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.

Analytic properties of Ohno function

Ohno's relation is a well-known relation on the field of the multiple zeta values and has an interpolation to complex function. In this paper, we call its complex function Ohno function and study it. We consider the region of absolute convergence, give some new expressions, and show new relations of the function. We also give a direct proof of the interpolation of Ohno's relation.

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 $\sigma \to+\infty$.

Absolute convergence factors of Lipschitz class functions for general Fourier series

The main aim of this paper is to investigate the sequences of positive numbers, for which multiplication with Fourier coefficients of functions f ∈ Lip ⁡ 1 {f\in\operatorname{Lip}1} class provides absolute convergence of Fourier series. In particular, we found special conditions for the functions of orthonormal system (ONS), for which the above sequences are absolute convergence factors of Fourier series of functions of Lip ⁡ 1 {\operatorname{Lip}1} class. It is established that the resulting conditions are best possible in certain sense.

Convergence or Divergence? Emission Performance in the Regional Comprehensive Economic Partnership Countries

Emission convergence is a fundamental ground for cooperative CO2 emission mitigation. We investigate the emission convergence in the Regional Comprehensive Economic Partnership (RCEP) countries using a modified dynamic β-convergence model. From 2000 to 2017, the per capita emissions of the RCEP countries and its subgroups show a statistically significant diverging pattern. Nonetheless, upon accounting for multiple inputs and outputs using data envelopment analysis, we find that two out of the three emission performance indicators show statistically significant absolute convergence. The carbon emission efficiency (CEE) of the 15 RCEP countries grew from 0.5719 in 2000 to 0.6725 in 2017 and will converge at a value of 0.8187, while the carbon–population performance (CPP) increases from 0.4534 to 0.5690 and will converge at 0.7831. Furthermore, using a conditional β-convergence model, we find that trade volume has no significant effect on the growth rates of CEE and CPP, but can accelerate their speed of convergence, which indicates that the establishment of the RCEP may facilitate the convergence of its 15 member countries on CEE and CPP. Our findings suggest that emission mitigation agreement in the RCEP countries is feasible. CEE- or CPP-based indicators can be used for emission budget allocation.

Dirichlet series associated to sum-of-digits functions

We study the Dirichlet series [Formula: see text], where [Formula: see text] is the sum of the base-[Formula: see text] digits of the integer [Formula: see text], and [Formula: see text], where [Formula: see text] is the summatory function of [Formula: see text]. We show that [Formula: see text] and [Formula: see text] have analytic continuations to the plane [Formula: see text] as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions [Formula: see text] and [Formula: see text] to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.

Nexus Among Energy Consumption Structure, Energy Intensity, Population Density, Urbanization And Carbon Intensity: A Heterogeneous Panel Evidence Considering Differences In Electrification Rates

In this paper, the clustering method is used to divide the 30 provinces of the country into high, medium and low electrification rates according to the electrification rate from 2000 to 2017. The heterogeneous panel technology is used to analyze the relationship of energy consumption structure, energy intensity, population density, urbanization rate and carbon intensity. According to Cross-sectional dependence(CD) test and cross-section Im-Pesaran-Shin (CIPS) test results, the data of each panel are not in the form of same order single integer, so α convergence analysis, β absolute convergence, and β conditional convergence analysis are required. The results show that the carbon intensity of the four panels shows an α convergence; the β absolute convergence shows there is a “catch-up effect”; β conditional convergence indicates that the carbon intensity approaches their respective steady state levels; there is a long-term equilibrium relationship of energy consumption structure, energy intensity, population density and carbon intensity in all panels, but the urbanization rate has a significant impact on carbon intensity only in areas with high electrification rates. Finally, based on the results of empirical research, policy recommendations for reducing the carbon intensity in different regions are proposed.

A Two-Dimensional DOA Estimation Method Based on Virtual Extension of Sparse Array

The interelement spacing of a coprime array breaks through the half-wavelength limitation, so that a larger array aperture can be obtained with a fixed number of array elements. In this paper, the symmetry of the noncircular signal is used to virtually expand the L-shaped array into an orthogonal cross array. Furthermore, the virtual received signal of the augmented array is obtained by the second-order statistic of the received data. Decoupling and dimension reduction of elevation and azimuth are realized by a z-axis subarray and y-axis subarray. Finally, the sparse reconstruction of the signal is realized by the minimum absolute convergence and selection operator method. This method can enlarge the aperture and freedom of array and has higher accuracy and resolution of DOA estimation. It has the advantages of automatic parameter pairing without additional pairing operation and is effective for coherent and incoherent signals. The final numerical simulation results prove the effectiveness of the method in this paper.