BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Bruce Berndt (University of Illinois) DTSTART:20200916T160000Z DTEND:20200916T170000Z DTSTAMP:20260423T035613Z UID:EIMINT/2 DESCRIPTION:Title: T he Circle Problem of Gauss\, the Divisor Problem of Dirichlet\, and Ramanu jan's Interest in Them\nby Bruce Berndt (University of Illinois) as pa rt of EIMI Number Theory Seminar\n\n\nAbstract\nLet $r_2(n)$ denote the nu mber of representations of the positive integer $n$ as a sum of two square s\, and let $d(n)$ denote the number of positive divisors of $n$. Gauss a nd Dirichlet were evidently the first mathematicians to derive asymptotic formulas for $\\sum_{n\\leq x}r_2(n)$ and $\\sum_{n\\leq x}d(n)$\, respect ively\, as $x$ tends to infinity. The magnitudes of the error terms for t he two asymptotic expansions are unknown. Determining the exact orders of the error terms are the Gauss Circle Problem and Dirichlet's Divisor Prob lem\, respectively\, and they represent two of the most famous and difficu lt unsolved problems in number theory.\n\nBeginning with his first letter to Hardy\, it is evident that Ramanujan had a keen interest in the Divisor Problem\, and from a paper written by Hardy and published in 1915\, shor tly after Ramanujan arrived in England\, we learn that Ramanujan was also greatly interested in the Circle Problem. In a fragment published with hi s Lost Notebook\, Ramanujan stated two doubly infinite series identities i nvolving Bessel functions that we think Ramanujan derived to attack these two famous unsolved problems. The identities are difficult to prove. Unfo rtunately\, we cannot figure out how Ramanujan might have intended to use them. We survey what is known about these two unsolved problems\, with a c oncentration on Ramanujan's two marvelous and mysterious identities. Join t work with Sun Kim\, Junxian Li\, and Alexandru Zaharescu is discussed.\n LOCATION:https://researchseminars.org/talk/EIMINT/2/ END:VEVENT END:VCALENDAR