Thomas Reuss

My research lies in the area of analytic number theory. In particular, I am interested in applications of the determinant method, for example to the distribution of power-free integers.My most recent research problem was to improve Heath-Brown's error term in the asymptotic formula for pairs of consecutive square-free integers. An integer is called square-free if it is not divisible by a square of a prime. The task then is to find a formula for the number of such pairs up to a certain number , say while the error term of such a formula should be as small as possible. This problem can then be reduced to estimating the number of integer solutions of the equation inside a certain box. The sharper the bound for the number of solutions, the smaller the error term in the formula. The strongest tool available to us to tackle such a problem seems to be the determinant method which got originally employed by Bombieri and Pila and was then further developed by Heath-Brown. To apply the method here, one transforms the original problem to estimating the number of rational points close to the curve . One then splits the range of into a finite number of intervals and obtains an auxiliary equation for each such interval by the use of the determinant method. Thus, every solution of the original equation will satisfy one of the auxiliary equations. This extra information is the key idea to deduce a very good bound on the error term. Furthermore, I used the ideas described here to generalize the problem to asymptotic formulas for square-free integers in arbitrary arithmetic progressions. This improved a previously established error term by Tsang.