeros< /a>\nby Dan Goldston (San José State University) as part of FRG Grad Semi nar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstra ct\nI will discuss joint work with Hung Bui\, Micah Milinovich\, and Hugh Montgomery on differences between consecutive zeros of the Riemann zeta-fu nction that are smaller than the average spacing between zeros. We assume the Riemann Hypothesis. One result is that by using the pair correlation m ethod one can prove there is a positive proportion of consecutive zeros cl oser than 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 method may not be finding any small gaps between zeros at all - here w e require a gap between two numbers to have non-zero length because that i s what a gap is. We refer to differences between consecutive zeros includi ng differences equal to zero as “spacings”. There are three methods kn own to deal with close zeros\, and all three actually produce small spacin gs between zeros rather than small gaps. (One method is unconditional\, th e other two assume RH.) For small gaps\, or differences between distinct z eros\, the three methods only produce gaps larger than the average spacing . Our second result is based on a new fourth method that on RH proves ther e are small gaps between zeros closer than 0.991 times the average spacing between 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 difficult problem.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/6/ END:VEVENT BEGIN:VEVENT SUMMARY:George Dickinson (University of Manchester) DTSTART:20221219T170000Z DTEND:20221219T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/7 DESCRIPTION:Title: Second moments of Dirichlet L-functions\nby George Dicki nson (University of Manchester) as part of FRG Grad Seminar (Averages of o f L-functions and Arithmetic Stratification)\n\n\nAbstract\nThe asymptotic formulae for moments of L-functions are well studied objects in analytic number theory as they are useful tools when investigating the L-functions themselves. Often especially useful are the moments that have been twisted by a Dirichlet polynomial\, and the longer the twist the better. However\ , finding formulae gets more difficult as the length increases. In this ta lk\, we will compare methods for finding different types of twisted second moments of Dirichlet L-functions\, as well as looking at some of their ap plications.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Emma Bailey (CUNY) DTSTART:20221031T160000Z DTEND:20221031T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/8 DESCRIPTION:Title: Large values of $\\zeta$ on the critical line\nby Emma B ailey (CUNY) as part of FRG Grad Seminar (Averages of of L-functions and A rithmetic Stratification)\n\n\nAbstract\nSelberg’s central limit theorem tells us that typically $|\\zeta(1/2 + it)|$ is of size $\\exp(\\sqrt{\\l og \\log T})$ for $t\\in [T\, 2T]$. One can ask about /atypical/ values\, or about large deviations to Selberg’s central limit theorem. By explori ng a connection between $\\zeta$ and branching random walks\, we are able to show that the Gaussian tail extends to the right\, on the scale of the variance. In this talk I will focus on the connection to branching random walks and show how this probabilistic interpretation allows us to understa nd large values of zeta. This is based on joint work with Louis-Pierre Arg uin.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Daodao Yang (TU Graz) DTSTART:20221107T170000Z DTEND:20221107T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/9 DESCRIPTION:Title: Large values of derivatives of the Riemann zeta function and related problems\nby Daodao Yang (TU Graz) as part of FRG Grad Semina r (Averages 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 and Landau in 1910. Resonance methods are modern tools to produce lar ge values of zeta and L-functions. GCD sums are one of important ingredien ts\, which naturally appears in a Diophantine approximation problem consid ered by Hardy and Littlewood in 1922. I will talk on producing large value s of derivatives of zeta and L-functions via resonance methods. On the oth er hand\, I will talk on conditional upper bounds and asymptotic formulas when assuming RH (GRH) and a conjecture of Granville-Soundararajan on char acter sums. If time permits\, the log-type GCD sums and related spectral n orms will be discussed.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Jakob Streipel (University of Maine) DTSTART:20221114T160000Z DTEND:20221114T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/10 DESCRIPTION:Title: Using second moments to count zeros\nby Jakob Streipel (University of Maine) as part of FRG Grad Seminar (Averages of of L-functi ons and Arithmetic Stratification)\n\n\nAbstract\nUsing Selberg's somewhat strange looking version of the argument principle\, it is possible to cou nt zeros of families of L-functions using upper bounds on second moments. We will explore this argument principle\, how one uses it\, and some appli cations of it to various zero counting problems\, old and new.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Dunn (Caltech) DTSTART:20221121T170000Z DTEND:20221121T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/11 DESCRIPTION:Title: Bias in cubic Gauss sums: Patterson's conjecture\nby Al exander Dunn (Caltech) as part of FRG Grad Seminar (Averages of of L-funct ions and Arithmetic Stratification)\n\n\nAbstract\nWe prove\, in this join t work with Maksym Radziwill\, a 1978 conjecture of S. Patterson (conditi onal on the Generalised Riemann hypothesis) concerning the bias of cubic G auss sums. This explains a well-known numerical bias in the distribution o f cubic Gauss sums first observed by Kummer in 1846.\n\nOne important byp roduct of our proof is that we show Heath-Brown's cubic large sieve is sha rp under GRH. This disproves the popular belief that the cubic large siev e can be improved.\n\nAn important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large si eve with correction by a non-trivial asymptotic main term.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Aled Walker (King's College\, London) DTSTART:20221128T170000Z DTEND:20221128T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/12 DESCRIPTION:Title: Correlations of sieve weights and distributions of zeros\nby Aled Walker (King's College\, London) as part of FRG Grad Seminar (A verages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn this talk\, we will briefly review Montgomery's pair correlation conject ure for the zeros of the Riemann zeta function\, before discussing a (cond itional) partial lower bound on the Fourier transform of this pair correla tion function: the so-called 'form factor' $F_T(x)$. The methods\, based i n part on ideas of Goldston and Gonek\, utilise some new correlation estim ates for Selberg sieve weights.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Asif Zaman (University of Toronto) DTSTART:20221205T170000Z DTEND:20221205T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/13 DESCRIPTION:Title: Random multiplicative functions and a simplified model\ nby Asif Zaman (University of Toronto) as part of FRG Grad Seminar (Averag es of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nOver th e past few years\, there has been a lot of interest in random multiplicati ve functions and their partial sums. This subject has many intriguing ques tions and connections to other areas of number theory and probability. In joint work with Soundararajan\, we have introduced a simplified model of p artial sums of random multiplicative functions and established a result pa rallel to Harper’s breakthrough on better-than-squareroot cancellation. In this expository talk\, I will review some of the history of random mult iplicative functions\, and illustrate how random multiplicative functions connect to our simplified model.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Dobner (University of Michigan) DTSTART:20221212T170000Z DTEND:20221212T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/14 DESCRIPTION:Title: Optimization and moment methods in number theory\nby Al exander Dobner (University of Michigan) as part of FRG Grad Seminar (Avera ges of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nA comm on technique in analytic number theory is to turn a number theoretic probl em into some sort of optimization problem which is hopefully more tractabl e. A well known example is the Selberg sieve method which turns classical sieving problems into a quadratic optimization problem. This technique al so appears in conjunction with the so-called moment method from probabilit y theory. In this talk I'll summarize several instances of this including finding primes in bounded intervals\, finding small/large gaps between zet a zeros\, and finding large values of Dirichlet series.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Nathan Ng (University of Lethbridge) DTSTART:20230123T170000Z DTEND:20230123T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/15 DESCRIPTION:Title: The eighth moment of the Riemann zeta function\nby Nath an Ng (University of Lethbridge) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn recent wor k (https://arxiv.org/abs/2204.13891)\, Quanli Shen\, Peng-Jie Wong\, and I have shown that the Riemann hypothesis and a conjecture for quaternary ad ditive divisor sums implies the conjectured asymptotic for the eighth mome nt of the Riemann zeta function. This builds on earlier work on the sixth moment of the Riemann zeta function (Ng\, Discrete Analysis\, 2021). One key difference is that sharp bounds for shifted moments of the zeta functi on on the critical line are required. In this talk\, I will discuss some of the ideas that go into the proof.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/15/ END:VEVENT BEGIN:VEVENT SUMMARY:David Farmer (American Institute of Mathematics) DTSTART:20230130T170000Z DTEND:20230130T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/16 DESCRIPTION:Title: The zeta function when it is particularly large\nby Dav id Farmer (American Institute of Mathematics) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\n What does the zeta function look like in a neighborhood of its largest val ues? Nobody knows for sure\, because particularly large values have never been computed. We will give a plausible answer by combining theorems fro m analytic number theory\, first principles reasoning\, and examples of ra ndom characteristic polynomials.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Matilde Lalín (Université de Montréal) DTSTART:20230417T160000Z DTEND:20230417T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/17 DESCRIPTION:Title: The distribution of values of cubic $L$-functions at $s=1$< /a>\nby Matilde Lalín (Université de Montréal) as part of FRG Grad Semi nar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstra ct\nWe investigate the distribution of values of cubic Dirichlet $L$-funct ions at $s=1$. Following ideas of Granville and Soundararajan\, and Dahl a nd Lamzouri for quadratic $L$-functions\, we model values of $L(1\,\\chi)$ with the distribution of random Euler products $L(1\,\\mathbb{X})$ for ce rtain family of random variables $\\mathbb{X}(p)$ attached to each prime. We obtain a description of the proportion of $|L(1\,\\chi)|$ that are larg er or that are smaller than a given bound\, and yield more light into the Littlewood bounds. Unlike the quadratic case\, there is a clear asymmetry between lower and upper bounds for the cubic case.\n\nThis is joint work w ith Pranendu Darbar\, Chantal David\, and Allysa Lumley.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Keshav Aggarwal (Alfréd Rényi Institute of Mathematics) DTSTART:20230206T170000Z DTEND:20230206T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/19 DESCRIPTION:Title: Bound for the existence of prime gap graphs\nby Keshav Aggarwal (Alfréd Rényi Institute of Mathematics) as part of FRG Grad Sem inar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstr act\nGiven a sequence $\\mathbf{D}$ of non-negative integers\, it is inter esting to know whether there exists a graph with vertices of degrees equal ing the integers in $\\mathbf{D}$. If that happens\, we say $\\mathbf{D}$ is graphic. Clearly\, if the sequence is graphic\, then the sum of its mem bers must be even. However\, it is not self-evident whether a given sequen ce is graphic. There are exponentially many different realizations for alm ost every graphic degree sequence. At the same time\, the number of all gr aphic degree sequences is infinitesimal compared to the number of integer partitions of the sum of the degrees. Therefore it is incredibly hard to c ome up with an interesting (or non-trivial) graphic degree sequence.\n\nLe t us call a simple graph on $n>2$ vertices a prime gap graph if its vertex degrees 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 large enough $n$. In a joint work with Robin Frot\, we make their wor k effective.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Rizwanur Khan (University of Mississippi) DTSTART:20230213T170000Z DTEND:20230213T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/20 DESCRIPTION:Title: The fourth moment of Dirichlet L-functions and related prob lems\nby Rizwanur Khan (University of Mississippi) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nA bstract\nI will discuss asymptotics for the fourth moment of Dirichlet L-f unctions and related problems\, especially with regards to simplifying exi sting approaches and sharpening the error terms in these asymptotics. This is joint work with Zeyuan Zhang.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Katy Woo (Princeton University) DTSTART:20230220T170000Z DTEND:20230220T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/21 DESCRIPTION:Title: Small scale distribution of primes in four-term arithmetic progressions\nby Katy Woo (Princeton University) as part of FRG Grad S eminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbs tract\nIn 1985\, Maier demonstrated that there are short intervals with ex ceptionally large or small numbers of primes. In this talk\, I will discus s adapting Maier's matrix method to look at the small scale distribution o f primes in three-term and four-term arithmetic progressions. I aim to hig hlight the similarities and differences in the proofs for the two cases\; the former uses the classical circle method\, whereas the latter requires tools from ergodic theory. This is based on joint work with Mayank Pandey. \n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Ian Whitehead (Swarthmore College) DTSTART:20230313T160000Z DTEND:20230313T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/22 DESCRIPTION:Title: Multiple Dirichlet Series and Moments of L-functions\nb y Ian Whitehead (Swarthmore College) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nWeyl grou p multiple Dirichlet series are multivariable analogues of Dirichlet L-fun ctions. Their meromorphic continuation leads to asymptotics for moments in families of L-functions\, most notably the family of quadratic Dirichlet L-functions. In this talk I will present work of Diaconu-Goldfeld-Hoffstei n and Chinta-Gunnells which constructs multiple Dirichlet series associate d with various moments of quadratic L-functions. There is an important dis tinction between series with finite groups of functional equations\, where meromorphic continuation is proven\, and series with infinite groups of f unctional equations\, where it is an open question. If time permits\, I wi ll discuss work of Diaconu-Pașol\, Sawin\, and myself which takes an axio matic approach to defining these series.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Peter Zenz (Brown University) DTSTART:20230320T160000Z DTEND:20230320T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/23 DESCRIPTION:Title: On the Distribution of Holomorphic Cusp Forms and Applicati ons\nby Peter Zenz (Brown University) as part of FRG Grad Seminar (Ave rages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nArit hmetic Quantum Chaos (AQC) is an active area of research at the intersecti on of number theory and physics. One major goal in AQC is to study the mas s distribution of Hecke Maass cusp forms on hyperbolic surfaces as the Lap lace eigenvalue tends to infinity. In this talk we will focus on analogous questions for holomorphic Hecke cusp forms. We review solved and open con jectures in this direction\, like the Quantum Unique Ergodicity Conjecture and the Random Wave Conjecture. We then divert our attention to similar q uestions\, when restricted to certain subsets of the fundamental domain. F inally\, we elaborate on how to use some of the mentioned distribution res ults to detect real zeros of holomorphic cusp forms low in the fundamental domain.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Aleksander Simonič (UNSW Canberra) DTSTART:20230410T160000Z DTEND:20230410T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/24 DESCRIPTION:Title: Some conditional estimates for functions in the Selberg cla ss\nby Aleksander Simonič (UNSW Canberra) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\ nIn this talk I will present recent progress in obtaining conditional (GRH ) estimates for $(L'/L)(s)$ and $\\log{L(s)}$\, when $L$ is an element of the Selberg class of functions and $s$ is not too close to the critical li ne. We are able to obtain effective results while assuming the strong $\\l ambda$-conjecture and a polynomial Euler product representation for $L$. I f time permits\, I will also briefly touch on similar results for $s$ bein g close to the critical line. This is a joint work with N. Palojärvi.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Eun Hye Lee (Stony Brook University) DTSTART:20230424T160000Z DTEND:20230424T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/25 DESCRIPTION:Title: The Shintani Zeta Functions\nby Eun Hye Lee (Stony Broo k University) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nCounting number fields is a cent ral interest in number theory. In this talk\, I will introduce Shintani ze ta functions\, the counting functions for the number of cubic fields\, and survey some of the results on them. Time permitting\, I will also discuss some key points of some of the proofs.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Xiannan Li (Kansas State University) DTSTART:20230227T170000Z DTEND:20230227T180000Z DTSTAMP:20260404T095029Z 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-fu nctions and Arithmetic Stratification)\n\n\nAbstract\nThe behavior of quad ratic twists of modular L-functions is at the critical point is related bo th to coefficients of half integer weight modular forms and data on ellipt ic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions\, previously available conditionally on the Generalized Riemann Hypothesis by the work of Soundararajan and Young. Ou r proof depends on deriving an optimal large sieve type bound.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Kiseok Yeon (Purdue University) DTSTART:20230501T160000Z DTEND:20230501T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/27 DESCRIPTION:Title: The Hasse principle for random projective hypersurfaces via the circle method\nby Kiseok Yeon (Purdue University) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\ n\nAbstract\nIn this talk\, we introduce a framework via the circle method in order to confirm the Hasse principle for random projective hypersurfac es over $\\mathbb{Q}$. In particular\, we mainly give a motivation for dev eloping this framework by providing the overall history of the problems of confirming the Hasse principle for projective hypersurfaces over $\\mathb b{Q}$. 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 introduce an auxiliary mean value theorem which plays a crucial role i n our argument and may be of independent interest.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Jaime Hernandez Palacios (University of Mississippi) DTSTART:20230529T160000Z DTEND:20230529T170000Z DTSTAMP:20260404T095029Z 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 Seminar (Averages of of L-functions and Arithmetic Stratification)\n \n\nAbstract\nThere is a great deal of evidence\, both theoretical and exp erimental\, 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 small normalized gaps between the ordinates of (high) zeros zeta and L-functions. Previous results are known for zeta and L-functions of degree s 1 and 2. We discuss some new results for higher degrees\, including Dede kind zeta-functions associated to Galois extensions of the rational number s and principal automorphic L-functions.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandro Fazzari (American Institute of Mathematics) DTSTART:20230508T160000Z DTEND:20230508T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/29 DESCRIPTION:Title: Averages of long Dirichlet polynomials with modular coeffic ients\nby Alessandro Fazzari (American Institute of Mathematics) as pa rt of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratifi cation)\n\n\nAbstract\nWe study the moments of L-functions associated with primitive cusp forms\, in the weight aspect. In particular\, we present r ecent joint work with Brian Conrey\, where we obtain an asymptotic formula for the twisted $r$th moment of a long Dirichlet polynomial approximation of such L-functions. This result\, which is conditional on the Generalize d Lindelöf Hypothesis\, agrees with the prediction of the recipe by Conre y\, Farmer\, Keating\, Rubinstein and Snaith.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Ofir Gorodetsky (University of Oxford) DTSTART:20230310T170000Z DTEND:20230310T180000Z DTSTAMP:20260404T095029Z 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 (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\n Smooth numbers are integers whose prime factors are all small (smaller tha n some threshold $y$). In the 80s they became important outside of pure ma th\, because 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-differ ential equation. \nAll of the above is also true for smooth polynomials\, which are defined similarly and have practical applications.\n\nWe'll surv ey these topics and discuss recent results whose proofs rely on relating t he number of smooth numbers to the Riemann zeta function and its zeros.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Emily Quesada-Herrera (Technische Universität Graz) DTSTART:20230605T160000Z DTEND:20230605T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/32 DESCRIPTION:Title: On the vertical distribution of the zeros of the Riemann ze ta-function\nby Emily Quesada-Herrera (Technische Universität Graz) a s part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stra tification)\n\n\nAbstract\nIn 1973\, assuming the Riemann hypothesis (RH)\ , Montgomery studied the vertical distribution of zeta zeros\, and conject ured that they behave like the eigenvalues of some random matrices. We wil l discuss some models for zeta zeros – starting from the random matrix m odel but going beyond it – and related questions\, conjectures and resul ts on statistical information on the zeros. In particular\, assuming RH an d a conjecture of Chan for how often gaps between zeros can be close to a fixed non-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 ma trix models alone do not accurately predict the actual behavior (based on joint work with Meghann Moriah Lugar and Micah B. Milinovich).\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Vorappan (Fai) Chandee (Kansas State University) DTSTART:20230515T160000Z DTEND:20230515T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/33 DESCRIPTION:Title: The eighth moment of $\\Gamma_1(q)$ L-functions\nby Vor appan (Fai) Chandee (Kansas State University) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\n In this talk\, I will discuss my joint work with Xiannan Li on an uncondit ional asymptotic formula for the eighth moment of $\\Gamma_1(q)$ L-functio ns\, which are associated with eigenforms for the congruence subgroups $\\ Gamma_1(q)$. Similar to a large family of Dirichlet L-functions\, the fami ly of $\\Gamma_1(q)$ L-functions has a size around $q^2$ while the conduct or is of size $q$. An asymptotic large sieve of the family is available b y the work of Iwaniec and Xiaoqing Li\, and they observed that this family of harmonics is not perfectly orthogonal. This introduces certain subtlet ies in our work.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Igor Shparlinski (UNSW\, Sydney) DTSTART:20230522T160000Z DTEND:20230522T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/34 DESCRIPTION:Title: Bilinear forms with Kloosterman and Salie Sums and Moments of L-functions\nby Igor Shparlinski (UNSW\, Sydney) as part of FRG Gra d Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\n Abstract\nWe present some recent results on bilinear forms with complete a nd incomplete Kloosterman and Salie sums. These results are of independent interest and also play a major role in bounding error terms in asymptotic formulas for moments of various L-functions. We then describe several res ults about non-correlation of Kloosterman and Salie sums between themselve s and also with some classical number-theoretic functions such as the Mobi us function\, the divisor function and the sum of binary digits\, etc. Som e open problems will be outlined as well.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Mayank Pandey (Princeton University) DTSTART:20230918T160000Z DTEND:20230918T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/35 DESCRIPTION:Title: $L^1$ means of exponential sums with multiplicative coeffic ients\nby Mayank Pandey (Princeton University) as part of FRG Grad Sem inar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstr act\nWe discuss some recent results on the $L^1$ norm of exponential sums with multiplicative functions\, with specific results for the Mobius and L iouville functions. Joint work with Maksym Radziwill.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Henryk Iwaniec (Rutgers University) DTSTART:20230925T160000Z DTEND:20230925T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/36 DESCRIPTION:Title: Integer Parts Mutually Coprime\nby Henryk Iwaniec (Rutg ers University) as part of FRG Grad Seminar (Averages of of L-functions an d Arithmetic Stratification)\n\n\nAbstract\nOne of the problems which is t he subject of this talk concerns the density of integers $n$ for which the integer parts $[(n+i)^c]$ are pairwise coprime with $i=1\,...\,k$. Here $ c$ is any constant\, $1Short Second Moment Bound for GL(2) L-functions in the Leve l Aspect\nby Agniva Dasgupta (Texas A&M University) as part of FRG Gra d Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\n Abstract\nWe will discuss my recent work on moments of $L$-functions at th e central point. Early results in this area were concerned with 'full' mom ents\, by studying expressions like $\\int_{T}^{2T} \\left \\vert L(f\,\\f rac12+it) \\right \\vert^k dt$\, or $\\sum_{\\chi (\\text{mod }q)} \\left \\vert L(f\\otimes \\chi\, \\frac12) \\right \\vert^k$. A 1978 paper of Iw aniec proved a Lindelöf-on-average upper bound on a 'short' fourth moment \, by showing $\\int_{T}^{T+T^{\\frac23}} \\left \\vert{\\zeta(\\frac12+it )} \\right \\vert ^4 \\ll T^{\\frac23 + \\varepsilon}$. Good (1982) proved a similar upper bound for a short (second) moment for level 1 cusp forms. We prove a level-aspect analogue to Good's result. We assume $q=p^3$ fo r an odd prime $p$\, and for the short second moment\, we consider the twi sts of a level 1 cusp form along a coset of subgroup of the characters mod ulo $q^{\\frac23}$.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Winston Heap (Norwegian University of Science and Technology (NTNU )) DTSTART:20231023T160000Z DTEND:20231023T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/38 DESCRIPTION:Title: Simultaneous extreme values of zeta and L-functions\nby Winston Heap (Norwegian University of Science and Technology (NTNU)) as p art of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratif ication)\n\n\nAbstract\nWe use a modification of the resonance method to d emonstrate simultaneous large values of L-functions on the critical line. The method extends to other families and can be used to show both simultan eous large and small values. Joint work with Junxian Li.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Curran (University of Oxford) DTSTART:20231106T170000Z DTEND:20231106T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/39 DESCRIPTION:Title: Correlations of the Riemann zeta function\nby Michael C urran (University of Oxford) as part of FRG Grad Seminar (Averages of of L -functions and Arithmetic Stratification)\n\n\nAbstract\nShifted moments o f the Riemann zeta function\, introduced by Chandee\, are natural generali zations of the moments of zeta. While the moments of zeta capture large va lues of zeta\, the shifted moments capture how the values of zeta are corr elated along the half line. I will describe recent and forthcoming work gi ving sharp bounds for shifted moments assuming the Riemann hypothesis\, im proving previous work of Chandee and Ng\, Shen\, and Wong.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Julia Stadlmann (University of Oxford) DTSTART:20231113T170000Z DTEND:20231113T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/40 DESCRIPTION:Title: Primes in arithmetic progressions to smooth moduli\nby Julia Stadlmann (University of Oxford) as part of FRG Grad Seminar (Averag es of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nThe twi n prime conjecture asserts that there are infinitely many primes $p$ for w hich $p+2$ is also prime. This conjecture appears far out of reach of curr ent mathematical techniques. However\, in 2013\, Zhang achieved a breakthr ough\, showing that there exists some positive integer $h$ for which $p$ a nd $p+h$ are both prime infinitely often. Equidistribution estimates for p rimes in arithmetic progressions to smooth moduli were a key ingredient of his work.\n\nIn this talk\, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinemen t of the q-van der Corput method can be used to improve on equidistributio n estimates of the Polymath project for primes in APs to smooth moduli.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Steve Lester (King's College London) DTSTART:20231120T170000Z DTEND:20231120T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/41 DESCRIPTION:Title: Around the Gauss circle problem\nby Steve Lester (King' s College London) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nHardy conjectured that the e rror term arising from approximating the number of lattice points lying in a radius-$R$ disc by its area is $O(R^{1/2+o(1)})$. One source of support for this conjecture is a folklore heuristic that uses i.i.d. random varia bles to model the lattice points lying near the boundary and square-root c ancellation of sums of these random variables. In this talk I will examine this heuristic and discuss how lattice points near the circle interact wi th one another. This is joint work with Igor Wigman.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael Yiasemides (London School of Economics) DTSTART:20231127T170000Z DTEND:20231127T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/42 DESCRIPTION:Title: Lattice Points in Function Fields\, and Hankel Matrices.\nby Michael Yiasemides (London School of Economics) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAb stract\nIn this talk we discuss the function field analogue of lattice poi nts in thin elliptic annuli. We will begin with a general introduction to lattice points in the classical setting\, including briefly highlighting c onnections to physics and various areas of number theory\; before introduc ing the function field analogue and stating our results on the mean and va riance of lattice points in elliptic annuli. Our approach is unique to the function field setting\, and it translates the problem to one involving H ankel matrices over finite fields. We will summarise this approach\, befor e finishing by highlighting some interesting connections between Hankel ma trices and number theory in function fields.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrés Chirre (Pontificia Universidad Católica del Perú) DTSTART:20231211T170000Z DTEND:20231211T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/44 DESCRIPTION:Title: Remarks on a formula of Ramanujan\nby Andrés Chirre (P ontificia Universidad Católica del Perú) as part of FRG Grad Seminar (Av erages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn this talk\, we will discuss a well-known formula of Ramanujan and its rela tionship with the partial sums of the Möbius function. Under some conject ures\, we analyze a finer structure of the involved terms. It is a joint w ork with Steven M. Gonek.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Maxim Gerspach (KTH Royal Institute of Technology) DTSTART:20231218T170000Z DTEND:20231218T180000Z DTSTAMP:20260404T095029Z 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 G rad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n \nAbstract\nIn this talk\, I will discuss heuristics for low moments of qu adratic character sums\, i.e. for low powers (between 0 and 2) of quadrati c character sums averaged over the conductor. I will begin by talking abou t the rational setting and then go over to the function field setting. The se heuristics are backed up by rigorous results in a random model that I w ill describe. Moreover\, I will touch upon the extent to which these heuri stics can be made rigorous in the deterministic setting.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Sacha Mangerel (Durham University) DTSTART:20231030T160000Z DTEND:20231030T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/48 DESCRIPTION:Title: Large order Dirichlet characters and an analogue of a conje cture of Vinogradov\nby Sacha Mangerel (Durham University) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification )\n\n\nAbstract\nLet $q$ be a large prime. According to an old and famous conjecture 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 re sults\, due essentially to Burgess\, are satisfactory only when $c > 1/4$\ , due to the potential obstruction\, difficult to rule out\, that $(n/q) = +1$ for "many" initial integers $n$. \n\nIn this talk I will discuss a ge neralisation of Vinogradov's conjecture to other primitive Dirichlet chara cters $\\chi$ modulo $q$\, seeking the least $n$ for which $\\chi(n)$ is n ot $1$. I will explain some recent work of mine that shows\, using techniq ues from elementary additive combinatorics\, that when the order $d$ of $\ \chi$ is a prime that grows with $q$:\n\ni) the aforementioned obstruction does not occur\, \nii) the analogue of Vinogradov's conjecture for $\\chi $ does hold\, and moreover \niii) for each $c > 0$ and $q$ large enough wi th respect to $c$\, $\\chi(n) = 1$ occurs rarely when $n < q^c$. \n\nThese results are connected with averaged cancellation in short sums of $\\chi$ over $n < q^c$ for arbitrarily small $c > 0$\, going beyond Burgess' esti mate.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Vivian Kuperberg (ETH Zürich) DTSTART:20240226T170000Z DTEND:20240226T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/49 DESCRIPTION:Title: Sums of odd-ly many fractions\nby Vivian Kuperberg (ETH Zürich) as part of FRG Grad Seminar (Averages of of L-functions and Arit hmetic Stratification)\n\n\nAbstract\nIn this talk\, I will discuss new bo unds on constrained sets of fractions. Specifically\, I will discuss the a nswer to the following question\, which arises in multiple areas of number theory: for an integer $k \\ge 2$\, consider the set of $k$-tuples of red uced fractions $\\frac{a_1}{q_1}\, \\dots\, \\frac{a_k}{q_k} \\in I$\, whe re $I$ is an interval around $0$.\nHow many $k$-tuples are there with $\\s um_i \\frac{a_i}{q_i} \\in \\mathbb Z$?\n\nWhen $k$ is even\, the answer i s well-known: the main contribution to the number of solutions comes from "diagonal'' 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 prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of th e distributions of primes and reduced residues.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexandre de Faveri (Stanford University) DTSTART:20240212T170000Z DTEND:20240212T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/51 DESCRIPTION:Title: An inequality for GL(3) Fourier coefficients\nby Alexan dre de Faveri (Stanford University) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nWe prove a certain 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://stable.researchseminars.org/talk/frggradseminar/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Jackie Voros (University of Bristol) DTSTART:20240219T170000Z DTEND:20240219T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/52 DESCRIPTION:Title: On the average least negative Hecke eigenvalue\nby Jack ie Voros (University of Bristol) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nIn this talk we discuss the first sign change of Fourier coefficients of newforms\, or equivalently Hecke eigenvalues. We will see this to be an analogue of the least quadratic non-residue problem\, of which the average was investigate d by Erdős in 1961. In fact\, we will see that the average least negative prime Hecke eigenvalue holds the same (finite) value as the average least quadratic non-residue\, under GRH. This is mainly due to the fact that He cke eigenvalues at primes are equidistributed with respect to the Sato-Tat e measure\, a consequence of the Sato-Tate conjecture that was proven in 2 011. We further explore the so-called vertical Sato-Tate conjecture to sho w the average least Hecke eigenvalue has a finite value unconditionally.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Prahlad Sharma (Max Planck Institute for Mathematics\, Bonn) DTSTART:20240318T160000Z DTEND:20240318T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/53 DESCRIPTION:Title: Counting special points on quadratic surfaces\nby Prahl ad Sharma (Max Planck Institute for Mathematics\, Bonn) as part of FRG Gra d Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\n Abstract\nWe show how the modern versions of the circle method can be comb ined with the equidistribution of quadratic roots\, allowing us to count s pecial points on quadratic surfaces. For example\, we will obtain asymptot ic for integer points on general quadratic surfaces with prime coordinates and in short intervals.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Trevor Wooley (Purdue University) DTSTART:20240129T170000Z DTEND:20240129T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/54 DESCRIPTION:Title: Primes as sums of k-th powers\, and Freiman's theorem\n by Trevor Wooley (Purdue University) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nSuppose t hat one seeks to apply the circle method to the problem of representing a large integer n as the sum of a prime number and a number of k-th powers. The Weyl sum over the prime is small on a set of minor arcs\, but the comp lementary set of major arcs is incompatible with conventional technology f or handling the corresponding Weyl sums over the k-th powers. In this talk we explain progress on this problem that delivers conclusions with only s lightly more than 2k of these k-th powers. The key idea is to obtain parti al information concerning moments on minor arcs of large height well beyon d the conventional range accessible to Poisson summation. Similar ideas yi eld progress on such problems as that of Freiman concerning sums of mixed powers. This is work joint with Joerg Bruedern.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Seth Hardy (University of Warwick) DTSTART:20240304T170000Z DTEND:20240304T180000Z DTSTAMP:20260404T095029Z UID:frggradseminar/55 DESCRIPTION:Title: Bounds for exponential sums with random multiplicative coef ficients\nby Seth Hardy (University of Warwick) as part of FRG Grad Se minar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbst ract\nThe study of exponential sums with multiplicative coefficients is cl assical in analytic number theory\, yet our understanding of them is far f rom complete. This is unsurprising\, seeing as multiplicative functions al one are often difficult objects to grasp. However\, in recent years\, our understanding of random multiplicative functions has flourished\, and pion eering work has been conducted by Benatar\, Nishry\, and Rodgers to uncove r how exponential sums behave when their coefficients are given by random multiplicative functions. In this talk\, we will introduce random multipli cative functions\, discuss some of the literature surrounding them\, and o utline recent work on conjecturally sharp lower bounds for the maximum siz e of exponential sums involving them.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Quanli Shen (Shandong University\, Weihai) DTSTART:20240325T160000Z DTEND:20240325T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/56 DESCRIPTION:Title: The fourth moment of quadratic Dirichlet L-functions II \nby Quanli Shen (Shandong University\, Weihai) as part of FRG Grad Semina r (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract \nI will discuss the fourth moment of quadratic Dirichlet L-functions wher e we prove an asymptotic formula with four main terms unconditionally. Pre viously the asymptotic formula was established with the leading main term under the generalized Riemann hypothesis. This work is based on Li's rece nt work on the second moment of quadratic twists of modular L-functions. I t is joint work with Joshua Stucky.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Steve Gonek (University of Rochester) DTSTART:20240409T160000Z DTEND:20240409T170000Z DTSTAMP:20260404T095029Z 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 ( Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nW e prove that Dirichlet $L$-functions corresponding to Dirichlet character s for $\\mathbb{F}_{q}[x]$ with $q$ odd are universal in the following sen se. 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 prod ucts of a fixed set of prime polynomials $Q_1\, Q_2\, \\ldots\, Q_r \\in \\mathbb F_q[x]$. 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 mat rices\nby Siegfred Baluyot (American Institute of Mathematics) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratifica tion)\n\n\nAbstract\nIn the late 90's\, Keating and Snaith used random mat rix theory to predict the exact leading terms of conjectural asymptotic fo rmulas for all integral moments of the Riemann zeta-function. Prior to the ir work\, no number-theoretic argument or heuristic has led to such exact predictions for all integral moments. In 2015\, Conrey and Keating revisit ed the approach of using divisor sum heuristics to predict asymptotic form ulas for moments of zeta. Essentially\, their method estimates moments of zeta using lower twisted moments. In this talk\, I will discuss a rigorous random matrix theory analogue of the Conrey-Keating heuristic. This is on going joint work with Brian Conrey.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Ertan Elma (University of Lethbridge) DTSTART:20240311T160000Z DTEND:20240311T170000Z DTSTAMP:20260404T095029Z 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://stable.researchseminars.org/talk/frggradseminar/59/ END:VEVENT BEGIN:VEVENT SUMMARY:Jonathan Keating (University of Oxford) DTSTART:20240416T160000Z DTEND:20240416T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/60 DESCRIPTION:Title: Joint Moments\nby Jonathan Keating (University of Oxfor d) as part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n\nAbstract\nI will discuss the evaluation of the joint moments of the characteristic polynomials of random unitary matrices and t heir derivatives\, and in this context the joint moments of the Riemann ze ta-function and its derivates\, on the critical line.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/60/ END:VEVENT BEGIN:VEVENT SUMMARY:Ertan Elma (University of Lethbridge) DTSTART:20240401T160000Z DTEND:20240401T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/61 DESCRIPTION:Title: A discrete mean value of the Riemann zeta function and its derivatives\nby Ertan Elma (University of Lethbridge) as part of FRG G rad Seminar (Averages of of L-functions and Arithmetic Stratification)\n\n \nAbstract\nIn this talk\, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Diri chlet polynomials. Assuming the Riemann Hypothesis\, we obtain a lower bou nd for the 2kth moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kü bra Benli and Nathan Ng.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/61/ END:VEVENT BEGIN:VEVENT SUMMARY:Brian Conrey (American Institute of Mathematics) DTSTART:20240429T160000Z DTEND:20240429T170000Z DTSTAMP:20260404T095029Z UID:frggradseminar/62 DESCRIPTION:Title: Averages of L-functions and arithmetic stratification: a re port on the FRG\nby Brian Conrey (American Institute of Mathematics) a s part of FRG Grad Seminar (Averages of of L-functions and Arithmetic Stra tification)\n\n\nAbstract\nI will give an update on some of the work that has been done since the FRG started.\n LOCATION:https://stable.researchseminars.org/talk/frggradseminar/62/ END:VEVENT END:VCALENDAR