In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

Chronic HCV infection and associated liver cancer impose a heavy burden on the healthcare system. Direct acting antivirals eliminate HCV, unless it is drug resistant, and partially reverse liver disease, but they cannot cure HCV-related cancer. Possible remedy could be a multi-component immunotherapeutic vaccine targeting both HCV-infected and malignant cells, also those not infected with HCV. To meet this need we developed a two-component DNA vaccine based on the highly conserved core protein of HCV to target HCV-infected cells, and a renowned tumor associated antigen telomerase reverse transcriptase (TERT) based on the rat TERT, to target malignant cells. Their synthetic genes were expression-optimized, and HCV core was truncated after aa 152 (Core152opt) to delete the domain interfering with immunogenicity. Core152opt and TERT DNA were highly immunogenic in BALB/c mice, inducing IFN-γ/IL-2/TNF-α response of CD4+ and CD8+ T cells. Also, DNA-immunization with TERT enhanced cellular immune response against luciferase encoded by a co-delivered plasmid (Luc DNA). However, DNA-immunization with Core152opt and TERT mix resulted in abrogation of immune response against both components. A loss of bioluminescent signal after co-delivery of TERT and Luc DNA into mice indicated that TERT affects the in vivo expression of luciferase directed by the immediate early cytomegalovirus and interferon-β promoters. Panel of mutant TERT variants was created and tested for their expression effects. TERT with deleted N-terminal nucleoli localization signal and mutations abrogating telomerase activity still suppressed the IFN-β driven Luc expression, while the inactivated reverse transcriptase domain of TERT and its analogue, enzymatically active HIV-1 reverse transcriptase, exerted only weak suppressive effects, implying that suppression relied on the presence of the full-length/nearly full-length TERT, but not its enzymatic activity. The effect(s) could be due to interference of the ectopically expressed xenogeneic rat TERT with biogenesis of mRNA, ribosomes and protein translation in murine cells, affecting the expression of immunogens. HCV core can aggravate this effect, leading to early apoptosis of co-expressing cells, preventing the induction of immune response.

Let β= (((24!)!)!)!, and let {${\mathcal P}_{n^2+1}$} denote the set of all primes of the form {$n^2+1$}. Let ${\mathcal M}$ denote the set of all positive multiples of elements of the set {${\mathcal P}_{n^2+1} \cap (\beta,\infty)$}. The set {${\mathcal X}=\{0,\ldots,\beta\} \cup {\mathcal M}$} satisfies the following conditions: (1) ${\rm card}({\mathcal X})$ is greater than a huge positive integer and it is conjectured that ${\mathcal X}$ is infinite, (2)} we do not know any algorithm deciding the finiteness of ${\mathcal X}$, (3)~a~known and short algorithm for every {$n \in N$} decides whether or not {$n \in {\mathcal X}$}, (4) a known and short algorithm returns an integer~$n$ such that ${\mathcal X}$ is infinite if and only if ${\mathcal X}$ contains an element greater than $n$. The following problem is open: {\em simply define a set {${\mathcal X} \subseteq N$} such that ${\mathcal X}$ satisfies conditions (1)-(4), and \underline{we do not know any representation of~~${\mathcal X}$~~as a finite union of sets whose definitions are simpler than} \underline{the definition of ${\mathcal X}$}} {\tt (5)}. Let {$f(1)=2$}, {$f(2)=4$}, and let {$f(n+1)=f(n)!$} for every integer {$n \geqslant 2$}. For a positive integer $n$, let {$\Psi_n$} denote the following statement: {\em if a system of equations ${\mathcal S} \subseteq \Bigl\{x_i!=x_k: i,k \in \{1,\ldots,n\}\Bigr\} \cup \Bigl\{x_i \cdot x_j=x_k: i,j,k \in \{1,\ldots,n\}\Bigr\}$ has only finitely many solutions in positive integers \mbox{$x_1,\ldots,x_n$}, then each such solution \mbox{$(x_1,\ldots,x_n)$} satisfies {$x_1,\ldots,x_n \leqslant f(n)$}.} We prove that for every statement $\Psi_n$ the bound {$f(n)$} cannot be decreased. The author's guess is that the statements {$\Psi_1,\ldots,\Psi_9$} are true. We prove that the statement $\Psi_9$ implies that the set ${\mathcal X}$ of all {non-negative} integers $k$ whose number of digits belongs to {${\mathcal P}_{n^2+1}$} satisfies conditions {\tt (1)-(5)}.