Differences and similarities between multiplicatively and additively universal entire functions and universality properties of compositionally non-normal families of holomorphic functions

该研究项目的目标是发展乘法和加法泛整函数之间的异同，并分析组成非正规族的普适性。在计划项目的第一阶段，将阐述乘法和加法泛整函数的附加性质。在这个研究领域内，已经有一些文献关于Julia方向、泛函的有界性和指数衰减性的研究结果。然而，类似于加法泛整函数的乘法泛整函数是否具有任意小的先验增长，或者这些函数是否在整个平面上受到一定的增长限制的问题仍然是悬而未决的。该项目第一阶段的主要课题是回答这一研究问题。由于Riemann Zeta函数在临界带的右半部分满足一定的可加性普适性，并且如果Riemann假设为真，那里没有零点，那么乘性普适性函数是否也可以是零的，这一点需要进一步澄清。在项目的第二阶段，应该研究组成非正则全纯函数族允许普适性的程度。全纯函数族的合成泛函的存在意味着相应的后合成函数族不再是正规族。然而，反向蕴含并不是普遍成立的；由指数函数组成的非正规族在这里是一个激励的例子，该指数函数在论证中用正整数进行标度。因此，一个问题就产生了，一个合成的非正规族的全纯函数族可以具有什么程度的普适性。通过再次区分乘法和加法普遍性之间可能的区别和相似之处，这个研究问题的答案将在项目的第二阶段得到解决。随着研究项目的成功实施，我期待着在新的数学环境中取得重大的科学发展。

The objective of the research project is to develop differences and similarities between multiplicatively and additively universal entire functions and to analyse universality properties of compositionally non-normal families.In the first phase of the planned project, additional properties of multiplicatively and additively universal entire functions shall be elaborated. Within this research area, there already exist a number of results in the literature concerning Julia directions, boundedness and exponential decay of universal functions. However, the question whether multiplicatively universal entire functions, analogously to additively universal entire functions, may have arbitrarily small transcendental growth or if these functions are subject to certain growth restrictions on the whole plane is still open. The main subject of the first phase of the project is to answer this research question. As the Riemann zeta function fulfils a certain additive universality property on the right half of the critical strip and, if the Riemann hypothesis is true, has no zeros there, it furthermore shall be clarified if multiplicatively universal functions may also be zero-free.In the second phase of the project, it shall be investigated in how far compositionally non-normal family of holomorphic functions allow universality. The existence of a compositionally universal function for a normal family of holomorphic functions implies that the corresponding family of post-compositions is not a normal family any more. However, the reverse implication does not hold in general; the non-normal family consisting of the exponential function which is scaled in the argument with positive integers serves as a motivating example here. Therefore, the question arises which degree of universality a compositionally non-normal family of holomorphic functions can have. An answer to this research question, by again distinguishing possible differences and similarities between multiplicative and additive universality, shall be developed in the second phase of the project.As a result of the successful implementation of the research project, I expect a significant scientific development in a new mathematical environment.