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On some Dirichlet series. (English) Zbl 1230.11095

Steuding, Rasa (ed.) et al., New directions in value-distribution theory of zeta and \(L\)-functions. Proceedings of the conference, Würzburg, Germany, October 6–10, 2008. Aachen: Shaker Verlag (ISBN 978-3-8322-8818-1/pbk). 171-248 (2009).
From the abstract: “The author proves some rather general value-distribution theorems for finite sets of Dirichlet series satisfying certain natural hypotheses, with applications to the universality of such series.”
In this long and interesting paper the author uses his thorough knowledge of almost periodic functions to prove the existence of a joint distribution for finitely many general Dirichlet series \[ \sum_{n\geq 0}A_{n,j}e^{-\lambda_{n,j}s},\quad \text{where}\;\lambda_{n+1,j}>\lambda_{n,j}\geq 0 \] (provided some conditions are satisfied: the series is absolutely convergent in \(\operatorname{Re}(s)>\alpha\) and meromorphic in \(\operatorname{Re}(s)>\alpha_1\), where \(\alpha_1\leq \alpha\), \(\sup \frac 1T \int_{-T}^T |f(\sigma + it)|^2 dt\) is finite in \(\sigma>\alpha_1\), \(\sigma\) is not an abscissa of a pole, and a polynomial growth condition in \(\sigma>\alpha_1\), \(t\) sufficiently large).
In the first two sections, the author uses his previous results [in: Analytic and probabilistic methods in number theory. Proceedings of the 4th international conference in honour of J. Kubilius, Palanga, Lithuania, September 25–29, 2006. Vilnius: TEV. 109–142 (2007; Zbl 1151.11039)] to provide a solution for the problem of the existence of a joint distribution (in \(H(D)\), the space of functions analytic in some domain \(D\)) of a finite set of finitely many series \(\sum_{n\geq 0}A_{n,j}e^{-\lambda_{n,j}s}\), \(1\leq j\leq k\), satisfying the conditions mentioned above.
Next, he proves results on the (joint) distribution of general Dirichlet series in the meromorphic case.
Then, he proves a “fundamental proposition”, giving the estimate \[ \limsup_{T\to\infty} \frac{1}{2T} \int_{-T}^{+T}\left(\sup_{s\in K} \left| f^{(r)}(s+iu)\right|\right)^2 du \leq C_r \sum_{n\geq 0} |A_n|^2\, e^{-2\lambda_n\sigma_1} \] for the \(r\)th derivative of a function satisfying the conditions above. \(K\) is a compact set contained in the strip \[ S=S_{(\beta_1,\beta_2)}=\{s\in\mathbb C : \beta_1< \operatorname{Re}(s)<\beta_2,\;\alpha_1\leq \beta_1<\beta_2\}, \] which must not contain any of the poles of \(f\), and \(\sigma_1={\frac 12 \cdot(\min_{z\in K}\operatorname{Re}(z)+\beta_1)}\), and \(C_r\) may depend on \(\beta_1\), \(\beta_2\), \(m\), \(r\), and \(K\).
Then the author gives (in some cases) necessary and sufficient conditions for universality (see below).
In an appendix he explains the notations and definitions, related to the theory of Besicovitch almost periodic functions, used in the paper (compact groups associated with a sequence of real numbers, dual of \(\mathbb Q\), abstract measures associated to Dirichlet series, almost periodic functions).
The author’s necessary condition for universality is as follows.
Let \(\sum_{n\geq 0} A_n e^{-\lambda_ns}\), absolutely convergent for \(\sigma = \operatorname{Re}(s)\), define a function \(f(s)\), meromorphic in \(\sigma >\alpha_1\), where \(\alpha_1<\alpha\), which has a finite set \(Z\) of poles and satisfies
(i) \(\sup\frac 1T\int_{-T}^T |f(\sigma+it)|^2 dt = K_\sigma\;\text{is finite for}\;\sigma>\alpha_1\), \(\sigma\not\in\operatorname{Re}(Z),\) (ii) \(\exists\, m\in\mathbb Z\), \(m>0\), \(D>0\), \(U_0>0\), so that \(|f(\sigma +it)| < D |\frac12 t|^{m-\frac12}\) in \(\sigma>\alpha_1\), \(|t|>_0\), \(\sigma \not\in \operatorname{Re}(Z)\).
The compact set \(K\) is as above, and \(h\) is a continuous function \(h: K\to\mathbb C\). If, for any \(\varepsilon >0\), \[ \limsup_{T\to\infty} \frac1{2T}\text{meas} \left\{t\in\mathbb R,\;|t|\leq T\:\, \sup_{s\in K} |f(s+it)-h(s)|\leq\varepsilon\right\}>0, \] then for any sequence \(\{E_\ell\}_{\ell\in\mathbb N}\), \(E_\ell \subset\mathbb N\), so that \[ \{n\in\mathbb N;\;0\leq n\leq\ell\} \subset E_\ell\;\text{ and } \;\sum_{n\in E_\ell} |A_n| e^{-\lambda_n\sigma_1} < \infty, \] the condition \[ \liminf_{t\to\infty}\inf_{t\in\mathbb R} \sup_{s\in K}\left| \sum_{n\in E_\ell} A_n e^{-\lambda_n(s+it) } - h(s) \right| = 0 \] holds.
In the next section a universality result for Euler products is given.
If the Euler product \(F(s)= \prod_{q\in\mathbb N}\sum_{r=0}^\infty a_{r\,u_q} e^{-sr\,u_q}\), absolutely convergent for \(\operatorname{Re}(s) = \sigma >\alpha\), defines a function \(F(s)\) meromorphic in \(\operatorname{Re}(s) > \alpha_1\), \(\alpha_1\leq\alpha\) with a finite set \(Z\) of poles, which satisfies the conditions (i) and (ii) above, if \(\forall\, q\in\mathbb N\) and \(s\in S_{(\beta_1,\beta_2)}\) one has \(\sum_{r=0}^\infty a_{ru_q} e^{-sr\,u_q}\not=0\), and if \(K \subset S_{(\beta_1,\beta_2)}\) is compact, \(S_{(\beta_1,\beta_2)}\cap Z=\emptyset\), \(h\colon K\to\mathbb C\) is continuous, and \[ \liminf_{y\to\infty}\inf_{t\in\mathbb R}\sup_{s\in K} |F_{y-}(s+it)-h(s)|=0, \] where \(F_{y-}(s)= \prod_{q\in\mathbb N_{y-}}\sum_{r=0}^\infty a_{r\,u_q} e^{-sr\,u_q}\), then the set \[ \left\{ t\in\mathbb R\:\;\sup_{s\in K} |F(s+it) -h(s)| \leq\varepsilon\right\} \] has a positive lower density, for any \( \varepsilon >0\).
It follows as a corollary that a necessary and sufficient condition for the universality of the Euler product \(F(s)\) in \(S_{(\beta_1,\beta_2)}\) is that for any compact subset \(K\subset S_{(\beta_1,\beta_2)}\) with connected complement and any function \(h(s)\), continuous on \(K\) and analytic on \(K^\circ\), the equality \[ \liminf_{y\to\infty}\inf_{t\in\mathbb R}\sup_{s\in K} |F_{y-}(s+it)-h(s)|=0 \] is fulfilled.
In the next section, as a typical application, the universality of the “periodic pseudo–Lerch zeta function” \[ L_{\lambda,\,\frac{r+\alpha}k}(s) = \sum_{m=0}^\infty \frac{e^{i\lambda_{km+r}}}{\left(m + \frac{(r+\alpha)}k\right)^s} \] is proved. The author states that there is no difficulty in obtaining joint universality results for the continuous case as well as for the discrete case.
For the entire collection see [Zbl 1186.11002].

MSC:

11K70 Harmonic analysis and almost periodicity in probabilistic number theory
11M35 Hurwitz and Lerch zeta functions
11M41 Other Dirichlet series and zeta functions
30K10 Universal Dirichlet series in one complex variable
42A75 Classical almost periodic functions, mean periodic functions
60F17 Functional limit theorems; invariance principles

Citations:

Zbl 1151.11039
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