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

Bateman, Paul T.; Diamond, Harold G.

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].

Reviewer: Ramūnas Garunkštis (Vilnius)

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

Keywords:

oscillation theorem; boundary behavior of series; Dirichlet series; Dirichlet \(L\)-function; \(k\)-th power-free integers; Möbius function; survey