Let $j\ge 2$ be a given integer. Let $H_k^{*}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\ge 2$ for the full modulo group $\mathrm{SL}(2,\mathbb{Z})$. For $f\in H_k^{*}$, denote by ${\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)$ the $n$th normalized Fourier coefficient of $j$th symmetric power $L$-function ($L(s,{\mathrm{sym}}^{j}f)$) attached to $f$. We are interested in the average behaviour of the sum $$\sum _{\genfrac{}{}{0pt}{}{n=a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x}{(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}}{\lambda }_{{\mathrm{sym}}^{j}f}^2\left(n\right),$$ where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$. Denote by ${\lambda }_f\left(n\right)$ the $n$th normalized Fourier coefficient of $f$. We are interested in the average behaviour of the sum $$\sum _{a^2+b^2\le x}{\lambda }_f^j(a^2+b^2)$$ for $x\ge 1$, where $a,b\in \mathbb{Z}$ and $j\ge 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$-functions and Rankin-Selberg $L$-functions.

Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_1$, $k_2$ and $k_3$ for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$, respectively, and let ${\lambda }_f\left(n\right)$, ${\lambda }_g\left(n\right)$ and ${\lambda }_h\left(n\right)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions ${\lambda }_g\left(n\right)$, ${\lambda }_h\left(n\right)$ twisted by ${\lambda }_f\left(n\right)$ and establish the following results: $$\sum _{n\le x}{\lambda }_f\left(n\right){\lambda }_g{\left(n\right)}^i{\lambda }_h{\left(n\right)}^j{\ll }_{f,g,h,\epsilon }{x}^{1-1/2^{i+j}+\epsilon }$$ for any $\epsilon >0$, where $1\le i\le 2$, $j\ge 5$ are any fixed positive integers.

Let $X$ denote either ${\mathrm{ℂ\mathbb{P}}}^m$ or ${ℂ}^m$. We study certain analytic properties of the space ${ℰ}^n(X,gp)$ of ordered geometrically generic $n$-point configurations in $X$. This space consists of all $q=(q_1,\cdots ,q_n)\in X^n$ such that no $m+1$ of the points $q_1,\cdots ,q_n$ belong to a hyperplane in $X$. In particular, we show that for a big enough $n$ any holomorphic map $f:{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ commuting with the natural action of the symmetric group $𝐒\left(n\right)$ in ${ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$ is of the form $f\left(q\right)=\tau \left(q\right)q=(\tau \left(q\right)q_1,\cdots ,\tau \left(q\right)q_n)$, $q\in {ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)$, where $\tau :{ℰ}^n({\mathrm{ℂ\mathbb{P}}}^m,gp)\to \mathrm{𝐏𝐒𝐋}(m+1,ℂ)$ is an $𝐒\left(n\right)$-invariant holomorphic map. A similar result holds true for mappings of the configuration...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

We prove that for a normed linear space $X$, if $f_1:X\to ℝ$ is continuous and semiconvex with modulus $\omega$, $f_2:X\to ℝ$ is continuous and semiconcave with modulus $\omega$ and $f_1\le f_2$, then there exists $f\in C^{1,\omega }\left(X\right)$ such that $f_1\le f\le f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega \left(t\right)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C_{loc}^{1,\omega }$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

For an integer $k\ge 2$, let ${\left(n\right)}_n$ be the $k-$generalized Pell sequence which starts with $0,...,0,1$ ($k$ terms) and each term afterwards is given by the linear recurrence $n=2n-1+n-2+\cdots +n-k$. In this paper, we find all $k$-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${\left(P_n^{\left(2\right)}\right)}_n$. ...

Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n\in S_{k+1/2}(N,{\chi }_0)$ be a nonzero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus ${\chi }_0$, where the Fourier coefficients $a\left(n\right)$ are real. Bruinier and Kohnen conjectured that the signs of $a\left(n\right)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${\left\{a\left(t{n}^2\right)\right\}}_n$, where $t$ is a squarefree integer such that $a\left(t\right)\ne 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that ${\left\{a\left(t{n}^2\right)\right\}}_n$ is equidistributed over any arithmetic progression $n\equiv dmodq$.

Let $k\ge 2$ and let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence defined by $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}$$ for $n\ge 2$ with initial conditions $$P_{-(k-2)}^{\left(k\right)}=P_{-(k-3)}^{\left(k\right)}=\cdots =P_{-1}^{\left(k\right)}=P_0^{\left(k\right)}=0,P_1^{\left(k\right)}=1.$$ In this study, we handle the equation $P_n^{\left(k\right)}=y^m$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\ge 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_n^{\left(k\right)}=y^m$ with $2\le y\le 1000$ has only one solution given by $P_7^{\left(2\right)}={13}^2.$

A natural number $n$ is said to be a $(k,r)$-integer if $n=a^{k}b$, where $k>r>1$ and $b$ is not divisible by the $r$th power of any prime. We study the distribution of such $(k,r)$-integers in the Piatetski-Shapiro sequence $\left\{\lfloor n^c\rfloor \right\}$ with $c>1$. As a corollary, we also obtain similar results for semi-$r$-free integers.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for ${\Gamma }_0\left(N\right)$ with normalized eigenvalues ${\lambda }_f\left(n\right)$ and let $X>1$ be a real number. We consider the sum $${𝒮}_k\left(X\right):=\sum _{n<X}\sum _{n=n_1,n_2,...,n_k}{\lambda }_f\left(n_1\right){\lambda }_f\left(n_2\right)...{\lambda }_f\left(n_k\right)$$ and show that ${𝒮}_k\left(X\right){\ll }_{f,ϵ}{X}^{1-3/\left(2\right(k+3\left)\right)+ϵ}$ for every $k\ge 1$ and $ϵ>0$. The same problem was considered for the case $N=1$, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k\ge 5$. Since the result is valid...

Let ${\Phi }_1,...,{\Phi }_{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U{\subset }^{N+m}$, let $𝕄$ be a $N$-dimensional $C^k$ submanifold of $U$ and define $$𝕋:=\{z\in 𝕄:{\Phi }_1\left(z\right),...,{\Phi }_{k+1}\left(z\right)\in T_z𝕄\}$$ where $T_z𝕄$ is the tangent space to $𝕄$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $𝕄$) of $𝕋$: If $z_0\in 𝕄$ is a $(N+k)$-density point (relative to $𝕄$) of $𝕋$ then all the iterated Lie brackets of order less or equal to $k$ $${\Phi }_{i_1}\left(z_0\right),[{\Phi }_{i_1},{\Phi }_{i_2}]\left(z_0\right),[[{\Phi }_{i_1},{\Phi }_{i_2}],{\Phi }_{i_3}]\left(z_0\right),...(h,i_h\le k+1)$$ belong to $T_{z_0}𝕄$. Such a property has been proved in [9] for $k=1$ and its proof in the...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

A famous theorem of Carleson says that, given any function $f\in L^p\left(𝕋\right)$, $p\in (1,+\infty )$, its Fourier series $\left(S_{n}f\left(x\right)\right)$ converges for almost every $x\in 𝕋$. Beside this property, the series may diverge at some point, without exceeding $O\left(n^{1/p}\right)$. We define the divergence index at $x$ as the infimum of the positive real numbers $\beta$ such that $S_{n}f\left(x\right)=O\left(n^{\beta }\right)$ and we are interested in the size of the exceptional sets $E_{\beta }$, namely the sets of $x\in 𝕋$ with divergence index equal to $\beta$. We show that quasi-all functions in $L^p\left(𝕋\right)$ have a multifractal behavior with respect to...