We consider the behavior of the power series ${}_0\left(z\right)={\sum }_{n=1}^{\infty }{\mu }^2\left(n\right)z^n$ as z tends to $e\left(\beta \right)=e^{2\pi i\beta }$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then ${}_0\left(e\left(\beta \right)r\right)=O\left({(1-r)}^{-1/2-\epsilon }\right)$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that ${}_0\left(e\left(\beta \right)r\right)=\Omega \left({(1-r)}^{-1+\delta }\right)$.

We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j\ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,{\mathrm{sym}}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $$S_j^{*}:=\sum _{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}{\lambda }_{{\mathrm{sym}}^{j}f}^2(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),$$ where $x$ is sufficiently large, and $$L(s,{\mathrm{sym}}^{j}f):=\sum _{n=1}^{\infty }\frac{{\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)}{n^s}.$$ When $j=2$, the error term which we obtain improves the earlier known result.

It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of $T$-conditionality. In this paper, the $T$-conditionality for $T$-power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication $I^T$ being $T$-conditional are obtained. Moreover, the sufficient conditions under which a power based implication $I^T$ is $T^{*}$-conditional are discussed, this discussions...

We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. This requires suitable decay and smoothness conditions on $A,V$. The estimate (2) is trivial when $A=0$, but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

We consider the quasilinear equation ${\Delta }_{p}u+K\left(\right|x\left|\right)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty$ when $r^{-\ell }K\left(r\right)$ for $\ell \ge -p$ is bounded near $\infty$.

Let $c_q\left(n\right)$ be the Ramanujan sum, i.e. $c_q\left(n\right)={\sum }_{d\left|\right(q,n)}d\mu (q/d)$, where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for ${\sum }_{n\le y}{\left({\sum }_{q\le x}{c}_q\left(n\right)\right)}^k$ (k = 1,2) are obtained. As an analogous problem, we evaluate ${\sum }_{n\le y}{\left({\sum }_{n\le x}c{̂}_q\left(n\right)\right)}^k$ (k = 1,2), where $c{̂}_q\left(n\right):={\sum }_{d\left|\right(q,n)}d\left|\mu (q/d)\right|$.

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 $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

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.

A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character...

The sequence of balancing numbers $\left(B_n\right)$ is defined by the recurrence relation $B_n=6B_{n-1}-B_{n-2}$ for $n\ge 2$ with initial conditions $B_0=0$ and $B_1=1.$ $B_n$ is called the $n$th balancing number. In this paper, we find all repdigits in the base $b,$ which are sums of four balancing numbers. As a result of our theorem, we state that if $B_n$ is repdigit in the base $b$ and has at least two digits, then $(n,b)=(2,5),(3,6)$. Namely, $B_2=6={\left(11\right)}_5$ and $B_3=35={\left(55\right)}_6.$

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.

We consider Weil sums of binomials of the form $W_{F,d}\left(a\right)={\sum }_{x\in F}\psi (x^d-ax)$, where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{\times }\left|\right)=1$, and $a\in F^{\times }$. If we fix F and d, and examine the values of $W_{F,d}\left(a\right)$ as a runs through $F^{\times }$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{\times }|$). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...