Tuesdays, 1 pm - 2 pm, Room 159 Altgeld Hall

This seminar is aimed primarily at the participants of the REU Number Theory Program, but anyone is welcome to attend. The talks should be of interest to local graduate students. In the seminar, local number theory faculty (and perhaps also graduate students) will give accessible talks on topics related to their research interests. In addition, later in the summer, some the REU participants themselves will give talks describing their own research. If necessary, a second weekly session will be added to accommodate these talks.

**Tuesday, June 12, 2 - 3 pm, 145 AH:**Professor Harold Diamond, Elementary methods in prime number theory**Tuesday, June 19, 2 - 3 pm, 145 AH:**Professor Heini Halberstam, Hardy-Ramanujan's theorem on the number of prime factors of an integer**Tuesday, June 26, 1 - 2 pm, 159 AH:**Professor Bruce Reznick, An Introduction to the Stern sequence.

**Abstract:**The Stern sequence (s(n)) is defined by: s(0) = 0, s(1) = 1; and s(2n) = s(n), s(2n+1) = s(n) + s(n+1), for n=2,3,.... This sequence has a surprising range of properties, applications and generalizations.**Tuesday, July 3, 1 - 2 pm, 159 AH:**Professor Ken Stolarsky, Beatty sequences.**Tuesday, July 10, 1 - 2 pm, 159 AH:**Professor Bruce Berndt, The Rogers-Ramanujan Continued Fraction.

**Abstract:**The Rogers-Ramanujan continued fraction was independently introduced by the English mathematician L. J. Rogers in 1894 and India's most famous mathematician Srinivasa Ramanujan early in the last century. Its value at 1 is the golden ratio, but many further explicit values are known, and they are often "beautiful radicals" (in the non-political sense). We will explain what continued fractions are, why they are interesting, and give a survey of many of the beautiful properties of the Rogers-Ramanujan continued fraction.**Tuesday, July 17, 1 - 2 pm, 159 AH:**Dr. Will Galway, Memory-efficient sieving, Dirichlet's divisor problem, and related problems

**Abstract:**We describe a new sieving algorithm that efficiently sieves for primes near $x$ using only $O(x^{1/3})$ bits of memory. Previously known algorithms have required roughly $x^{1/2}$ bits. The new algorithm is based on ideas used in Voronoi's derivation of an $O(x^{1/3} \ln x)$ error bound for the Dirichlet divisor problem.**Tuesday, July 24, 1 - 2 pm, 159 AH:**Professor Vishma Dumir, Minkowski's fundamental theorem and its applications**Tuesday, July 31, 1 - 2 pm, 159 AH:**Professor Paul Bateman, Representations of integers as sums of squares**Tuesday, July 31, 2 pm - 3 pm, 159 AH: Student Presentations, I:****2:00:**Evan Borenstein, An analog of the Rudin-Shapiro sequence**2:20:**David Dueber, Billiard problems and Farey fractions**2:40:**Alan Haynes, Farey fractions with odd denominators**Thursday, August 2, 2 pm - 3 pm, 159 AH: Student Presentations, II:****2:00:**Rich Astudillo, Sign changes in block product sequences**2:20:**David Weaver and Jason You, Numerical investigations on two conjectures of Gelfond**2:40:**Michael Comerford, Farey fractions with restricted denominators

*Last modified: Sun Jul 29 18:01:49 2001
ajh@uiuc.edu
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