6
votes
2answers
278 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
1
vote
1answer
54 views

Why does this equation work?

let $ P(x) := \sum_{p \leq x} Log [p]$, then we have $P(2^{k+1}) = \sum_{i=0}^k ( P(2^{i+1}) - P(2^i)) < 2 \cdot Log[2] \cdot (1 + 2 + 4 +... + 2^k) \leq 4 \cdot Log[2] \cdot 2^k$. Why does ...
2
votes
1answer
29 views

Find the minimum value of $S$

Let $a,b,c$ be real numbers greater than 1. Let $S=\log_a{bc}+\log_b{ac}+\log_c{ba}$ Then find the minimum value of $S$
1
vote
1answer
86 views

$\sum_{p\le x} \frac{1}{pq}$

I was given that $\sum_{p\le x} \frac{1}{p}$ = $\log\log x$+O(1). I need to show that $\sum_{pq\le x} \frac{1}{pq} = (\log \log x)^2 + O(\log \log x)$. Here we go: Break the sum into two sums: ...
1
vote
1answer
64 views

Find $b-d$ when $\log_ab={3\over2}$ and $\log_cd={5\over4}$

$a,b,c$ are three natural numbers such that $\log_ab={3\over2}$ and $\log_cd={5\over4}$. Given: $a-c=9$ Find $b-d$
1
vote
1answer
31 views

Confused about discrete logarithm question

For purposes of explaining the notation for those unfamiliar, if we fix a prime $q$, as well as $a,b$ nonzero integers $\mod{q}$, $L_a(b) = x$ is the solution to the equation $b = a^x \mod{p}$ We are ...
1
vote
1answer
38 views

What is the reasoning behind ways of splitting up this summation sign?

Some context: I've been studying Chebyshev's $\psi$ - function, which claims that $\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \log p$ where $p$ is prime and $\Lambda(n)$ is the von ...
2
votes
2answers
50 views

Solving a partial fractions

I have set up partial fractions so that$$Aln^3x-B(x+x^2)=1-x^2$$ and $$ Cln^3x+D(x+x^2)=1+x^2$$ to set up and solve the following $$\alpha(1+x)+ \gamma x= A+C$$ and from $$\frac {D lnx-B}{ln^3x} ...
0
votes
0answers
26 views

Suppose y is a prime and that b has order 3 modulo y. What is the order of b+1? [duplicate]

order is 6 by showing the expansion of (b+1)^6, but are there other solutions where I have to solve for all the pairs (y,b) that meets the conditions of the problem ?
1
vote
0answers
59 views

Estimation of a logarithmic sum

I need to estimate the sum $$ \underset{r=2}{\overset{t}{\sum}}\left(\frac{\log\log r}{r}\right)^{2}. $$ I tried to use the Abel's partial summation, and I got $$ \frac{(\log\log ...
3
votes
1answer
208 views

log(log(123456789101112131415…)))

How would you fin the integer closest to log(log(1234567891011121314...2013)) where the number is the concatenation of numbers 1 through 2013 inclusive. log() in this case is log base 10. Also, how ...
0
votes
3answers
56 views

What is the value of $ (a+b) $ where $ a\log_{1971}3 + b\log_{1971}{73} = 2012 $

I have two integers which are a and b . They satisfy the following equation which is $ a\log_{1971}3 + b\log_{1971}{73} = 2012 $ . I want to know the value of $ (a+b) $ . I have tried to solve ...
1
vote
1answer
103 views

Attempted exercise using Littlewood's theorem

This was an exercise to try to show we can use Littlewood's theorem$^1$ to prove that $$\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N} \frac{g(n)}{\log p_n} = 1 \hspace{30mm}(1)$$ If $\vartheta(p_k) ...
1
vote
2answers
66 views

Given $\{\log_ab \mid a,b\in \mathbb N, \mathrm{gcd}(a,b)=1,a,b≥3\}$ does the sum of any two $\log_ab$ form an irrational or rational number?

Given $\{\log_ab \mid a,b\in \mathbb N, \mathrm{gcd}(a,b)=1,a,b≥3\}$ does the sum of any two $\log_ab$ form an irrational or rational number? I know that $\log_ab$ is irrational, but does the sum of ...
1
vote
1answer
129 views

Littlewood's 1914 proof relating to Skewes' number

From Littlewood's 1914 theorem (paraphrase): I propose to show there are arbitrarily large values of x for which successively $\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$ $ ...
4
votes
2answers
135 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
4
votes
2answers
131 views

Estimating $\sum_{p_2 \leq x} (\log p_2)^2$

This was an exercise to use the approach here to estimate the sum $\sum_{p_2 \leq x} \log (p_2)^2,$ in which $p_2$ are numbers containing two prime factors (repetitions allowed). $\pi_2(x)$ is the ...
-1
votes
1answer
77 views

$\log$ transform of the fundamental theorem of arithmetic? [closed]

Taking the canonical form of the fundamental theorem of arithmetic in the form: $$n=\prod_{j=1}^\infty p^{m_j}_j \qquad ;m_j\in \Bbb N_0$$ Does anybody know about a $\log n$ transform of this? Note: ...
4
votes
0answers
64 views

Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see ...
5
votes
4answers
680 views

Summation of logs

Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
4
votes
0answers
94 views

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
0
votes
0answers
80 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
54 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
60 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 $n > 1$, $x \ge 1$: ...
4
votes
1answer
52 views

Looking for help in understanding Jitsuro Nagura's analysis of the upper bound for $\psi(x)$

I'm working on understanding Nagura's analysis of the upper bound for $\psi(x)$ which is done in Lemma 2. I am unclear on one step of his reasoning. With Lemma 1, he establishes for $x \ge 2000$: ...
5
votes
1answer
120 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) = ...
0
votes
1answer
114 views

Proving that a specific gamma function is a guaranteed lower bound for a factorial function

In reviewing Ramanujan's proof of Bertrand's postulate, Ramanujan observes that: $$\ln\Gamma(x) - 2\ln\Gamma(\frac{x+1}{2}) \le \ln(\lfloor{x}\rfloor!) - 2\ln(\lfloor\frac{x}{2}\rfloor!)$$ I have ...
0
votes
1answer
40 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
1
vote
0answers
34 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
8
votes
1answer
226 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 ...
1
vote
4answers
1k views

Why aren't logarithms defined for negative $x$?

Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...
36
votes
1answer
1k views

Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$

Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
3
votes
1answer
182 views

Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function?

In Ramanujan's proof of Bertrand's Postulate, Ramanujan states: $\log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x)$ where: $\vartheta(x) = \sum_{p \le x} ...
2
votes
1answer
91 views

Solving $ f(\log x)$

A generalization of the conjecture $$\pi(x+x^{\theta}) - \pi(x) \sim \frac{x^\theta}{\log x} $$ (Ingham, 1937 or earlier) might be $$\Delta \pi_k = \pi_k((x+1)^2) - \pi_k(x^2)\sim \frac{x}{\log ...
3
votes
1answer
393 views

Chebyshev's first $\vartheta(x)$ function question

This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: ...
17
votes
1answer
306 views

Approximation of $\log(x)$ as a linear combination of $\log(2)$ and $\log(3)$

I wonder if it's possible to approximate $\log(n)$, n integer, by using a linear combination of $\log(2)$ and $\log(3)$. More formally, given integer $n$ and and real $\epsilon>0$, is it always ...
1
vote
3answers
66 views

If $\log_{b}N$ is rational, what are the limitations on the possible values of $b$ and $N$?

If $\log_{b}N$ is rational, is there a set of values to which $b$ and $N$ must belong? Is there a set of values to which $b$ and $N$ cannot belong? Further, if it is presupposed that $b$ and $N$ are ...
14
votes
3answers
452 views

Are the logarithms in number theory natural?

I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the prime number counting business, somewhat unsettling. What is the reason for them? Has it maybe ...
1
vote
1answer
109 views

Number of digits in different number systems?

I know a similar question was asked before, but I wanted to know if this can be extended to any number system by a generic formula. For example, given a number X in number system A, how many digits ...
1
vote
1answer
215 views

Discrete logarithm to a primitive root base

I need to find out $\log_g {-1}$ in $\mathbb{Z}_n$ where $n$ is an odd prime and $g$ is a primitive root mod $n$. How do I do that? Thanks.