Are there a set of theorems dealing with "amount of divergence" series?


Let me explain by example. The Dirchlet $\eta$ series $\sum\_n (-1)^{n-1} n^{-x}$ converges when $x > 0$. We may say amount of divergences when $x>0$ is $0$. When $x=0$, the partial sum oscillates between $1$ and $0$, thus amount of divergence at $x=0$ is $1$ and more specifically the limit of oscillations is between $1$ and $0$. Are there theorems dealing with this kind of analysis?


I would like to examine if partial sum fluctuations stay within a certain range for various values of $x$ for a few series.


Thank you!



Are there theorems dealing with the "amount of oscillatory divergence" of series?
The study of divergent series up to the early twentieth century was masterfully summarized in Hardy's book [1], see also [2]. Significant extensions were developed by Boshernitzan in the 1980's, see [3], [4], [5], with some overlapping work by Rosenlicht [6].


[1] Hardy, Godfrey Harold. Divergent series. Vol. 334. American Mathematical Soc., 2000.


[2] Tucciarone, J., 1973. The development of the theory of summable divergent series from 1880 to 1925. Archive for history of exact sciences, 10(1), pp.1-40.


[3] Boshernitzan, Michael. "An extension of Hardy’s class L of “orders of infinity”." Journal d’Analyse Mathématique 39, no. 1 (1981): 235-255.


[4] Boshernitzan, Michael. "Hardy fields and existence of transexponential functions." Aequationes mathematicae 30, no. 1 (1986): 258-280.


[5] Boshernitzan, M., 1984. Discrete" Orders of Infinity". American Journal of Mathematics, 106(5), pp.1147-1198.


[6] Rosenlicht, Maxwell. "Growth properties of functions in Hardy fields." Transactions of the American Mathematical Society 299, no. 1 (1987): 261-272.


