Tagged Questions

4
votes
1answer
30 views

Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function

In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$. I have a question about Theorem 2.27 on page 22. My ...
2
votes
1answer
47 views

Prove that $n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}$.

Let $c \not= 1$ be a real positive number, and let $n$ be a positive integer. Prove that $$n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}.$$ My initial thought was to try and induct on $n$, but the ...
1
vote
1answer
64 views

Solution to a functional equation

Let $n,i$ be positive integers and $C$ a strictly positive real value. Consider the equation for $f$ : $$1*\ln(f(n)) = C * \sum_{3<i* \ln(i) < \sqrt{n}} \left(\ln[f( i* \ln(i) )-1)] - \ln[(f( i* ...
0
votes
0answers
58 views

Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.

In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that: $$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
0
votes
0answers
42 views

Do these inequalities regarding the gamma function and factorials work?

I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In a previous question, I asked whether the following inequality is ...
2
votes
0answers
55 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
0
votes
0answers
54 views

Inequality with natural numbers.

Let $d_1 \geq d_2 \geq \dots \geq d_n$, $n\geq 3 $ natural numbers. Prove that $$ \prod_{i=1}^n(d_i + 1 ) \leq \binom{ \sum_{i=1}^n d_i -d_n +n +2 }{n} - \sum_{j=1}^n \binom{ (\sum_{i=1}^n d_i) - d_j ...
5
votes
1answer
98 views

Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem

I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = ...
1
vote
0answers
26 views

conversion from psi function to prime counting function

Can we convert $\psi(x)$ to $\pi(x)$ without using integrals. Also if $\psi(x)>\psi(y)$ when we can say that $\pi(x)>\pi(y)$ . It seems that $\theta(x)>\theta(y)$ so $\pi(x)>\pi(y)$ but ...
8
votes
1answer
205 views

Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?

I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident. In particular, Ramanujan's does the following ...
2
votes
1answer
46 views

Calculate or bound infimum

Let $a_1, \ldots, a_n \in\mathbb R$ and nonnegative let $b\geq1$ and $c\in [0,1]$. Calculate or bound from above $$ \inf \left\{d>0: \sum_{i=1}^n \ln ...
6
votes
1answer
71 views

How to prove $\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$ where $a_i\in\mathbb N$ and $a_i\lt a_{i+1}$?

Let $a_1,a_2,\ldots ,a_n\in\mathbb N$ and $a_1\lt a_2\lt\cdots\lt a_n$. Then how to prove $$\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$$ Thanks in advance
2
votes
0answers
34 views

Lower bound on diophantine system of inequalities with all but one non-linear constraint

I have a system of $n+1$ diophantine inequalities, in the following form: $$f_{1}(x_1, x_2, \dots, x_n) \geq 0$$ $$f_{2}(x_1, x_2, \dots, x_n) \geq 0$$ $$\vdots$$ $$f_{m}(x_1, x_2, \dots, x_n) ...
2
votes
0answers
38 views

Primes of the form $\dfrac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
5
votes
1answer
163 views

$x^2+y^2=z^2(1+xy)$ prove $z=\min \{x;y;z\}$ (with $x,y,z \in \mathbb{Z^+}$)

$x,y,z \in \mathbb{Z^+}$ such that $x^2+y^2=z^2(1+xy)$. Prove $z=\min \{x;y;z\}$ $$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum $z>y \implies xy < x^2/z^2$, we have $xy \in Z ...
1
vote
3answers
45 views

inequality with one number and a sum of numbers

Let $x_1, \ldots, x_n $ be non-zero real number, such that $\sqrt{x_1^2 + \cdots + x_n^2}=1$. Show that for any $i = 1, \ldots, n$, $$|x_i| \leq \sqrt{\frac{x_1^2 + \cdots + x_n^2}{n}}= ...
1
vote
2answers
102 views

Two Problem: find max, min; number theory: find x, y

Find $x, y \in \mathbb{N}$ such that $$\left.\frac{x^2+y^2}{x-y}~\right|~ 2010$$ Find max and min of $\sqrt{x+1}+\sqrt{5-4x}$ (I know $\max = \frac{3\sqrt{5}}2,\, \min = \frac 3 2$)
4
votes
4answers
137 views

Floor function inequality

I am racking my brain trying to get this problem solved and I can't seem to break it... Let $m, n$ be positive integers, with $m > 1$. Prove ...
1
vote
3answers
102 views

bound for the product of numbers

Let $n \in N$. Fix $m \in [-n,n]$. I am curious, how to bound from above the following expression $$ (n-m)^{\frac{n-m}{2}+1}(n+m)^{\frac{n+m+1}{2}}\leq \quad ? $$ Thank you.
11
votes
2answers
234 views

Find the number of pairs $(m, n)$ of positive integers such that $\frac{ m}{n+1} < \sqrt{2} < \frac{m+1}{n}$

Find the number of pairs $(m, n)$ of positive integers such that $\frac{ m}{n+1} < \sqrt{2} < \frac{m+1}{n}$ Constraint: $m$ and $n$ are both less than or equal to 1000 I toiled over this ...
4
votes
1answer
70 views

inequality with numbers--when its true?

Help me please to understand when the inequality true. Let $n<N,$ where $n, N$ are natural numbers. For which $n$ and $N$ the following is true $$ n^{2n+1}\leq N^{N+1}? $$ Thank you.
0
votes
0answers
63 views

When inequality for binomial coefficients is true?

I've asked similar question here Inequality for binomial coefficients, but with slightly different assumptions. I am curious what happend if $m, k$ are fixed. Let $m \leq n, n \leq N$ and $0\leq k ...
1
vote
1answer
115 views

Inequality for binomial coefficients

Let $m \leq n, n \leq N$ and $0\leq k \leq m$. I am wondering what is the dependence of $n$ and $N$ that for all $m, k$ $$ \frac{{N-m \choose n-k}}{{N \choose n}}\leq 1. $$ Thank you for your help.
3
votes
2answers
203 views

Bounding the prime counting function

How can I get inequalities that bound the prime counting function if I have the following inequalities for some functions $f(x)$ and $g(x)$: $$ g(x)<\psi(x)<f(x), $$ where $\psi(x)$ is second ...
1
vote
1answer
28 views

Condition when inequality with numbers is true

Let $n\geq 1, m>1, k\leq n$. I am trying to find condition on $m,$ that $$ 4\sqrt{\pi}(2m)^{mn}\leq2^k $$ Thank you.
0
votes
1answer
81 views

Prime counting inequality

What is the largest constant $c$, such that $$ (c)x/\ln(x) < \pi(x)$$ For all integers $n$, also, if no one knows, do you think expecting an anwser is unreasonble?
0
votes
0answers
78 views

Proving a simple inequality

Can someone show that the inequality bellow holds? $$ f(n) \leq f(n+1) \ $$ Where $$ \frac{\sum\limits_{k=1}^n \Lambda(k) {k}/{n}\lceil{n}/{k}\rceil{}\{ n/k \}}{\sum\limits_{k=1}^n \Lambda(k)}=f(n)$$ ...
1
vote
0answers
50 views

Inequality help

Can someone help me prove the inequality, $$ \frac{\sum\limits_{k=1}^n \Lambda(k) \frac{k}{n}\lceil\frac{n}{k}\rceil\{ \frac{n}{k} \}}{\sum\limits_{k=1}^n \Lambda(k)}<\ \frac{\sum\limits_{k=1}^n ...
6
votes
0answers
121 views

Question about an upper bound

Let each of the positive integers $a_1 ,\dots, a_n$ be less than $m$ such that the least common multiple of any two of the positive integers $a_1 ,\dots, a_n$ is greater than the integer $m$. Then ...
1
vote
1answer
50 views

Inequality with numbers

It seems its a simple question, but I am confused. Let a be natural number and let b be some number $1\le b\le a$. Find an upper bound for $$ \frac{a^2+2b^2-4ab-a}{a(a-1)}. $$ I've got $$ ...
0
votes
3answers
86 views

Properties of Mediants

If $\frac{a}{c} > \frac{b}{d}$, then the mediant of these two fractions is defined as $\frac{a+b}{c+d}$ and can be shown to lie striclty between the two fractions. My question is can you prove ...
1
vote
3answers
208 views

Generalizing Bernoulli's inequality

We are already familiar with Bernoulli's inequality: $(1+px)\le(1+x)^p$ for $x\ge-1, p\ge1$. Can this be generalized to say something useful about $(1+p_1x_1+\cdots+p_nx_n)$? For instance, one would ...
1
vote
0answers
35 views

Minimal modulus for the finite field NTT

I need your support. Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity. I am using it to compute the convolution of two vectors of ...
2
votes
1answer
97 views

inequality proof of $x^{y-1} \ge xy$

How to prove $x^{y-1}\geq xy$ with $x,y\in \mathbb{R}$ with $x,y\geq 3$ . Do I need induction? Or is there an elegant way?
-2
votes
2answers
142 views

Simple ceiling function problem [closed]

Prove that $\lceil4n/3\rceil\le 4\lceil n/3\rceil$ for all integers $n$. Try to generalize this result to something where something other than 4 and 3 are used.
0
votes
3answers
845 views

Proof by induction of summation inequality

Prove by induction the summation of $\frac1{2^n}$ is greater than or equal to $1+\frac{n}2$. We start with $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$$ for all positive ...
0
votes
1answer
120 views

question about psi function in paper on primes

This question is from Nagura's 1952 paper on primes in short intervals. He uses $\psi(x) = \sum_{m=1}^{\infty}\phi(\sqrt[m]{x})$, in which $\phi(x) = \sum_{ p \leq x }\ln p$. Using close arguments ...
12
votes
1answer
246 views

Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$

How do I prove the following: $$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$ I tried induction, but I didn't know how to go on because I don't have a look at all numbers. ...
1
vote
2answers
183 views

Proof $\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$

I try to proof the following $$\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$$ with $r \geq 3$ and $r \in \mathbb{P}$. Do I have to make in induction over $r$ or any better ideas? Any help is ...
6
votes
1answer
194 views

Sum of divisor ratio inequality

Consider the divisors of $n$, $$d_1 = 1, d_2, d_3, ..., d_r=n$$ in ascending order and $r \equiv r(n)$ is the number of divisors of $n$. Is there any expression $f(n) < r(n)$ such that, ...
2
votes
1answer
333 views

Proof of Chebyshev's theorem

(a) Show that $\int_2^x\frac{\pi(t)}{t^2}dt=\sum_{p\leq x }\frac{1}{p}+o(1)\sim\log\log x.$ (b) Let $\rho(x)$ be the ratio of the two functions involved in the prime number theorem: ...
6
votes
6answers
2k views

Is a prime factor of a number always less than its square root?

I was going through the fundamental theorem in Number Theory where any non zero integer n can be represented as a product of distinct primes. A related problem with this theorem is to prove that for ...
2
votes
2answers
358 views

Non-negative integral solutions of $X_1+X_2+X_3+X_4<n$

The number of non-negative integral solutions of $X_1+X_2+X_3+X_4<n$ (where $n$ is a positive integer) is?
8
votes
2answers
284 views

$ \sum\limits_{i=1}^{p-1} \Bigl( \Bigl\lfloor{\frac{2i^{2}}{p}\Bigr\rfloor}-2\Bigl\lfloor{\frac{i^{2}}{p}\Bigr\rfloor}\Bigr)= \frac{p-1}{2}$

I was working out some problems. This is giving me trouble. If $p$ is a prime number of the form $4n+1$ then how do i show that: $$ \sum\limits_{i=1}^{p-1} \Biggl( ...
1
vote
1answer
190 views

Showing that $|\sqrt{3} - m/n| \geq 1/(5n^2)$

Can you help me to prove this inequality \begin{aligned} |\sqrt{3} - m/n| \geq 1/(5n^2) \end{aligned} where m and n are integers. Hint:$sqrt(3)$ is irrational.