In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by where the product is taken over all primes dividing (By convention, , which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions. The value of for the first few integers is: 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... (sequence in the OEIS). The function is greater than for all greater than 1, and is even for all greater than 2. If is a square-free number then , where is the divisor function. The function can also be defined by setting for powers of any prime , and then extending the definition to all integers by multiplicativity. This also leads to a proof of the generating function in terms of the Riemann zeta function, which is This is also a consequence of the fact that we can write as a Dirichlet convolution of . There is an additive definition of the psi function as well. Quoting from Dickson, R. Dedekind proved that, if is decomposed in every way into a product and if is the g.c.d. of then where ranges over all divisors of and over the prime divisors of and is the totient function. (Wikipedia).
Transcendental Functions 19 The Function a to the power x.mp4
The function a to the power x.
From playlist Transcendental Functions
What are the Inverse Trigonometric functions and what do they mean?
👉 Learn how to evaluate inverse trigonometric functions. The inverse trigonometric functions are used to obtain theta, the angle which yielded the trigonometric function value. It is usually helpful to use the calculator to calculate the inverse trigonometric functions, especially for non-
From playlist Evaluate Inverse Trigonometric Functions
Transcendental Functions 21 The Derivative of e to the power x.mp4
The derivative of e to the power x.
From playlist Transcendental Functions
Transcendental Functions 13 Derivatives of a Function and its Inverse.mov
The first derivative of a function and the inverse of that function.
From playlist Transcendental Functions
Transcendental Functions 3 Examples using Properties of Logarithms.mov
Examples using the properties of logarithms.
From playlist Transcendental Functions
A3 More graphs and their functions
We expand to transcendental functions such a trigonometric functions. Ply around with the Desmos calculator software and learn more about the how variables that can appear in trigonometric functions affect the graphs of those functions.
From playlist Biomathematics
Inverse Trigonometric Functions
We know about inverse functions, and we know about trigonometric functions, so it's time to learn about inverse trigonometric functions! These are functions where you plug in valid values that trig functions can possess, and they spit out the angles that produce them. There's a little more
From playlist Trigonometry
Abstract Algebra | A PID that is not a Euclidean Domain
We present an example of a principal ideal domain that is not a Euclidean domain. We follow the outline described in Dummit and Foote. In particular, we show that an integral domain D is a PID if and only if it has a Dedekind-Hasse Norm and that every Euclidean domain has a universal side
From playlist Abstract Algebra
Second part to this video: http://youtu.be/shEk8sz1oOw More links & stuff in full description below ↓↓↓ If the highest power of a function or polynomial is odd (e.g.: x^3 or x^5 or x^4371) then it definitely has a solution (or root) among the real numbers. Here's a nice proof demonstrate
From playlist David Eisenbud on Numberphile
Transcendental Functions 22 The integral of e to the power u.mp4
The integral of e to the power u.
From playlist Transcendental Functions
A crash course in Algebraic Number Theory
A quick proof of the Prime Ideal Theorem (algebraic analog of the Prime Number Theorem) is presented. In algebraic number theory, the prime ideal theorem is the number field generalization of the prime number theorem. It provides an asymptotic formula for counting the number of prime idea
From playlist Number Theory
Set Theory (Part 15b): Dedekind Cuts for Complicated Numbers
Please leave your questions, comments, and thoughts below! In this video, I try to give more examples of Dedekind cuts beyond the standard example of sqrt(2), such as e, pi, and sin(2). It is essential for the theory to be intelligible for there to be numerous examples. Unfortunately, not
From playlist Set Theory by Mathoma
Dedekind domains: Introduction
This lecture is part of an online graduate course on commutative algebra, and is an introduction to Dedekind domains. We define Dedekind domains, and give several examples of rings that are or are not Dedekind domains. This is a replacement video: as several alert viewers pointed out, t
From playlist Commutative algebra
Set Theory (Part 14): Real Numbers as Dedekind Cuts
Please feel free to leave comments/questions on the video and practice problems below! In this video, we will construct the real number system as special subsets of rational numbers called Dedekind cuts. The trichotomy law and least upper bound property of the reals will also be proven. T
From playlist Set Theory by Mathoma
Evaluate inverse of cosecant using a calculator
👉 Learn how to evaluate inverse trigonometric functions. The inverse trigonometric functions are used to obtain theta, the angle which yielded the trigonometric function value. It is usually helpful to use the calculator to calculate the inverse trigonometric functions, especially for non-
From playlist Evaluate Inverse Trigonometric Functions
Fundamentals of Mathematics - Lecture 33: Dedekind's Definition of Infinite Sets are FInite Sets
https://www.uvm.edu/~tdupuy/logic/Math52-Fall2017.html
From playlist Fundamentals of Mathematics
Umberto Bottazzini, The immense sea of the infinite - 10 aprile 2019
https://www.sns.it/it/evento/the-immense-sea-of-the-infinite Umberto Bottazzini (Università degli Studi di Milano) The immense sea of the infinite Abstract In a celebrated talk Hilbert stated that the infinite was nowhere to be found in the real, external world. Yet from time immemorial
From playlist Colloqui della Classe di Scienze
Difficulties with Dedekind cuts | Real numbers and limits Math Foundations 116 | N J Wildberger
Richard Dedekind around 1870 introduced a new way of thinking about what a real number `was'. By analyzing the case of sqrt(2), he concluded that we could associated to a real number a partition of the rational numbers into two subsets A and B, where all the elements of A were less than al
From playlist Math Foundations
Sigmoid functions for population growth and A.I.
Some elaborations on sigmoid functions. https://en.wikipedia.org/wiki/Sigmoid_function https://www.learnopencv.com/understanding-activation-functions-in-deep-learning/ If you have any questions of want to contribute to code or videos, feel free to write me a message on youtube or get my co
From playlist Analysis
A Mathematical Analysis Book so Famous it Has a Nickname
A Mathematical Analysis Book so Famous it Has a Nickname In this video I go over the famous book "Baby Rudin", also known as "Principles of Mathematical Analysis" written by Walter B. Rudin. This book is notoriously rigorous and has some pros and cons, which I go over carefully in the vid
From playlist Cool Math Stuff