  A sumtest for a discrete semimeasure $P$ is a function $f$ mapping bitstrings
to non-negative rational numbers such that \[
  \sum P(x)f(x) \le 1 \,.
  \] Sumtests are the discrete analogue of Martin-L\"of tests. The behavior of
sumtests for computable $P$ seems well understood, but for some applications
lower semicomputable $P$ seem more appropriate. In the case of tests for
independence, it is natural to consider upper semicomputable tests (see
[B.Bauwens and S.Terwijn, Theory of Computing Systems 48.2 (2011): 247-268]).
  In this paper, we characterize upper semicomputable sumtests relative to any
lower semicomputable semimeasures using Kolmogorov complexity. It is studied to
what extend such tests are pathological: can upper semicomputable sumtests for
$m(x)$ be large? It is shown that the logarithm of such tests does not exceed
$\log |x| + O(\log^{(2)} |x|)$ (where $|x|$ denotes the length of $x$ and
$\log^{(2)} = \log\log$) and that this bound is tight, i.e. there is a test
whose logarithm exceeds $\log |x| - O(\log^{(2)} |x|$) infinitely often.
Finally, it is shown that for each such test $e$ the mutual information of a
string with the Halting problem is at least $\log e(x)-O(1)$; thus $e$ can only
be large for ``exotic'' strings.
