Number - User contributions [en]

User:Vipul

2026-08-26T00:28:07Z

Vipul: /* Notes for stuff I plan to expand */

I'm Vipul, the person who came up with the original idea for this website.

==Notes for stuff I plan to expand==

===Euler totient function===

In [[Euler totient function]], to Measures of difference, add entries for <math>\ln (\varphi(n) / n) / \ln n</math>, <math>\ln (\varphi(n) / n) / \ln \ln n</math>, <math>\ln (\varphi(n) / n) / \ln \ln \ln n</math> for various <math>D</math>. I expect 0 at primes for all, and 0, 0, -1 at primorials since <math>(1 - 1/p)</math> for a primorial should have log approx <math>-\ln \ln \ln n</math>.

===Multiplicative and completely multiplicative functions===

* [[Multiplicative functions form abelian group under Dirichlet product]]: Characterize the group as an external direct product of countably many (prime-indexed) copies of <math>1 + xR[[x]]</math> under multiplication (a formal group law on the power series)
* [[Multiplicative function]]: Mention the above characterization
* [[Modified Dirichlet character]]: Make a page on these
* [[Completely multiplicative function]]
** Mention modified Dirichlet characters
** Discuss their generating role in all the "interesting" parts of the group of multiplicative functions (basically, everything that shows up in our study is a finite Euler product, or a convolution of finitely many completely multiplicative functions and their inverses)

User:Vipul

2026-08-25T19:19:31Z

Vipul: /* Notes for stuff I plan to expand */

2024-09-30T01:27:06Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.''' 
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]] 
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

MediaWiki:Sitenotice

2024-09-06T01:31:25Z

Vipul:

Number:Enabling site search autocompletion

2024-09-06T01:30:49Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Number). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in th..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Number).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

File:Site search autocompletion working.png

2024-09-06T01:30:37Z

Vipul:

File:Site search autocompletion broken.png

2024-09-06T01:30:18Z

Vipul:

Number:429 Too Many Requests error

2024-09-06T01:27:05Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

Vipul: /* Similar functions */

{{arithmetic function}}

==Definition==

Let <math>n</math> be a [[natural number]]. The '''Euler phi-function''' or '''Euler totient function''' of <math>n</math>, denoted <math>\varphi(n)</math>, is defined as following:

* It is the order of the multiplicative group modulo <math>n</math>, i.e., the multiplicative group of the ring of integers modulo <math>n</math>.
* It is the number of elements in <math>\{ 1,2, \dots, n \}</math> that are relatively prime to <math>n</math>.

===In terms of prime factorization===

Suppose we have the following prime factorization of <math>n</math>:

<math>n = p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}</math>.

Then, we have:

<math>\varphi(n) = \prod_{i=1}^r p_i^{k_i}\left(1 - \frac{1}{p_i}\right)</math>.

In other words:

<math>\frac{\varphi(n)}{n} = \prod_{i=1}^r \left(1 - \frac{1}{p_i} \right)</math>.

==Properties==

{| class="sortable" border="1"
! Property !! Satisfied? !! Statement with symbols
|-
| [[satisfies property::multiplicative function]] || Yes || If <math>m</math> and <math>n</math> are relatively prime [[natural number]]s, then <math>\varphi(mn) = \varphi(m)\varphi(n)</math>.
|-
| [[dissatisfies property::completely multiplicative function]] || No || It is not true for arbitrary natural numbers <math>m</math> and <math>n</math> that <math>\varphi(mn) = \varphi(m)\varphi(n)</math>. For instance, if <math>m = n = 2</math>, then <math>\varphi(m) = \varphi(n) = 1</math> whereas <math>\varphi(mn)</math> is 2.
|-
| [[satisfies property::divisibility-preserving function]] || Yes || If <math>m</math> and <math>n</math> are natural numbers such that <math>m</math>divides <math>n</math>, then <math>\varphi(m)</math> divides <math>\varphi(n)</math>.
|}

==Behavior==

===High and low points (relatively speaking)===

* '''Primes are high points''': We also have <math>\varphi(n) \le n - 1</math> for <matH>n > 1</math>. Equality occurs if and only if <math>n</math> is a [[prime number]].
* '''Primorials are low points''': Roughly, the numbers occurring as [[primorial]]s (products of the first few primes) have the lowest value of <math>\varphi(n)</math> relative to <math>n</math>, compared with other similarly sized numbers.

===Measures of difference===

We use the [[infinitude of primes]] for arguing about limit superiors. The limits discussed are in the limit as <math>n \to \infty</math>. Note that the last quotient is undefined for <math>n = 1</math>.

{| class="sortable" border="1"
! Measure !! Limit superior !! Explanation !! Limit inferior !! Explanation
|-
| <math>\varphi(n) - n</math> || -1 || maximum value of -1 occurs at primes || <math>-\infty</math> || Consider the sequence of powers 2. <math>\varphi(2^k) = 2^{k-1}</math>, so <math>\varphi(2^k) - 2^k = -2^{k-1} \to -\infty</math>.
|-
| <math>\frac{\varphi(n)}{n}</math> || 1 || At each prime <math>p</math>, value is <math>1 - (1/p)</math>. Limit is 1 as <math>p \to \infty</math> || 0 || Consider the sequence of primorials. The corresponding values of <math>\varphi(n)/n</math> are products of the values <math>1 - (1/p)</math> for the first few primes <math>p</math>. The limit of these is the infinite product <math>\prod_p \left(1 - \frac{1}{p}\right)</math> over all prime <math>p</math>. This diverges because the infinite sum of the reciprocals of the primes diverges.
|-
| <math>\frac{\ln \varphi(n)}{\ln n}</math> || 1 || For each prime <math>p</math>, the limit is <math>\ln(p - 1)/\ln p</math> ,which approaches 1. || 1 || We can show that for every <math>\varepsilon > 0</math>, there exists <math>N_\varepsilon</math> such that <math>\varphi(n) \ge n^{1 - \varepsilon}</math> for all <math>n \ge N_\varepsilon</math>.
|}

==Related arithmetic functions==

===Summatory functions===

The sum of the values of the totient function for all natural numbers up to a given number is termed the [[totient summary function]].

===Similar functions===

{| class="sortable" border="1"
! Name of arithmetic function !! Description !! Mathematical relation with Euler totient function
|-
| [[Universal exponent]] (also called Carmichael function)|| The [[groupprops:exponent of a group|exponent]] of the multiplicative group modulo <math>n</math>. Denoted <math>\lambda(n)</math>|| <math>\lambda(n)</math> divides <math>\varphi(n)</math>. This is related to the group-theoretic fact that [[groupprops:exponent divides order|exponent divides order]].
|-
| [[Dedekind psi-function]] || Defined as <math>\psi(n) = n \prod_{p|n} \left( 1 + \frac{1}{p}\right)</math> || Structural similarity in definition. Also, <math>\psi(n) \ge \varphi(n)</math> for all <math>n</math>, with equality occurring iff <math>n = 1</math>.
|}

===Relations expressed in terms of Dirichlet products===

Below are some [[Dirichlet product]]s of importance.

{| class="sortable" border="1"
! Statement !! Function !! Value of the Dirichlet product <math>\varphi *</math> the function !! Description !! Statement in ordinary notation !! Proof !! Corollaries
|-
| <math>\varphi * U = E</math>|| <math>U</math> || the function that sends every natural number to 1 || <math>E</math> || the function that sends every natural number to itself || <math>\sum_{d|n} \varphi(d) = n</math> || Direct combinatorial argument: each summand is the number of elements in <math>\{ 1, \dots, n \}</math> whose gcd with <math>n</math> with <math>n/d</math>. || <math>\varphi = E * \mu</math>, where <math>\mu</math> is the [[Mobius function]]. This follows from the fact that <math>\mu = U^{-1}</math>. Also, <math>\varphi * \sigma_1 = E * E</math>, follows from this and <math>\sigma_1 = E * U</math>.
|-
| <math>\varphi * \sigma_0 = \sigma_1</math> || <math>\sigma_0<math> || the [[divisor count function]] <math>\sigma = \sigma_1</math> || the [[divisor sum function]] || <math>\sum_{d|n} \varphi(d) \sigma_0(n/d) = \sigma(n)</math> || Proof: We have <math>\sigma_0 = U * U</math> and <math>\sigma_1 = E * U</math> by definition. Multiply both sides of the former by <math>\varphi</math> and use associativity to get <math>\varphi * \sigma_0 = (\varphi * U) * U</math>. Use the preceding identity to get <math>\varphi * \sigma_0 = E * U</math>, so <math>\varphi * \sigma_0 = \sigma_1</math>.
|}

The table of relationships can be conceptualized as follows, where the cell entries are the products of their row and column headers:

{| class="sortable" border="1"
! !! <math>\mu = U^{-1}</math> ([[Mobius function]]) !! <math>I</math> (Kronecker delta with 1, identity for Dirichlet product) !! <math>U</math> ([[all ones function]]) !! <math>\sigma_0 = U^2</math> ([[divisor count function]])
|-
| <math>E</math> (identity map, sends everything to itself) || <math>\varphi</math> || <math>E</math> || <math>\sigma_1</math> || no name
|}
===Inequalities===

* <math>\varphi(n) \ge \pi(n) - \omega(n)</math>: Here, <math>\pi(n)</math> is the [[prime-counting function]], and counts the number of primes less than or equal to <math>n</math>, while <math>\omega(n)</math> is the [[prime divisor count function]] of <math>n</math>.
* <math>\varphi(n) \le n - \sigma_0(n) + 1</math>: Here, <math>\sigma_0(n)</math> is the [[divisor count function]], counting the total number of divisors of <math>n</math>.
==Relation with properties of numbers==

* [[Prime number]]: A natural number <math>n</math> such that <math>\varphi(n) = n-1</math>.
* [[Polygonal number]]: A natural number <math>n</math> such that <math>\varphi(n)</math> is a power of <math>2</math>, or equivalently, such that the regular <math>n</math>-gon is constructible using straightedge and compass.

==Dirichlet series==

{{further|[[Formula for Dirichlet series of Euler phi-function]]}}

The [[Dirichlet series]] for the Euler phi-function is given by:

<math>\frac{\varphi(n)}{n^s}</math>.

Using the [[Dirichlet product]] identity <math>\varphi * U = E</math> and the fact that [[Dirichlet series of Dirichlet product equals product of Dirichlet series]], we get:

<math>\sum_{n \in \mathbb{N}} \frac{\varphi(n)}{n^s} \sum_{n \in mathbb{N}} \frac{1}{n^s} = \sum_{n \in \mathbb{N}} \frac{n}{n^s}</math>.

This simplifies to:

<math>\sum_{n \in mathbb{N}} \frac{\varphi(n)}{n^s} = \frac{\zeta(s - 1)}{\zeta(s)}</math>.

This identity holds not just for the formal Dirichlet series, but also for their analytic continuations, and is valid universally for the meromorphic functions.
==Algebraic significance==

The Euler phi-function of <math>n</math> is important in the following ways:

* It is the number of generators of the cyclic group of order <math>n</math>.
* It is the order of the multiplicative group of the ring of integers modulo <math>n</math> (in fact, this multiplicative group is precisely the set of generators of the additive group).