1. Let $X$ be a random variable such that $\mathrm{E} e^{s X}<\infty, s \in \mathbb{R}$ and

$$
\int_{\mathbb{R}}\left|\mathrm{E} e^{(s+i t) X}\right| d t<\infty, \quad s, t \in \mathbb{R}
$$

The latter for $s=0$ guarantees the existence of a density $f=f(x)$ for $X$. Under these conditions, suggest some method for obtaining upper bounds for $f$ on $\mathbb{R}$.
Solution. The finiteness of $\mathrm{E} e^{s X}$ is equivalent to the fact that the function $e^{s x} f(x)$ defines the density of some finite measure on $\mathbb{R}$, corresponding to the characteristic function $\mathrm{E} e^{(s+i t) X}$. Therefore, by the inversion formula,

$$
e^{s x} f(x)=\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-i x t} \mathrm{E} e^{(s+i t) X} d t
$$

Consequently,

$$
f(x) \leqslant \inf _{s} \frac{e^{-s x}}{2 \pi} \int_{\mathbb{R}}\left|\mathrm{E} e^{(s+i t) X}\right| d t
$$

Remark. Using the proposed upper bound for $f=f(x)$, in the work [89] (see also [27]), a very accurate estimate of the density $f=f(x)$ of the random harmonic series $X=\sum \pm \frac{1}{n}$ (the signs + and - are chosen with equal probability and independently for different $n$) was obtained, of the form

$$
\lim _{|x| \rightarrow \infty} \frac{\ln f(x)}{e^{|x|}}=-\frac{e^{2 \gamma} \pi}{4}
$$

where $\gamma=\lim \left[1+\frac{1}{2}+\ldots+\frac{1}{n}-\ln n\right]$ is the Euler constant. From this, it can be deduced that when the signs are randomly placed before the terms of the harmonic series, it is practically impossible to obtain a large number.