User:Vipul: Difference between revisions

Latest revision as of 00:28, 26 August 2026

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 $\ln (φ (n) / n) / \ln n$ , $\ln (φ (n) / n) / \ln \ln n$ , $\ln (φ (n) / n) / \ln \ln \ln n$ for various $D$ . I expect 0 at primes for all, and 0, 0, -1 at primorials since $(1 - 1 / p)$ for a primorial should have log approx $- \ln \ln \ln n$ .

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 $1 + x R [[x]]$ 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)

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 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)