All Questions
Tagged with pr.probability nt.number-theory
60 questions with no upvoted or accepted answers
1
vote
0
answers
233
views
Mellin transform of time-shifted function
The Mellin transform of a function $f(x)$ can be written as
$$
\mathcal M[f(x);z]=\int_0^\infty f(x)x^{z-1} dx
$$
Is there a simple expression for the Mellin transform of the function $f(x-x_0)$? ...
- 385
1
vote
0
answers
217
views
Calculating or estimating a combinatorial multivariate sum
Dear all,
I'm currently looking at a problem in which the following combinatorial product emerges:
$c(m_1,\dots,m_\lambda;n_1,\dots,n_\lambda)=\frac{m_1 !}{(m_1-n_1)!}\frac{(m_1+m_2-n_1)!}{(m_1+m_2-...
- 41
1
vote
0
answers
225
views
What is the limiting distribution of local minima of n mod i, for i up to $\sqrt{n}$, as $n \rightarrow \infty$?
The sequence n mod i
Consider the sequence n mod i for i=1...$\sqrt{n}$. If we draw the sequence as an xy-plot, we get a dense triangle (since n mod i < i). More precisely, the limiting density of ...
- 1,906
0
votes
0
answers
102
views
Formalizing the "pseudorandomness" of primes
Many conjectures about primes seem to revolve around the idea of "primes are random". So I thought about how this "randomness" may be formally defined, and came up with the ...
- 1
0
votes
0
answers
63
views
Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 47
0
votes
0
answers
55
views
Counting matrix paths for (n,m>2) matrices
Given a $n\times m$ matrix with $k$ elements inside it, I need to calculate the number of arrangements of those $k$ elements that form at least 1 path from the top to bottom matrix row composed of the ...
- 47
0
votes
0
answers
257
views
Unexpected autocorrelations in sequence of primes modulo 4
It is well known that there is a little bias in the distribution of prime residues modulo 4. But the bias eventually vanishes. I looked at the first million primes, and the counts are as follows:
...
- 3,098
0
votes
0
answers
167
views
Prime gap heuristics (follows up my question "Moments of merit")
I previously asked generally what people knew or conjectured concerning the moments of the probability distribution governing $M_n:= g_n/\ln(p_n)$, the normalized $n$th prime gap (or ``merit''). Greg ...
- 17.6k
0
votes
0
answers
337
views
What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?
This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
- 2,480
0
votes
0
answers
319
views
Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...