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A sumtest for a discrete semimeasure $P$ is a function $f$ mapping bitstringsto 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 ofsumtests for computable $P$ seems well understood, but for some applicationslower semicomputable $P$ seem more appropriate. In the case of tests forindependence, 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 anylower semicomputable semimeasures using Kolmogorov complexity. It is studied towhat 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 testwhose logarithm exceeds $\log |x| - O(\log^{(2)} |x|$) infinitely often.Finally, it is shown that for each such test $e$ the mutual information of astring with the Halting problem is at least $\log e(x)-O(1)$; thus $e$ can onlybe large for ``exotic'' strings.

Upper semicomputable sumtests for lower semicomputable semimeasures