Note to the behavior of the maximal term of Dirichlet series absolutely convergent in half-plane

By $S_0(\Lambda)$ denote a class of Dirichlet series $F(s)=\sum_{n=0}^{\infty}a_n\exp\{s\lambda_n\} (s=\sigma+it)$ withan increasing to $+\infty$ sequence $\Lambda=(\lambda_n)$ of exponents ($\lambda_0=0$) and the abscissa of absolute convergence $\sigma_a=0$.We say that $F\in S_0^*(\Lambda)$ if $F\in S_0(\Lambda)$ and $\ln \lambda_n=o(\ln |a_n|)$ $(n\to\infty)$. Let$\mu(\sigma,F)=\max\{|a_n|\exp{(\sigma\lambda_n)}\colon n\ge 0\}$ be the maximal term of Dirichlet series. It is proved that in order that $\ln (1/|\sigma|)=o(\ln \mu(\sigma))$ $(\sigma\uparrow 0)$ for every function $F\in S_0^*(\Lambda)$ it is necessary and sufficient that $\displaystyle \varlimsup\limits_{n\to\infty}\frac{\ln \lambda_{n+1}}{\ln \lambda_n}<+\infty. $For an analytic in the disk $\{z\colon |z|<1\}$ function $f(z)=\sum_{n=0}^{\infty}a_n z^n$ and $r\in (0, 1)$ we put $M_f(r)=\max\{|f(z)|\colon |z|=r<1\}$ and $\mu_f(r)=\max\{|a_n|r^n\colon n\ge 0\}$. Then from hence we get the following statement: {\sl if there exists a sequence $(n_j)$ such that $\ln n_{j+1}=O(\ln n_{j})$ and $\ln n_{j}=o(\ln |a_{n_{j}}|)$ as $j\to\infty$, then the functions $\ln \mu_f(r)$ and $\ln M_f(r)$ are or not are slowly increasing simultaneously.

Pseudostarlike and pseudoconvex solutions of a differential equation with exponential coefficients

Dirichlet series $F(s)=e^{s}+\sum_{k=1}^{\infty}f_ke^{s\lambda_k}$ with the exponents $1<\lambda_k\uparrow+\infty$ and the abscissa of absolute convergence $\sigma_a[F]\ge 0$ is said to be pseudostarlike of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F'(s)}{F(s)}-1\right|<\beta\left|\dfrac{F'(s)}{F(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0=\{s\colon \,\text{Re}\,s<0\}$. Similarly, the function $F$ is said to be pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$ if$\left|\dfrac{F''(s)}{F'(s)}-1\right|<\beta\left|\dfrac{F''(s)}{F'(s)}-(2\alpha-1)\right|$\ for all\ $s\in \Pi_0$. Some conditions are found on the parameters $b_0,\,b_1,\,c_0,\,c_1,\,\,c_2$ and the coefficients $a_n$, under which the differential equation $\dfrac{d^2w}{ds^2}+(b_0e^{s}+b_1)\dfrac{dw}{ds}+(c_0e^{2s}+c_1e^{s}+c_2)w=\sum\limits_{n=1}^{\infty}a_ne^{ns}$has an entire solution which is pseudostarlike or pseudoconvex of order $\alpha\in [0,\,1)$ and type $\beta \in (0,\,1]$. It is proved that by some conditions for such solution the asymptotic equality holds $\ln\,\max\{|F(\sigma+it)|\colon t\in {\mathbb R}\}=\dfrac{1+o(1)}{2}\left(|b_0|+\sqrt{|b_0|^2+4|c_0|}\right)$ as $\sigma \to+\infty$.

Tamagawa Products of Elliptic Curves Over ℚ

We explicitly construct the Dirichlet series $$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product m. Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)={0.5053\dots}$. As a corollary, we find that $L_{\mathrm{Tam}}(-1)={1.8193\dots}$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}$. We give an application of these results to canonical and Weil heights.