South East Regional Meeting On Numbers -- SERMON 2007
Abstracts of Talks
Nigel Boston: Factoring iterates mod pWe observe and explain patterns in the form of the factorizations of iterates of a given polynomial modulo a prime. Using these and other tools we describe their Galois groups which act naturally on certain rooted trees.
Matthew Boylan: Non-vanishing of weakly holomorphic modular form coefficients modulo l and applicationsLet f(z) be a half-integral weight modular form with integer coefficients a(n) whose poles (if it has any) are supported at the cusps. Fix a prime l. In this talk, we estimate the number of a(n)'s not divisible by l and give applications to the study of the ordinary partition function, p(n), and other functions of arithmetic interest whose generating functions are of this type. (Joint work with Scott Ahlgren, Univ. of Illinois).
Greg Dresden: Look, there's more to say about Conway's Look-and-say sequenceWe start with a simple game: write down some numbers (like 1,2,2,3) and then write down the description. In this case, the description would be: one 1, two 2's, and one 3, or in more compact terms, 1,1,2,2,1,3. Now, repeat. What do we end up with? What if we do this in base-2 or base-3? What are some possible variations? This topic was first covered in depth by John Conway, but we have some new results to announce.
Jon Grantham: An Unconditional Improvement to the Running Time of the Quadratic Frobenius TestDamgard and Frandsen have recently demonstrated an extension to the Quadratic Frobenius Test (a probably primality proving test) which has a reduced running time. This improvement can be extended to other versions of the test, but in general depends on the Extended Riemann Hypothesis. I will show how to achieve a speedup intermediate between the original and the new running times that does not depend on any unproven hypotheses.
Robert Juricevic: Explicit upper bounds for &prodp≤p&omega(n) p/(p-1)Let n≥3 be an integer, σ(n) be the sum of divisors function, and let g(n)=σ(n)/(n log(log n)). Robin (1984) proved that the Riemann Hypothesis is true if and only if g(n)<eγ for n≥ 5041, where γ is Euler's constant. A consequence of Robin's theorem is that under the assumption of the Riemann Hypothesis g(n)≤ g(180)=(1.0338...)eγ for n≥121. We describe an algorithm that for a give 0<ε<1 determines all of the exceptions to the inequality ∏p≤p&omega(n)p/(p-1)<eγ(1+ε)log(log n). By employing this algorithm, we prove unconditionally that σ(n)/(n log(log n)) ≤ σ(180)/(180 log(log 180)) = (1.0338...)eγ, for n≥ 121. This is joint work with Akbary and Friggstad, to appear in Contributions to Discrete Mathematics.
Michael Mossinghoff: Sign changes in sums of the Liouville functionLet λ(k) denote the Liouville lambda function, the completely multiplicative function defined by λ(p)=-1 for every prime p. In 1919, Polya noted that the Riemann hypothesis follows if the sum L(n)= ∑1≤k≤nλ(k) does not change sign for large n, and in 1948 Turan noted a similar property for the function T(n)= ∑1≤k≤nλ(k)/k In 1958, Haselgrove proved that both L(n) and T(n) change sign infinitely often, without determining any precise values where a sign change occurs. In 1960, Lehman found an integer n0>1 where L(n0)>0, but no specific integer n1 had been found where T(n1)<0. We describe a recent large computation that has determined the smallest such integer.
David Penniston: Arithmetic properties of Maass forms arising from theta seriesWe investigate the arithmetic properties of the Fourier coefficients of a certain class of Maass forms. As an application we obtain divisibility and distribution results on the coefficients of one of Ramanujan's mock theta functions.
Filip Saidak: Sharper Selberg's LemmaWe give an explicit connection between the remainder term in the prime number theorem and the error term in Selberg's prime number lemma, and we discuss some of its consequences.
Alice Silverberg: Point counting on CM elliptic curvesIn joint work with Karl Rubin, we generalize to the case of arbitrary imaginary quadratic fields earlier results of Gross and Stark on counting the number of points on elliptic curves with complex multiplication, and on finding models for Q-curves. As an application, we give an easy way to distinguish between the twists of an ordinary elliptic curve E over Fp in order to identify one with p +1-2U points, when p=U2+dV2 with half-integers U and V and E is constructed using the CM method.
Ethan C. Smith: Elliptic Curves, Modular Forms, and Sums of Hurwitz Class NumbersLet H(N) denote the Hurwitz class number. It is known that if p is a prime, then ∑|r|<2p1/2 H(4p-r2) = 2p. In this talk, we will investigate the behavior of this sum with the additional condition r=c mod m. Three different methods will be explored for determining the values of such sums. The key ingredients involved are
Gang Yu: Finite 2-bases of integersGiven a positive integer N, a set of intgers A is called a 2-basis for N if every n ∈ [0,N] ∩ Z can be represented as n=a+b, a,b ∈ A. In this talk I will give a lower bound for |A| as N → ∞ which improves the existing results.
Michael Zieve: Polynomial decompositionI will discuss the operation of composition on polynomials over a field K, namely f(x) o g(x) = f(g(x)). Every polynomial can be written as a composition of indecomposable polynomials, but this decomposition need not be unique. When K is the complex numbers, Ritt precisely described all sources of nonuniqueness; subsequently this result was extended to other fields of characteristic zero. After reviewing these results, I will present various theorems and examples in positive characteristic, where new phenomena are possible: for instance, there are odd polynomials which decompose into polynomials that are not linear changes of odd polynomials.
I will also present results on the difficult problem of computing the intersection of two subfields of K(x), as well as the reducibility of `variables-separated' polynomials f(x)-g(y).