All Questions
Tagged with logarithms number-theory
142
questions
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60
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Is there connection between digits of nth prime and natural logarithm? [closed]
I started with a small list of primes less than 100 and checked counted its digits then I wonder what will happen for larger values I made a plot for values up to 500,000 for this range the closest ...
- 113
3
votes
1
answer
80
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“Logarithm” with respect to Dirichlet convolution
Similar to a previous question of mine, I’m trying to find equivalents to logarithms in other rings in which we can carry out computations. If we consider an arithmetic function to be a function of ...
- 2,663
3
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0
answers
102
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How to tell if an infinite series sum will be rational or irrational?
I plugged in the following series into a calculator: $$\sum_{n=1}^\infty \ln(1+\frac{1}{n^2})$$ and got a result of approximately $1.29686$. That's nice and all, but I want to know: is this result ...
- 453
3
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1
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203
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Why is the average number of divisors of the first $N$ natural numbers approximately $\log(N)$?
In the textbook An Illustrated Theory of Numbers, the following problem appears:
I wrote up some Sage code to confirm this:
...
- 402
2
votes
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answers
95
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Strange dips in sequence $u_n=\log{|(n-1)^{\text{st}}\text{ difference of first $n$ primes}|}$
I made a sequence related to prime numbers.
In column A, I listed the sequence of prime numbers.
In column B, I listed the sequence of gaps between the prime numbers.
In column C, I listed the ...
- 25.7k
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4
answers
200
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Asymptotic behaviour of $\displaystyle\sum_{n \leq x} \frac{\log(n)}{n}$
This is my first question, I hope I won't make too many mistakes.
I have been given this question as an exercise, but I am struggling to find the solution. I have to calculate the asymptotic behaviour ...
- 21
-1
votes
1
answer
97
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Proving $x*\ln(\ln(x))>\ln(\ln(x\#))$ for $\infty>$x>e$
Let $x \in \mathbb{N}$.
For $\infty>x>e$,
Let $x$ be the largest prime in the factorization of the primorial $x\#$.
How does one go about proving $x*\ln(\ln(x))>\ln(\ln(x\#))$?
The plot of ...
- 591
3
votes
1
answer
129
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How to get even numbers from adding pairs? (best title I thought of)
if I want to make a subset of numbers for which sums of pairs involve all even numbers what is the subset with the smallest density?
for example {1 , 3 , 5} can pair to get 2 :1+1 , 4 :1+3 , 6 :3+3, 8:...
3
votes
1
answer
95
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$\log(1＋x)∈\mathbb{Z}_p[[X]]$ can be expressed as infinite product
Let $p$ be a prime number. There is a well known product formula for $\log(1+X)$ as a power series in $\mathbb{Z}_p[[X]]$ . Namely letting $\Phi_n(X)=(1+X)^{p^n}-1$ and $Q_n(X)=\Phi_{n+1}(X)/p\Phi_n(X)...
- 6,351
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43
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Can this inequality be solved for $q$ in terms of $n$ (or the other way around), if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
The following general inequality relating $k$ and $q$ is proved in this ...
- 8,010
1
vote
0
answers
36
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Can the $\log(z)$ function be evaluated as a sum over exponential terms?
I've noticed the exponential function $e^{-z}$ can be evaluated as a sum over $\log$ terms as follows where the evaluation limits $N$ and $f$ are both assumed to be positive integers.
$$e^{-z}=\...
- 7,631
2
votes
1
answer
217
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Can this formula for $\zeta(3)$ be proven or simplified further?
This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
- 7,631
1
vote
1
answer
47
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Solving a exponential floor flunction equation
I am trying to solve this question but can't simplify this further.
Question:
$$ 2^{\lfloor log_2{(x)} + \frac{1}{2} \rfloor} = 2^{\lfloor log_2(x-2^{\lfloor{ log_2{(\frac{x}{2})} + \frac{1}{2}}\...
- 11
-1
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1
answer
78
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Finding natural points on binary logarithm's graph? [closed]
Consider a binary logarithm function as $y =\log_2(ax + b)$ which is actually defined in $\mathbb{R}$. Now how can I find all the ponits on this function's graph whose both coordinates (x,y) belong to ...
- 1
2
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3
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160
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$\log(n) + 1 < \frac{n}{\log(n)}$ for Lagarias inequality
In Lagarias paper "An Elementary Problem Equivalent to the Riemann
Hypothesis" he uses the following inequality:
$\log(n) + 1 \leqslant\dfrac{n}{\log(n)}\;\;\;$ for $\;n\geqslant3\;.$
I've ...
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