In the paper, we construct and study the class of continuous on $[0, 1]$functions with continuum set of peculiarities(singular, nowhere monotonic, and non-differentiable functionsare among them).The representative of this class is the function $y = f(x)$defined by the Engel representation of argument:\begin{align*}x = \sum_{n=1}^\infty\frac{1}{(2+g_1)(2+g_1+g_2)\ldots(2+g_1+g_2+\ldots+g_n)}=: \Delta^E_{g_1g_2\ldots g_n\ldots},\end{align*}where $g_n = g_n(x) \in \{ 0, 1, 2, \ldots \}$, and convergent real series\[\sum_{n=0}^\infty u_n = u_0 + u_1 + \ldots + u_n + r_n = 1,\qquad\lvert u_n \rvert < 1,\ 0 < r_n < 1,\]by the following equality\[f(\Delta^E_{g_1(x)g_2(x)\ldots g_n(x)\ldots})= r_{g_1(x)} + \sum_{k=2}^\infty\biggl( r_{g_k(x)} \prod_{i=1}^{k-1} u_{g_i(x)} \biggr).\]We study local and global properties of function $f$:structural, extremal, differential, integral, and fractal properties.

This paper is concerned with proving some embeddings of the form \begin{equation*}A_{p_{1},q_{1}}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })\cdot A_{p_{2},q_{2}}^{r}(\mathbb{R}^{n},|\cdot |^{\alpha })\hookrightarrowA_{p,q_{1}}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha }).\end{equation*} The different embeddings obtained here are under certain restrictions on theparameters. Here $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha})$ stands foreither the Besov space $B_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ orthe Triebel-Lizorkin space $F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$.These spaces unify and generalize the classical Lebesgue spaces of powerweights, Sobolev spaces of power weights, Besov spaces and Triebel-Lizorkinspaces. Almost all our assumptions on $s,r,p_{1},p_{2},q_{1}$ and $q_{2}$are necessary. An application to the continuity of pseudodifferentialoperators with non-regular symbols on Triebel-Lizorkin spaces of powerweights is given.

In this paper, we study the average behaviour of the coefficients of triple product $L$-functions and some related $L$-functions corresponding to normalized primitive holomorphic cusp form $f(z)$ of weight $k$ for the full modular group $SL(2,\mathbbZ).$ Here we call $f(z)$ a primitive cusp form if it is an eigenfunction of all Hecke operators simultaneously.

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $\pi(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $\zeta(s)$.

We evaluate in closed form a large class of power series where the coefficients are products of special numbers, or products of special numbers and Laguerre polynomials. A number of examples is presented involving harmonic, hyperharmonic, Fibonacci, Lucas, Bernoulli, Euler, and derangement numbers. We also obtain a new convolution formula for the Laguerre polynomials.

We show that the Fourier coefficients of the symmetric power $L$-functions of modular forms are $O((\log n)^{-\frac{1}{2}+\epsilon})$ except for a density zero set.

We establish quantitative results for sign changes in certain subsequences of primitive Fourier coefficients of a non-zero Siegel cusp form of arbitrary degree over congruence subgroups. As a corollary of our result for degree two Siegel cusp forms, we get sign changes of its diagonal Fourier coefficients. In the course of our proofs, we prove the non-vanishing of certain type of Fourier-Jacobi coefficients of a Siegel cusp form and all theta components of certain Jacobi cusp forms of arbitrary degree over congruence subgroups, which are also of independent interest.