arXiv subjects
Publications and source records attributed to Mehdi Hassani.
We obtain global explicit numerical bounds, with best possible constants, for the differences $\frac{1}{n}\sum_{k\leq n}ω(k)-\log\log n$ and$ \frac{1}{n}\sum_{k\leq n}Ω(k)-\log\log n$, where $ω(k)$ and $Ω(k)$ refer to the number of distinct prime divisors, and the total number of prime divisors of $k$, respectively.
In this paper we obtain asymptotic expansion for the geometric mean of the values of positive strongly multiplicative function $f$ satisfying $f(p)=α(d)\,p^d+O(p^{d-δ})$ for any prime $p$ with $d$ real and $α(d),δ>0$.
In this short note, we introduce an analogue of Wilson's theorem for all nonzero elements $a_1,a_2,...,a_{q-1}$ of a finite filed $\mathbb{F}$ with $|\mathbb{F}|=q\geq 3$, as follows: $$ \sum_{1\leq i_1< i_2<...< i_k\leq q-1}a_{i_1}a_{i_2}... a_{i_k}=\left\lfloor\frac{k}{q-1}\right\rfloor(-1)^q\hspace{10mm}(k=1,2,..., q-1), $$ which the left hand side of above formula is the $k-$th elementary symmetric polynomial evaluated at $a_1,a_2,...,a_{q-1}$. Specially, letting $\mathbb{F}=\mathbb{Z}_p$ with $p\geq 3$, reproves Wilson's theorem and yields some Wilson type identities. Finally, we obtain an analogue of Wolstenholme's theorem for nonzero elements of a finite filed.
We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $(\sum_{m\le n} φ(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $φ$ denotes Euler's totient function. The main part of the proof consists in showing that when $R$ is a large integer, the sequence of the residues of $\lfloor m^c \rfloor$ modulo $R$ contains blocks of consecutive values which are in an arithmetic progression.
Let $x\ge 2$. The $ψ$-form of the prime number theorem is $ψ(x) =\sum\sb{n \le x}Λ(n) =x +O\bigl(x\sp{1-H(x)} \log\sp{2} x\big)$, where $H(x)$ is a certain function of $x$ with $0< H(x) \le \tfrac{1}{2}$. Turán proved in 1950 that this $ψ$-form implies that there are no zeros of $ζ(s)$ for $\Re(s) > h(t)$, where $t=\Im(s)$, and $h(t)$ is a function related to $H(x)$ with $0< h(t) \le \tfrac{1}{2}$, but both $H(x)$ and $h(t)$ are very close to 1. We prove results similar to Turán's, with $H(x)$ and $ h(t)$ in some altered forms without the restriction that $H(x)$ and $h(t)$ are close to 1. The proof involves slightly revising and applying Turán's power sum method and using the Lindelöf hypothesis in the zero growth rate form, which is proved recently.
This submission has been withdrawn by arXiv admins due to fraudulent affiliation claims by the original submitter.
In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.
In this note, we give some explicit upper and lower bounds for the summation $\sum_{0<γ\leq T}\frac{1}γ$, where $γ$ is the imaginary part of nontrivial zeros $ρ=β+iγ$ of $ζ(s)$.
In this paper, mainly using the convexity of the function $\frac{a^x-b^x}{c^x-d^x}$ and convexity or concavity of the function $\ln\frac{a^x-b^x}{c^x-d^x}$ on the real line, where $a>b\geq c>d>0$ are fixed real numbers, we obtain some important relations between various important means of these numbers. Also, we apply the obtained results to Ky Fan type inequalities and get some new refinements.
In this paper, we introduce 3-dimensional $L-$summing method, which is a rearrangement of the summation $\sum A_{abc}$ with $1\leq a,b,c\leq n$. Applying this method on some special arrays, we obtain some identities on the Riemann zeta function and digamma function. Also, we give a Maple program for this method to obtain identities with input various arrays and out put identities concerning some elementary functions and hypergeometric functions. Finally, we introduce a further generalization of $L-$summing method in higher dimension spaces.
In this note, first we refine Mandl's inequality. Then, we consider the product $p_1p_2... p_n$ and we refine some known lower bounds for it, and we find some upper bounds for it by using Mandl's inequality and its refinement and the AGM-Inequality.
In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log n}\right\rfloor\leqπ\big((n+1)^2\big)-π(n^2). $$
In this paper we count the number of paths and cycles in complete graphs by using the number $e$. Also, we compute the number of derangements in same way. Connection by $e$ yields some nice formulas for the number of derangements, such as $D_n=\lfloor\frac{n!+1}{e}\rfloor$ and $D_n=\lfloor(e+e^{-1})n!\rfloor-\lfloor en!\rfloor$, and using these relations allow us to compute some incomplete gamma functions and hypergeometric summations; these connections are hidden in the heart of a nice polynomial that we call it derangement function and a simple ordinary differential equation concerning it.
In this short note, we introduce an Euler analogue of Wilson's theorem; $a_1a_2... a_{ϕ(n)}\equiv (-1)^{ϕ(n)+1}~({\rm mod}~n)$ say, where ${\rm gcd}(a_i,n)=1$.
In this paper, considering the concept of Universal Multiplication Table, we show that for every $n\geq 2$, the inequality: $$ M(n)=#\{ij|1\leq i,j\leq n\}\geq\frac{n^2}{\mathfrak{N}(n^2)}, $$ holds true with: $$ \mathfrak{N}(n)=n^{\frac{\log 2}{\log\log n}(1+\frac{387}{200\log\log n})}. $$
In this note, we are going to introduce some recurrence divisibility tests for all primes except than 2 and 5.