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Register a new userOn a sum involving certain arithmetic functions and the integral part function
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In this short, we study sums of the shape $sum_{nleqslant x}{f([x/n])}/{[x/n]},$ where $f$ is Euler totient function $varphi$, Dedekind function $Psi$, sum-of-divisors function $sigma$ or the alternating sum-of-divisors function $beta.$ We improve previous results when $f=varphi$ and derive new estimates when $f=Psi, f=sigma$ and $f=beta.$
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Denote by $tau$ k (n), $omega$(n) and $mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $omega$, 2 $omega$ , $mu$ 2 , $tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $epsilon$ (x $theta$ f +$epsilon$) for x $rightarrow$ $infty$, where $theta$ $omega$ = 53 110 , $theta$ 2 $omega$ = 9 19 , $theta$ $mu$2 = 2 5 , $theta$ $tau$ k = 5k--1 10k--1 and $epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{`e}s.
We obtain reasonably tight upper and lower bounds on the sum $sum_{n leqslant x} varphi left( leftlfloor{x/n}rightrfloorright)$, involving the Euler functions $varphi$ and the integer parts $leftlfloor{x/n}rightrfloor$ of the reciprocals of integers.
We define a new parameter $A_{k,n}$ involving Ramanujans theta-functions for any positive real numbers $k$ and $n$ which is analogous to the parameter $A_{k,n}$ defined by Nipen Saikia cite{NS1}. We establish some modular relation involving $A_{k,n}$ and $A_{k,n}$ to find some explicit values of $A_{k,n}$. We use these parameters to establish few general theorems for explicit evaluations of ratios of theta functions involving $varphi(q)$.
Let $mathcal{S}$ denote the family of all functions that are analytic and univalent in the unit disk $mathbb{D}:={z: |z|<1}$ and satisfy $f(0)=f^{prime}(0)-1=0$. In the present paper, we consider certain subclasses of univalent functions associated with the exponential function, and obtain the sharp upper bounds on the initial coefficients and the difference of initial successive coefficients for functions belonging to these classes.
By $(mathbb{Z}^+)^{infty}$ we denote the set of all the infinite sequences $mathcal{S}={s_i}_{i=1}^{infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let $mathcal{S}_n:={s_1, ..., s_n}$ and $H_f(mathcal{S}_n):=sum_{k=1}^{n}frac{1}{f(k)^{s_{k}}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $mathcal{S}$ of positive integers, $H_f(mathcal{S}_n)$ is never an integer if $nge 2$. Now let deg$f(x)ge 2$. Clearly, $0<H_f(mathcal{S}_n)<zeta(2)<2$. But it is not clear whether the reciprocal power sum $H_f(mathcal{S}_n)$ can take 1 as its value. In this paper, with the help of a result of ErdH{o}s, we use the analytic and $p$-adic method to show that for any infinite sequence $mathcal{S}$ of positive integers and any positive integer $nge 2$, $H_f(mathcal{S}_n)$ is never equal to 1. Furthermore, we use a result of Kakeya to show that if $frac{1}{f(k)}lesum_{i=1}^inftyfrac{1}{f(k+i)}$ holds for all positive integers $k$, then the union set $bigcuplimits_{mathcal{S}in (mathbb{Z}^+)^{infty}} { H_f(mathcal{S}_n) | nin mathbb{Z}^+ }$ is dense in the interval $(0,alpha_f)$ with $alpha_f:=sum_{k=1}^{infty}frac{1}{f(k)}$. It is well known that $alpha_f= frac{1}{2}big(pi frac{e^{2pi}+1}{e^{2pi}-1}-1big)approx 1.076674$ when $f(x)=x^2+1$. Our dense result infers that when $f(x)=x^2+1$, for any sufficiently small $varepsilon >0$, there are positive integers $n_1$ and $n_2$ and infinite sequences $mathcal{S}^{(1)}$ and $mathcal{S}^{(2)}$ of positive integers such that $1-varepsilon<H_f(mathcal{S}^{(1)}_{n_1})<1$ and $1<H_f(mathcal{S}^{(2)}_{n_2})<1+varepsilon$.
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