Sums of two squares and a power

We extend results of Jagy and Kaplansky and the present authors and show that for all $k\geq 3$ there are infinitely many positive integers $n$, which cannot be written as $x^2+y^2+z^k=n$ for positive integers $x,y,z$, where for $k\not\equiv 0 \bmod 4$ a congruence condition is imposed on $z$. These examples are of interest as there is no congruence obstruction itself for the representation of these $n$. This way we provide a new family of counterexamples to the Hasse principle or strong approximation.Comment: 6 pages, to appear in the memorial volume "From Arithmetic to Zeta-Functions - Number Theory in Memory of Wolfgang Schwarz

Exponential sums with coefficients of certain Dirichlet series

Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.Comment: 13 page

Renormalization : A number theoretical model

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep 2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200

Breaking classical convexity in Waring's problem: Sums of cubes and quasi-diagonal behaviour

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46588/1/222_2005_Article_BF01231451.pd