Annals of Communications in Mathematics

AbstractWe provide a proof of the theorem that, assuming Zorn’s lemma, two Hamel bases of a vector space have equal cardinality.

ABSTRACT.We describe an algorithm to compute in \( O\!\left(e^{c\sqrt{k\log k}}\right) \) binary operations, for some absolute constant \( c > 0 \), expressions like\[\sum_{1 \le n \le 2^{\alpha}}e^{\frac{2\pi i n^{2}}{2^{k}}}\, n^{a}\,\,\,{\rm and}\sum_{1 \le n \le 2^{\alpha}} \sum_{1 \le m \le 2^{\beta}}e^{\frac{2\pi i n m}{2^{k}}}\, n^{a} m^{b},\] where \( \alpha, \beta = O(k) \) and \( a,b \) are fixed (small) nonnegative integers. The error terms in these computations are \( O(e^{-ck}) \).Keywords. Theta series, partial sums, integer factorisation.