On the transfinite density of sequences and its applications to Dirichlet series

For an increasing to $\infty$ sequence $(\lambda_n)$ of positive numbers let $\displaystyle n(t)=\sum\limits_{\lambda_n\le t}1,\ N(x)=\int\nolimits_{0}^{x}\dfrac{n(t)}{t}dt, \ L_k(t)=\sum\limits_{\lambda_n\le t}\prod\limits_{j=0}^{k-1}\dfrac{1}{\ln_j \lambda_n}$ for $k\ge 1$ and $t\ge t_k=\exp_k (0)$, where $\ln_j x$ is the $j$-th iteration of the logarithm and $\exp_k (x)$ is the $k$-th iteration of the exponent. The quantities $D(0)=\varlimsup\limits_{t\to+\infty}\frac{n(t)}{t}$ and $\overline{D}^*=\varlimsup\limits_{t\to+\infty}\frac{1}{t}\int\nolimits_0^t \frac{n(x)}{x}dx$ are called the upper density and upper average density of $(\lambda_n)$ respectively. Moreover, let $D_k(0)=\varlimsup\limits_{t\to+\infty}\frac{L_k(t)}{\ln_k t}$ be the upper $k$-logarithmic density and $D=\lim\limits_{k\to\infty}D_k(0)$ be the maximal transfinite density of $(\lambda_n)$.In the works of many authors devoted to lacunary power series andDirichlet series, estimates of the canonical product $\Lambda(z)=\prod\limits_{n=0}^{\infty}1+z^2/\lambda^2_n)$ are used, which is an entire function if $D(0)<+\infty$. Here various properties of $k$-logarithmic densities are studied and the estimate $\displaystyle\varlimsup\limits_{r\to+\infty}\frac{\ln \Lambda(r)}{r}\le \pi D$ is proved. This allows us to replace$\overline{D}^*$ with $D$ in many results of G. Polya, S. Mandelbrojt and other authors.