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summatory
† ˈsummatory, a. Obs. rare. [ad. mod.L. summātōrius, f. med.L. summāt-: see summate and -ory.] summatory arithmetic, summatory calculus: see quots.1704 C. Hayes Treat. Fluxions 60 The fundamental Rule in Summatory Arithmetick, to find the Flowing Quantity of a given Fluxion. 1710 J. Harris Lex. Tech... Oxford English Dictionary
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Divisor summatory function
In number theory, the divisor summatory function is a function that is a sum over the divisor function. Definition The divisor summatory function is defined as where is the divisor function. wikipedia.org
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Representing expression through a Summatory I need represent this expression $L_{u_3}u(k-1)+L_{u_2}u(k-2)+L_{u_1}u(k-3)$ by using a summatory, $L_{u}$ is a vector that contains d elements $L_u=\begin{pmatrix}L_{u_1} L...
$$\sum_{j=1}^3L_{u_j}u(k-(4-j))$$ should do the trick.
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Dirichlet series inversion
A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral f by the continued DGF formula as It is often also convenient to express formulas for the summatory functions over the Dirichlet inverse function of wikipedia.org
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Probability on a summatory of exponentials functions with the same λ, knowing that they exceed a number As the title says, How do I calculate the probability of the sum of random variables being more than a number, k...
Have you ever heard of a Poisson process? That is essentially what you're asking about. If people walk into a shop at rate $\lambda$, the time between people is $\text{Exp}(\lambda)$, so you can imagine drawing exponential random variables and letting those be the distances between marks you put on ...
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Prime omega function
we suggest a variant of the summatory functions estimated in the above results for sufficiently large . To be completely precise, let the odd-indexed summatory function be defined as where denotes Iverson bracket. wikipedia.org
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O-notation property - sum of the first n powers growth I read here that in the tenth property: < The sum of the first $nr^{th}$ powers grows as the $(r+1)^{th}$ power This is not very intuitive to me, why is that? ...
For intuition, think about converting your sum to an integral. $\sum_{i=0}^n i^r \approx \int_0^n i^r di=\frac 1{r+1}n^{r+1}$ If you try it for small values, for $r=0$ you get the sum of $n \ 1$'s, which is $n$ (growing like $r+1=1$). For $r=1$ you get the triangular numbers $\frac {n(n+1)}2$, (grow...
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Liouville function
Conjectures on weighted summatory functions The Pólya problem is a question raised made by George Pólya in 1919. integers x where (as above) we have the special cases and These -weighted summatory functions are related to the Mertens function, or weighted summatory wikipedia.org
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Creating a summatory list without iteration Let list $S_k$ be an arbitrary list of numbers (may not necessarily be ordered). List $S_{k+1}$ is created via the cumulative sum of elements from list $S_k$. For example ...
Using $S_k(i)$ to indicate the $i^{th}$ term of $S_k$, then $$S_n(j) = \sum_{i \le j} {n-k-1+j-i \choose j-i} S_k(i)$$ so you only need to do weighted sums over the original sequence.
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conjecture about Totient summatory function (I am sorry that I am poor at English) Definition of Totient summatory function $\Phi \left ( n \right )$ is $\Phi \left ( n \right )= \sum_{k=1}^{n}\varphi \left ( k \rig...
From $\varphi(n) = \sum_{d | n} \mu(d) \frac{n }{d}$ $$\sum_{n=1}^\infty \varphi(n) n^{-s} =\frac{\zeta(s-1)}{\zeta(s)}$$ Since $\frac{\zeta(s-1)}{\zeta(s)}$ has a dominating pole of order $1$ at $s=2$ and $\varphi(n) \ge 0$ we obtain $$\sum_{n < x} \varphi(n) \sim \frac{x^2}{\zeta(2)}$$ The PNT mea...
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Summatory problem | Ordinary least square estimator How I can transform the first expression in the second? \begin{align} \hat{\beta}_{1} & =\frac{n\sum X_{i}Y_{i}-\sum X_{i}\sum Y_{i}}{n\sum X_{i}^{2}-\left(\sum X_{...
$\sum_iX_i=n\bar X$, so $$\begin{align*} \sum_i(X_i-\bar X)^2&=\sum_iX_i^2-2\bar X\sum_iX_i+\sum_i\bar X^2\\\ &=\sum_iX_i^2-2\bar X(n\bar X)+n\bar X^2\\\ &=\sum_iX_i^2-n\bar X^2\;. \end{align*}$$ Moreover, $$\begin{align*} \sum_i(X_i-\bar X)(Y_i-\bar Y)&=\sum_iX_iY_i-\bar X\sum_iY_i-\bar Y\sum_iX_i+...
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How to solve for a variable inside a summatory? I have a business where we buy cashier checks. The amount paid for a check is calculated as follows: $$T = m -\frac{m*t*d_{i}}{30} - m*c,$$ where $T$ is the total paid,...
You mean, something like this? \begin{align} T &= \sum_{i=1}^n \left( m_{i} - \frac{m_i t d_i}{30} -m_i c\right) \\\ &= \sum_{i=1}^n m_i(1-c) -\frac{t}{30} \sum_{i=1}^n m_i d_i \\\ \\\ \implies t &= 30 \cdot \dfrac{\displaystyle -T +\sum_{i=1}^n m_i(1-c)}{\displaystyle \sum_{k=1}^n m_k d_k} \\\ \end...
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Multivariate marginal distribution I am trying to deduce the following formula, from "Information-Theoretically Secure Secret-Key Agreement by NOT Authenticated Public Discussion" (Definition 4, page 9): $P_{\tilde{X...
The notation is a little strange, let me use $\tilde{x}$ for the values corresponding to variable $\tilde{X}$. In general, for any four random variables, we have: $$P_{\tilde{X},Y}( \tilde{x},y) = \sum_{x} \sum_{z} P_{X,\tilde X,Y,X}(x,\tilde{x},y,z) = \sum_{x} \sum_{z} P_{X,Y,Z}(x,y,z) P_{\tilde{X}...
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polynomial growth I'm currently trying to understand the link between $L(s,f)$ having an abscissa of convergence < $\infty$ and the grown of f(n). Here is what I have so far: 1. Let F(x) be the summatory function o...
Theorem 1.3 of Montgomery and Vaughan's **Multiplicative Number Theory, I** states: Let $\sigma_c$ be the abscissa of convergence of $\sum_{n=1}^\infty f(n)n^{-s}$. If $\sigma_c<0$, then $F(x)$ is bounded. If $\sigma_c\ge0$, then $F(x)$ grows roughly like $x^{\sigma_c}$, in that $$ \limsup_{x\to\inf...
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How to prove $D(n)<2n(\log\log n)$? How to prove $D(n)<2n(\log\log{n})$ for all sufficiently large $n$ where $D(n)$ is the Divisor summatory function.
There is an unconditional result of Robin (1984) that, for integers $n \geq 13,$ $$ \sigma(n) < e^\gamma \; n \, \log \log n + \frac{ 0.6482316\ldots \; n}{\log \log n} $$ with the constant $0.64...$ chosen for equality at $n=12.$ From Robin's result, the version with coefficient $2$ is true for $n ...
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