Asymptotic formulas for certain arithmetic functions

This is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere. The last named author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P18079-N12.     

Combinatorial Stokes formulas via minimal resolutions

We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Zk of order k. We then demonstrate how such a chain map induces a “Zk-combinatorial Stokes theorem,” which in turn implies “Dold's theorem” that there is no equivariant map from an n-connected to an n-dimensional free Zk-complex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k=2 for this is classical; it involves Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier's work (2006).     

Combinatorial numbers in binary recurrences

We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.      

Explicit formulas for the codimension of certain T-ideals

Sofia, Bulgaria. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 30–36, November–December, 1988.     

Determinant formulas for class numbers in function fields

In this paper, by extending Kucera's idea to the function field case, we obtain several determinant formulas involving the real class number and the relative class number of any subfield of cyclotomic function fields. We also provide several examples using these determinant formulas.

     

General formulas for the smoothed analysis of condition numbers

We provide estimates on the volume of tubular neighborhoods around a subvariety Σ of real projective space, intersected with a disk of radius σ. The bounds are in terms of σ, the dimension of the ambient space, and the degree of equations defining Σ. We use these bounds to obtain smoothed analysis estimates for some conic condition numbers. To cite this article: P. Bürgisser et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Self-recursion formulas of certain monstrous functions

We find self-recursion formulas for 20 monstrous functions (Thompson series) and one monster-like function of even level. We see that their Fourier coefficients can be determined either by the first four, or by the first, third and fifth.     

Summation formulas involving generalized harmonic numbers

By means of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.     

On non-existence of certain error formulas for ideal interpolation

We answer a question raised by Carl de Boor regarding the existence of certain “good” error formulas for ideal interpolation. We will show that, for a specific form of ideal interpolation by linear polynomials in two variables, such formulas do not exist.     

New upper bound formulas with parameters for Ramsey numbers

In this paper, we obtain some new results R(5,12)?848, R(5,14)?1461, etc., and we obtain new upper bound formulas for Ramsey numbers with parameters.