Questions related to real and complex logarithms.
20
votes
0answers
413 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 $x \ge 1331$, ...
8
votes
0answers
219 views
Understanding Ramanujan's approach in his proof of Bertrand's Postulate
I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$
What would be wrong with this approach for ...
5
votes
0answers
582 views
Choosing the branch of a logarithm
The problem: I am integrating complex logarithms over an angle $\phi$ over $[0,2\pi]$. It is quite complex (pun not intended) and I called Mathematica in to aid me. I am calculating an energy of a ...
4
votes
0answers
55 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 ...
4
votes
0answers
78 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 ...
4
votes
0answers
92 views
Prove that $\log_{2b+c}a+\log_{2c+a}b+\log_{2a+b}c\ge\frac{3}{2}$
Let $a\ge 3,b\ge3,c\ge3$. Prove that: $$\log_{2b+c}a+\log_{2c+a}b+\log_{2a+b}c\ge\frac{3}{2}$$
I don't know what to do. Rewrite to $\ln(x)$ or $e^x$
But it's not work
4
votes
0answers
349 views
Logarithm of a complex number as intersections of two logarithmic spirals
In Penrose's book "The Road to Reality" page 97 figure 5.9 he shows the values of the complex logarithm in a diagram as the intersection of two logarithmic spirals. Can you please explain how these ...
2
votes
0answers
54 views
Convexity of polylogarithms
I want to prove the following proposition:
The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$.
And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
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 $n > 1$, $x \ge 1$:
...
2
votes
0answers
29 views
Discrete logarithm - strange polynomials
If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number ...
2
votes
0answers
105 views
Log-concave functions whose sums are still log-concave: possible to find a subset?
Rationale:
I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest.
It is well-known that sums of ...
2
votes
0answers
56 views
Is there a (hopefully free) graphing utility BESIDES mathematica that can graph log polar coordinates?
This is my first post to the forum. I have had a good experience with other stackexchange fora, so I have high hopes for this one.
I have looked online and can not seem to find the software described ...
2
votes
0answers
70 views
What this graph $x^{\log y}=y^{\log x}$
What is the graph of $x^{\log y}=y^{\log x}$? This question appears on GRE exam.
2
votes
0answers
152 views
Exponential average of logs
I'm working on a sound recognition algorithm where an "exponential moving average" is used for "adapting" to sound levels. It turns out that taking an average of logs works better than simple sums ...
2
votes
0answers
178 views
Can the Baker-Campbell-Hausdorff formula for $\ln(AB)$ be simplified for similar, diagonizable matrices?
Given two similar, diagonizable square matrices $A$ and $B$ that do not commute, can the Baker-Campbell-Hausdorff formula be simplified exploiting the similarity to obtain a nice expression for ...
1
vote
0answers
34 views
simplification of a natural log of a trigonometric function
hope you are all well.
I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression:
$$k\cos X \cdot 4\ln(\cos X)$$
where $k$ is a constant and $X$ ...
1
vote
0answers
28 views
What's the most straight forward way to show that a function is increasing?
I am trying to show that:
$$\frac{2}{n}\log\Gamma(\frac{x}{2}) - \log\Gamma(\frac{x+n-1}{n})$$
is an increasing function for $x \ge 5$ and $n > 2$
One way to do this would be to show that ...
1
vote
0answers
24 views
transforming a straight band into a logarithmic spiral
I want to plot the labels and the graduations of an historical timeline onto a logarithmic spiral.
If this timeline is on the $x$-axis, $-\infty$ would project to the center of the spiral, $+\infty$ ...
1
vote
0answers
28 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 ...
1
vote
0answers
101 views
Using the gamma function as an upper and lower bound to the logarithm of a factorial function.
I am trying to find an upper and lower bound for the following function:
$$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$
where
...
1
vote
0answers
41 views
Logarithmic simplification of a sum of power terms
It is entirely possible that there is no solution to this problem, but here goes...
I have a number of equation of states for fluids that have terms that are of the form
${\phi _r} = ...
1
vote
0answers
52 views
Is this “Elegant” algorithm for logarithm by Zeckendorf representation, the same as an 'efficient' algorithm?
The algorithm here which computes the exponent $b$ given a base $a$, and given $n$ = $a$^$b$, appears no better to me than simply counting the number of times we divide $n$ through by the base $a$ ...
1
vote
0answers
79 views
Complex Logarithm
For what values of $p$ is the following valid?
$$\log(z^p) = p\log(z)$$
where
$$\log(z) = \ln(|z|) + i[\arg(z)+2\pi n]$$
where $n$ is an integer.
I heard the expression above should not be valid for ...
1
vote
0answers
21 views
Compute $ \operatorname{Li}_{3}\left(\frac{1}{2} \right) $
Where could I find a proof of the identity
$$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)$$
?
1
vote
0answers
38 views
$f(n) = n^2 \lceil \log n \rceil$ is time constructible
I have a question, I want to show, that:
$$f(n) = n^2 \lceil \log n \rceil $$
is time-constructible. I have no idea how to do this. I know that $n^2$ is time-constructible and I know that $\log n$ ...
1
vote
0answers
34 views
Calculating the number of states in power of 10's
This is part of a problem that was assigned to me. It might seem elementary, but I need a hint on how to start this problem.
I have 10 billion bits that can either be on or off at any time.
...
1
vote
0answers
17 views
Log return of two different timeseries
Lets say I have a single timeseries, the simple return would be T/T-1-1
the log return would be ln(T/T-1)
But let's say I have two different time series, T and R
The values are close, but still ...
1
vote
0answers
41 views
Use of natural logarithm transformation on weighted index series
I have a value computed as sum of powers, e.g. $x^5+y^8+z^2$.
The exponent represents the weight for variables, $x, y$ and $z$ in the example above.
Applying natural logarithm on $x^5+y^8+z^2$, I get ...
1
vote
0answers
59 views
What are the Puiseux series for the inverse of $exp(z)(z-a)(z-b)(z-c)$ expanded at the singularities?
Let $a,b,c$ be real variables.
Let $z$ be a complex number and $g(z) = exp(z)(z-a)(z-b)(z-c)$.
Let $f(z)$ be the functional inverse of $g(z)$ such that $f(g(z)) = g(f(z)) = z$.
Now $f(z)$ must have ...
1
vote
0answers
34 views
Triangular exponentation logarithm and inverse
The generalized formula of triangular exponentation on real numbers field is
$x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $
It's my ...
1
vote
0answers
49 views
lower bounds for maximum computing times for integer factorisation
Supposing that n were known to have two prime factors, and that the computer had a database of all the primes $<\sqrt{n}$. Then, unless n is square, one factor would be $<\sqrt{n}$. If an ...
0
votes
0answers
61 views
Floor of log equation $S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - \frac{10^{\lfloor\log_{10}(x)\rfloor+1}-10}{9}$
I must find 'x' and I don't know how to solve the following equation.
Does it have a solution? How can I solve it?
$$
S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - ...
0
votes
0answers
58 views
Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.
In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that:
$$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 ...
0
votes
0answers
31 views
Does it follow that if $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$, $\log(\lfloor\frac{x}{2}\rfloor!) \le \log\Gamma(\frac{x+1}{2})$?
The answer seems to be yes.
Here's my reasoning.
Let $\{x\} = x - \lfloor{x}\rfloor$
Assume $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$
$$\log(\lfloor\frac{x}{2}\rfloor!) = ...
0
votes
0answers
16 views
Question concerning expansion of the log function.
I'll get straight to it.
$\ln(x)=\int\frac{1}{x}dx
=\int\frac{1}{1-(1-x)}dx$
And $\frac{1}{1-(1-x)}=\sum_{n=o}^{\infty}(1-x)^n$
Am I correct so far? Because on wikipedia, the series ...
0
votes
0answers
32 views
How to formulize this logarithmic expression?
If you have this list of numbers {$1, 2, 4, 7, 11, 16, 21, 27, 33, ...$}
Where the list starts at 1, then the next number is $1 + x + \left\lceil\log_2{x}\right\rceil$ where $x$ is the current ...
0
votes
0answers
58 views
Wondering if I could simplify $\quad e^{-\ln\left(\prod_i(x_i+1)^{\large\frac{\theta+1}{\theta}}\right)}$
I have the following that I am working with:
$$\large e^{-\ln\left(\prod_i(x_i+1)^{\large\frac{\theta+1}{\theta}}\right)}$$
I was wondering if there was a way I could simplify: ...
0
votes
0answers
58 views
equivalent calculator keys to: inv ln
What are the equivalent keys on a calculator to inv + ln?
My calculator doesn't have a inv key:
0
votes
0answers
65 views
Proof for the upper bound on entropy $H(S)$?
I was trying to prove the upper bound on $H(S)$ using the inequalities $\ln(x)\le(x-1)$ and $\ln(1/x)\ge(1-x)$ for independent and memory less source symbols $s_1,\dots,s_q$ .
I am trying to prove ...
0
votes
0answers
28 views
Comparing Exponential Depreciation with Linear Expense
I have a vehicle whose original costs is $\$26,000$ and depreciates at an annual rate of $5.5\%$. The person put a down payment of $6000$ on the vehicle and monthly payments sum up to $4800$ per year. ...
0
votes
0answers
52 views
What does taking the logarithm of a variable mean?
Question with regards to taking the logarithm of a variable (Statistics Question)
Say you have a bar graph displaying data for an example "Cost of Computer Orders by the Population" and you are ...
0
votes
0answers
28 views
Complex exponentials, Polylogarithms
How can I rewrite $\text{Li}{s}(e^{2\pi i\frac{a}{b}})$, in terms of hurwitz zeta function, I know I have to break it up into congruence classes and sum each one independtly but not sure how I would ...
0
votes
0answers
53 views
Vernier scale on logarithmic scale
I know that a vernier scale can be used to accurately read a linear scale, such as in vernier calipers. I'm wondering if there is a way the methods behind a vernier scale could be adapted for usage ...
0
votes
0answers
39 views
checking whether $\log(x+1):(0,+\infty)\to (0,+\infty)$ is a function, then whether it is onto or 1-1 or both
$f:\mathbb{R}_+\to \mathbb{R}_+, f(x)=\log(x+1)$
How to start? Should i start computing $f(1), f(2) ....$ and then plotting them on the graph
0
votes
0answers
101 views
significance of logarithms in computing large numbers
I'm considering the use of various strategies that allow very large numbers to be computed rather than simply giving errors. One way apparently is to use logarithms. Taking the log of every term in ...
0
votes
0answers
36 views
Finding correlation through plotting logarithms
I have a problem in which I am to find the correlation between two sets of data. Plotting them as logarithms makes for a constant increase, except for one of the data points (the first), as seen in ...
0
votes
0answers
27 views
Log base 2 and relation of apparent to real distance when looking at horizon
I once saw a math video where the professor explained how to convert the height from the bottom of a rectangle (sort of like a viewfinder I guess) to the horizon, to the real distance from the bottom ...
0
votes
0answers
56 views
Question about an asymptotic analysis proof in Ball Collision Decoding paper.
On page 21 of Daniel Bernstein's paper "Smaller decoding exponents: ball-collision decoding" he presents a proof that I have a few questions about.
$P,Q,R,L$ and $W$ are all positive and close to ...
0
votes
0answers
90 views
How To Create a Non-Linear Output from a Linear Input?
I'm not even sure how to ask this question, so bear with me for a second.
Given a linear input value, such as floating point numbers between 0 and 1, how can I produce an output that favors higher ...
