61 research outputs found
Sums of two squares and a power
We extend results of Jagy and Kaplansky and the present authors and show that
for all there are infinitely many positive integers , which cannot
be written as for positive integers , where for
a congruence condition is imposed on . These
examples are of interest as there is no congruence obstruction itself for the
representation of these . 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
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