  Citation distributions are lognormal. We use 30 lognormally distributed
synthetic series of numbers that simulate real series of citations to
investigate the consistency of the h index. Using the lognormal cumulative
distribution function, the equation that defines the h index can be formulated;
this equation shows that h has a complex dependence on the number of papers
(N). We also investigate the correlation between h and the number of papers
exceeding various citation thresholds, from 5 to 500 citations. The best
correlation is for the 100 threshold but numerous data points deviate from the
general trend. The size-independent indicator h/N shows no correlation with the
probability of publishing a paper exceeding any of the citation thresholds. In
contrast with the h index, the total number of citations shows a high
correlation with the number of papers exceeding the thresholds of 10 and 50
citations; the mean number of citations correlates with the probability of
publishing a paper that exceeds any level of citations. Thus, in synthetic
series, the number of citations and the mean number of citations are much
better indicators of research performance than h and h/N. We discuss that in
real citation distributions there are other difficulties.
