Irrationality Measure Of $π^2$
The note provides a simple proof of the irrationality measure $μ(π^2)=2$ of the real number $π^2$. The current estimate gives the upper bound $μ(π^2)\leq 5.0954 \ldots$.
math.GM↗
arXiv subjects
N. A. Carella
Publications and source records attributed to N. A. Carella.
Prescribed Primitive Roots And The Least Primes
Let $q\ne \pm1,v^2$ be a fixed integer, and let $x\geq 1$ be a large number. The least prime number $p \geq3 $ such that $q$ is a primitive root modulo $p$ is conjectured to be $p\ll (\log q)(\log \log q)^3),$ where $\gcd(p,q)=1$. This note proves the existence of small primes $p\ll(\log x)^c$, where $c>0$ is a constant, a close approximation to the conjectured upper bound.
math.GM↗
Approximate Atkin-Serre Conjecture
Let $λ(m)$ be the $m$th coefficient of a modular form $f(z)=\sum_{m\geq 1} λ(m)q^m$ of weight $k\geq 4$, let $p^n$ be a prime power, and let $\varepsilon>0$ be a small number. An approximate of the Atkin-Serre conjecture on the lower bound of the form $\left |λ\left (p^n\right )\right | \geq p^{(k-1)n/2-2k+2\varepsilon}$ is presented in this note.
math.GM↗
Euler Totient Function And The Largest Integer Function Over The Shifted Primes
Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ φ(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq x}φ([x/p])=(6/π^2)x\log \log x+c_0x+O\left (x(\log x)^{-1}\right) $, where $c_0$ is a constant, is evaluated in this note.
math.GM↗
Sum of Divisors Function And The Largest Integer Function Over The Shifted Primes
Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ σ(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ \sum_{p\leq x}σ([x/p])=c_0x\log \log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ \sum_{p\leq x}σ([x/(p+a)])=c_0x\log \log x+O(x) $ for any fixed integer $a$.
math.GM↗
Prime Values of Quadratic Polynomials
This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq x^{1/2},\, n \text{ odd}}Λ(qn^2+a)\gg qx^{1/2}/2φ(q)$ is achieved for $q\ll (\log x)^b$, where $ b\geq 0 $ is a constant.
math.GM↗
Average Orders of the Euler Phi Function, The Dedekind Psi Function, The Sum of Divisors Function, And The Largest Integer Function
Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ φ(n)$ be the Euler totient function. The result $ \sum_{n\leq x}φ([x/n])=(6/π^2)x\log x+O\left ( x(\log x)^{2/3}(\log\log x)^{1/3}\right ) $ was proved very recently. This note presents a short elementary proof, and sharpen the error term to $ \sum_{n\leq x}φ([x/n])=(6/π^2)x\log x+O(x) $. In addition, the first proofs of the asymptotics formulas for the finite sums $ \sum_{n\leq x}ψ([x/n])=(15/π^2)x\log x+O(x\log \log x) $, and $ \sum_{n\leq x}σ([x/n])=(π^2/6)x\log x+O(x \log \log x) $ are also evaluated here.
math.GM↗
Simultaneous Elements Of Prescribed Multiplicative Orders
Let $u\ne \pm 1$, and $v\ne \pm 1$ be a pair of fixed relatively prime squarefree integers, and let $d\geq 1$, and $e \geq1$ be a pair of fixed integers. It is shown that there are infinitely many primes $p\geq 2$ such that $u$ and $v$ have simultaneous prescribed multiplicative orders $\text{ord}_pu=(p-1)/d$ and $\text{ord}_pv=(p-1)/e$ respectively, unconditionally. In particular, a squarefree odd integer $u>2$ and $v=2$ are simultaneous primitive roots and quadratic residues (or quadratic nonresidues) modulo $p$ for infinitely many primes $p$, unconditionally.
math.GM↗
Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$
Let $p$ be a large prime, and let $k\ll \log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues $n_p$ modulo $p$ shows that $n_p\ll (\log p)(\log \log p)$.
math.GM↗
Simple Zeros Of The Zeta Function
This note studies the Laurent series of the inverse zeta function $1/ζ(s)$ at any fixed nontrivial zero $ρ$ of the zeta function $ζ(s)$, and its connection to the simplicity of the nontrivial zeros.
math.GM↗
Elementary Proof of the Siegel-Walfisz Theorem
This note offers an elementary proof of the Siegel-Walfisz theorem for primes in arithmetic progressions.
math.GM↗
Irrationality Exponents For Even Zeta Constants
Let $k\geq 1$ be a small fixed integer. The rational approximations $\left |p/q-π^{k} \right |>1/q^{μ(π^k)}$ of the irrational number $π^{k}$ are bounded away from zero. A general result for the irrationality exponent $μ(π^k)$ will be proved here. The irrationality exponents for the even parameters $2k$ correspond to those for the even zeta constants $ζ(2k)$. The specific results and numerical data for a few cases $k=2$ and $k=3$ are also presented and explained.
math.GM↗
Counting Primes Rationally And Irrationally
The recent technique for estimating lower bounds of the prime counting function $π(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $μ(ζ(s)) \geq 2$ of special values of the zeta function claims that $π(x) \gg \log \log x/\log \log \log x$. This note improves the lower bound to $π(x) \gg \log x$, and extends the analysis to the irrationality measures $μ(ζ(s)) \geq 1$ for rational ratios of zeta functions.
math.GM↗
The Zeta Quotient $ζ(3)/ π^3$ is Irrational
This note proves that the first odd zeta value does not have a closed form formula $ζ(3)\ne r π^3$ for any rational number $r \in \mathbb{Q}$. Furthermore, assuming the irrationality of the second odd zeta value $ζ(5)$, it is shown that $ζ(5)\ne rπ^5$ for any rational number $r \in \mathbb{Q}$.
math.GM↗
Primes In Fractional Sequences
The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these sequences of fractional numbers contain the set of primes, and the set primes in arithmetic progressions as $x \to \infty$ respectively. Furthermore, the corresponding error terms for these sequences are improved. Other results considered are the fractional sequences of integers such as the sequence $\left \{[x/n]^2+1:n \leq x\right \}$ generated by the quadratic polynomial $n^2+1$, and the sequence $\left \{[x/n]^3+2:n \leq x\right \}$ generated by the cubic polynomial $n^3+2$. It is shown that each of these sequences of fractional numbers contains infinitely many primes as $x \to \infty$.
math.GM↗
Note On Elliptic Groups of Prime Orders
Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $E(\mathbb{F}_p)$ be the elliptic group of order $\#E(\mathbb{F}_p)=n$. The number of primes $p\leq x$ such that $n$ is prime is expected to be $π(x,E)=δ(E)x/\log^2 x+o(x/\log^2 x)$, where $δ(E)\geq 0$ is the density constant. This note proves a lower bound $π(x,E) \gg x/\log^2 x$.
math.GM↗
Generalized Fibonacci Primitive Roots
This note generalizes the Fibonacci primitive roots to the set of integers. An asymptotic formula for counting the number of integers with such primitive root is introduced here.
math.GM↗
Note On Elliptic Primitive Points
Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $P \in E(\mathbb{Q})$ be a point of infinite order. The number of elliptic primes $p \leq x$ for which $\langle P\rangle=E(\mathbb{F}_p)$ is expected to be $π(x,E,P)=δ(E,P)x/\log x+o(x/\log x)$, where $δ(E,P)\geq 0$ is a constant. This note proves the lower bound $π(x,E,P) \gg x/\log x$.
math.GM↗