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.

引文分布是对数正态分布。我们使用30个对数正态分布的合成数字序列，模拟真实的引文序列，以研究h索引的一致性。使用对数正态累积分布函数，可以公式化定义h指数的方程式；该方程式表明h对论文数（N）具有复杂的依赖性。我们还研究了h与超过5到500次引用的各种引用阈值的论文数量之间的相关性。最佳相关性是针对100阈值，但是许多数据点偏离了总体趋势。大小无关指标h / N与发表论文超过任何引用阈值的可能性无关。与h指数相反，引文总数与超过10和50引文阈值的论文数具有高度相关性；平均引文数与发表论文超过任何引文水平的概率相关。因此，在合成系列中，引用次数和平均引用次数是比h和h / N好得多的研究绩效指标。我们讨论了在实际引用分布中还有其他困难。