## 1 Introduction  
Over the past decade, the interest in warm dense matter (WDM) has been steadily growing. A WDM regime can be characterized by high values of density and temperature, exceeding by several orders of magnitude those in solids, so WDM is usually non-ideal and degenerate. The study of WDM is required in many astrophysical applications, such as planet interiors [1; 2; 3], brown and white dwarfs [4; 5], and compact stars [6; 7]. Furthermore, WDM can be realized in experiments on inertial confinement fusion [8; 9] and interaction of intense laser beams with dense plasma [10; 11]. To understand this exotic state of matter, the study of its structure and thermodynamic properties is required. One of the most important structural quantities is a pair correlation function (PCF), characterizing the average spatial distribution of particles [12]. The analysis of PCFs gives information about the ordering of particles and hence allows to understand the state of the system: gas-like, liquid-like, or crystalline. Besides that, PCFs can be used for the calculation of average potential energy, which is an important quantity in thermodynamics. Another important thermodynamic quantity is a momentum distribution function (MDF). Characterizing the probability density to have certain momenta values for the particles, an MDF is essential in calculations of many other thermodynamic quantities such as internal energy, scattering parameters, the rate constants of chemical and nuclear reactions, and their thresholds [13; 14; 15; 16]. Due to a strong non-ideality and degeneracy, studying WDM is very challenging. The most powerful and developed analytical methods are usually based on the perturbative expansions or other asymptotic approaches. In this way, the perturbative approximationsof the MDFs for a Coulomb system leads to a \"quantum tail\" exceeding the Maxwell distribution in the weakly non-ideal and non-degenerate case [17; 18; 19]. However, these analytical methods are not applicable if any appropriate small parameter does not exist, and exactly this situation is common in the case of WDM. Therefore, numerical simulation is the most promising instrument in the studies of WDM, such as quantum Monte Carlo methods for relatively simple systems (particles in external field [20; 21], hydrogen and electron-hole plasma [22; 23; 24; 25], etc.) and the density functional theory (DFT)-based methods for more complex and realistic materials. Being the most powerful and universal numerical tools for studying WDM, DFT-based methods, however, require accurate knowledge of the exchange-correlation (xc) functional, characterizing the exchange and correlation interactions between electrons. To construct such an xc-functional, one has to study a many-electron subsystem in thermodynamic equilibrium and calculate its xc-energy in a wide range of WDM conditions [26]. One of the most fruitful and popular models of the electron subsystem is a uniform electron gas (UEG), which has to be considered under WDM conditions for DFT applications [27; 28]. Moreover, the UEG is a well-known quantum analog of a one-component plasma [29] and a simple model of alkali metals [30], so the study of UEG allows one to understand general features of the plasmas in the WDM regime. The UEG consists of electrons on a neutralizing rigid background with a uniformly distributed density of the electrical charges. The thermodynamic state of an unpolarized UEG can be described with two dimensionless parameters -- the Brueckner parameter \(r_{s}=(4\pi na_{0}^{3}/3)^{-1/3}\) and the reduced temperature \(\theta=kT/E_{F}\), where \(E_{F}\) is the Fermi energy, \(n\) is the electron density, \(a_{0}\) is the Bohr radius. Furthermore, the coupling strength \(\Gamma=(e^{2})/(r_{s}kT)\) and the degeneracy parameter \(\chi=n\lambda^{3}\), where \(\lambda=\sqrt{2\pi\hbar^{2}/(kT)}\) is the thermal wavelength, are often used. PCFs and MDFs in a weakly non-ideal case have been studied with standard perturbative methods [31; 32].