Unsolved problems in number theory | Zeta and L-functions | Analytic number theory | Millennium Prize Problems | Conjectures
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann, after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is 1/2. Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit. (Wikipedia).
What is the Riemann Hypothesis?
This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation
From playlist Mathematics
Understanding and computing the Riemann zeta function
In this video I explain Riemann's zeta function and the Riemann hypothesis. I also implement and algorithm to compute the return values - here's the Python script:https://gist.github.com/Nikolaj-K/996dba1ff1045d767b10d4d07b1b032f
From playlist Programming
A (compelling?) reason for the Riemann Hypothesis to be true #SOME2
A visual walkthrough of the Riemann Zeta function and a claim of a good reason for the truth of the Riemann Hypothesis. This is not a formal proof but I believe the line of argument could lead to a formal proof.
From playlist Summer of Math Exposition 2 videos
Sir Michael Atiyah | The Riemann Hypothesis | 2018
Slides for this talk: https://drive.google.com/file/d/1DNHG9TDXiVslO-oqDud9f-9JzaFCrHxl/view?usp=sharing Sir Michael Francis Atiyah: "The Riemann Hypothesis" Monday September 24, 2018 9:45 Abstract: The Riemann Hypothesis is a famous unsolved problem dating from 1859. I will present a
From playlist Number Theory
The Riemann Hypothesis, Explained
The Riemann hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. In fact, the person who solves it will win a $1 million prize from
From playlist Explainers
The Field With One Element and The Riemann Hypothesis (Full Video)
A crash course of Deninger's program to prove the Riemann Hypothesis using a cohomological interpretation of the Riemann Zeta Function. You can Deninger talk about this in more detail here: http://swc.math.arizona.edu/dls/ Leave some comments!
From playlist Riemann Hypothesis
Some identities involving the Riemann-Zeta function.
After introducing the Riemann-Zeta function we derive a generating function for its values at positive even integers. This generating function is used to prove two sum identities. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist The Riemann Zeta Function
More identities involving the Riemann-Zeta function!
By applying some combinatorial tricks to an identity from https://youtu.be/2W2Ghi9idxM we are able to derive two identities involving the Riemann-Zeta function. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist The Riemann Zeta Function
The Riemann Hypothesis and a New Math Tool (a new Indeterminate form)
In this video, you will see a mistake made by many(*) mathematicians. Also, you will see a simple proof for a new(**) indeterminate form that has an incredible connection to the Riemann hypothesis. Lastly, you will see a route to a new promising math tool to solve problems like the Rieman
From playlist Summer of Math Exposition 2 videos
"What is the Riemann Hypothesis and why does it matter?" by Ken Ono
The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of one-half. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize. In this lecture, Ken Ono wi
From playlist Number Theory Research Unit at CAMS - AUB
Youness Lamzouri: Large character sums
Abstract : For a non-principal Dirichlet character χ modulo q, the classical Pólya-Vinogradov inequality asserts that M(χ):=maxx|∑n≤xχ(n)|=O(q‾√log q). This was improved to q‾√log log q by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, th
From playlist Number Theory
The Riemann Hypothesis is one of the Millennium Prize Problems and has something to do with primes. What's that all about? Rather than another hand-wavy explanation, I've tried to put in some details here. Some grown-up maths follows. More information: http://www.claymath.org/publications
From playlist My Maths Videos
"How to Verify the Riemann Hypothesis for the First 1,000 Zeta Zeros" by Ghaith Hiary
An overview of algorithms and methods that mathematicians in the 19th century and the first half of the 20th century used to verify the Riemann hypothesis. The resulting numerical computations, which used hand calculations and mechanical calculators, include those by Gram, Lindelöf, Backlu
From playlist Number Theory Research Unit at CAMS - AUB
Weil conjectures 1 Introduction
This talk is the first of a series of talks on the Weil conejctures. We recall properties of the Riemann zeta function, and describe how Artin used these to motivate the definition of the zeta function of a curve over a finite field. We then describe Weil's generalization of this to varie
From playlist Algebraic geometry: extra topics
CTNT 2018 - "L-functions and the Riemann Hypothesis" (Lecture 1) by Keith Conrad
This is lecture 1 of a mini-course on "L-functions and the Riemann Hypothesis", taught by Keith Conrad, during CTNT 2018, the Connecticut Summer School in Number Theory. For more information about CTNT and other resources and notes, see https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2018 - "L-functions and the Riemann Hypothesis" by Keith Conrad
Riemann Hypothesis - Numberphile
Featuring Professor Edward Frenkel. Here is the biggest (?) unsolved problem in maths... The Riemann Hypothesis. More links & stuff in full description below ↓↓↓ Prime Number Theorem: http://youtu.be/l8ezziaEeNE Fermat's Last Theorem: http://youtu.be/qiNcEguuFSA Prof Edward Frenkel's boo
From playlist Edward Frenkel on Numberphile
Provided to YouTube by DistroKid The Riemann Hypothesis · Humus I Thought You Had a Backup Plan ℗ 973913 Records DK Released on: 2018-10-05 Auto-generated by YouTube.
From playlist And 1, and 2, and 1,2,4,1!