## On the oscillation theorems of Pringsheim and Landau.(English)Zbl 0964.11037

Bambah, R. P. (ed.) et al., Number theory. Basel: Birkhäuser. Trends in Mathematics. 43-54 (2000).
The paper under review gives an interesting survey on applications and extensions of the well known Landau’s Oscillation Theorem:
If $$\sum_1^{\infty}a_n n^{-s}$$ is a Dirichlet series with real coefficients (respectively $$\int_{1-}^{\infty}x^{-s}dF(x)$$ a Mellin transform, where $$F$$ is a right continuous real function supported on $$[1,\infty)$$ which is of bounded variation on each finite interval $$[0, X]$$) that converges for $$\operatorname{Re} s>\alpha$$ and if the associated analytic function is regular at the point $$\operatorname{Re} s=\alpha$$ then either the series (transform) converges to the left of the point $$s=\alpha$$ or else the sequence $$a_n$$ is not ultimately of one sign ($$F$$ is not ultimately monotonic). Equivalently, if $$a_n$$ or $$dF$$ is of one sign from some point onward, then the abscissa of convergence exeeds $$\alpha$$ or the associated function must have a singularity at the point $$s=\alpha$$.
The work contains the following chapters: 1. Introduction; 2. Proof of Landau’s Theorem; 3. Some Applications of Landau’s Theorem in Number Theory; subchapters: Nonvanishing of $$L$$-functions; Schnirelmann Density of $$k$$-th Power-free Integers; A Power Series Involving the Möbius Function; 4. An $$L^2$$ Theorem of Wiener and Wintner; 5. Quantitative Versions of Landau’s Oscillation Theorem.
All eight theorems in this survey are presented with elegant proofs.
For the entire collection see [Zbl 0935.00033].

### MSC:

 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11N25 Distribution of integers with specified multiplicative constraints 30B30 Boundary behavior of power series in one complex variable; over-convergence