Zeta Explained Course

A math course explaining the Riemann zeta function

Curated by: Zeta Explained (121 videos)

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Tracks in this Playlist

• Zeta Explained #01: Intro and the Basel Problem
• Zeta Explained #02: Zeta in the Complex Plane
• Zeta Explained #03: Analytic Properties of Zeta
• Zeta Explained #04: The Euler Product Formula
• Zeta Explained #05: Proof that Zeta has No Zeros with Re(s)＞1
• Zeta Explained #06: The Logarithmic Derivative and the Von Mangoldt Function
• Zeta Explained #07: Intro to Analytic Continuation
• Zeta Explained #08: The Dirichlet Eta Function
• Zeta Explained #09: Riemann–Stieltjes Integration
• Zeta Explained #10: Euler–Maclaurin Summation
• Zeta Explained #11: The Laurent Series of the Zeta Function at s=1
• Zeta Explained #12: The Explicit Formula for Zeta at Even Integers
• Zeta Explained #13: Proof that the Zeta Function Has No Zeros on the Line Re(s) = 1
• Zeta Explained #14: Intro to the Gamma Function
• Zeta Explained #15: Riemann's Functional Equation Connects ζ(s) and ζ(1-s)
• Zeta Explained #16: Proof of Riemann's Functional Equation
• Zeta Explained #17: Riemann's Xi Function
• Zeta Explained #18: Growth Rates of Complex Functions (More Zeros = Faster Growth)
• Zeta Explained #19: Proof that the Number of Zeros in the Critical Strip Is O(T log T)
• Zeta Explained #20: Hadamard's Factorization of the Zeta Function
• Zeta Explained #21: The Riemann–von Mangoldt Formula Counts Zeros of the Zeta Function
• Zeta Explained #22: The Classical Zero-Free Region Inside the Critical Strip
• Zeta Explained #23: Intro to Exponential Sums
• Zeta Explained #24: An Approximate Functional Equation for Zeta
• Zeta Explained #25: The Lindelöf Hypothesis
• Zeta Explained #26: The Phragmén–Lindelöf Principle and the Order of Zeta in the Critical Strip
• Zeta Explained #27: The Prime Number Theorem (1/2)
• Zeta Explained #28: The Prime Number Theorem (2/2)
• Zeta Explained #29: Visualizing the Riemann Hypothesis
• Zeta Explained #30: The Riemann–Siegel Z Function Under the Riemann Hypothesis
• Zeta Explained #31: Dirichlet Series
• Zeta Explained #32: Möbius Inversion
• Zeta Explained #33: The Mertens Conjecture (and the Riemann Hypothesis)
• Zeta Explained #34: How the Mertens Conjecture was Disproved
• Zeta Explained #35: Diophantine Approximation
• Zeta Explained #36: Omega Results for Re(s) ≥ 1
• Zeta Explained #37: Zeta Can Get Close to Zero in Re(s) ≥ 1
• Zeta Explained #38: Mean Value Theorems of Zeta
• Zeta Explained #39: Riemann's Global Equation for ζ
• Zeta Explained #40: Riemann's Proof of the Functional Equation
• Zeta Explained #41: Riemann's Second Proof of the Functional Equation (via Jacobi Theta Functions)
• Zeta Explained #42: Riemann's ξ Function and the Origin of the Riemann Hypothesis
• Zeta Explained #43: Riemann's Connection Between ζ and Primes
• Zeta Explained #44: Primes as Waves
• Zeta Explained #45: Von Mangoldt's Explicit Formula for ψ(x)
• Zeta Explained #46: Zeta at Zero
• Zeta Explained #47: The Truncated Explicit Formula for ψ(x)
• Zeta Explained #48: Proof of the Prime Number Theorem
• Zeta Explained #49: An Omega Bound on the Prime Number Theorem
• Zeta Explained #50: Littlewood's Log Log Log Bound on the Prime Number Theorem
• Zeta Explained #51: Skewes' Number and Gauss's False Conjecture on Primes
• Zeta Explained #52: Prime Gaps
• Zeta Explained #53: Zeta Gaps (Are ≤12)
• Zeta Explained #54: Littlewood's Log Log Log Bound on Zeta Gaps
• Zeta Explained #55: S(T) and the Accuracy of the Riemann–von Mangoldt Formula
• Zeta Explained #56: S(T) as a Divergent Sum of Prime Waves
• Zeta Explained #57: The Duality of Primes and Zeta Zeros
• Zeta Explained #58: The Riemann–Siegel Z and θ Functions
• Zeta Explained #59: Gram Points and Gram's Law
• Zeta Explained #60: Hardy's Proof of Infinitely Many Zeros on Critical Line
• Zeta Explained #61: The Hardy–Littlewood Approximate Functional Equation
• Zeta Explained #62: The Cosine Sum on the Critical Line
• Zeta Explained #63: The Riemann–Siegel Formula
• Zeta Explained #64: Computational Methods for Zeta
• Zeta Explained #65: Example Computations and How to Find a Zero
• Zeta Explained #66: Edwards' Speculation on the Riemann Hypothesis
• Zeta Explained #67: Historical Verification of the Riemann Hypothesis (1859–1936)
• Zeta Explained #68: Turing's Method of Verifying the Riemann Hypothesis (1953–Present)
• Zeta Explained #69: Deep Dive into Gamma (Weierstrass Product Formula)
• Zeta Explained #70: Proving Gamma Formulas (Euler's Reflection & Legendre's Duplication)
• Zeta Explained #71: Laplace's Method for Stirling's Approximation
• Zeta Explained #72: The Saddle-Point Method for Stirling's Approximation
• Zeta Explained #73: The Riemann–Siegel Formula via Saddle-Point Method
• Zeta Explained #74: Titchmarsh's Proof of Infinitely Many Zeros on Critical Line
• Zeta Explained #75: The Visible Grid Problem
• Zeta Explained #76: Euler's Totient Function and its Dirichlet Series
• Zeta Explained #77: Square-Free and k-Free Numbers
• Zeta Explained #78: Powerful Numbers
• Zeta Explained #79: Liouville's Function
• Zeta Explained #80: Pólya's (False) Conjecture
• Zeta Explained #81: Divisor Functions and Dirichlet Series
• Zeta Explained #82: Proof of Ramanujan's Divisor Formula
• Zeta Explained #83: Using Zeta to Solve Arithmetic Identities
• Zeta Explained #84: Perron's Inversion Formula
• Zeta Explained #85: Landau's Theorem on Dirichlet Series
• Zeta Explained #86: Prime Progressions and L-Functions (Part 1)
• Zeta Explained #87: Prime Progressions and L-Functions (Part 2)
• Zeta Explained #88: Prime Progressions and L-Functions (Part 3)
• Zeta Explained #89: Abel's Summation Formula and Abel's Inequality
• Zeta Explained #90: Weyl Differencing
• Zeta Explained #91: The Hardy–Littlewood 1/6 Bound on the Lindelöf Hypothesis
• Zeta Explained #92: van der Corput's A Process
• Zeta Explained #93: van der Corput's B Process
• Zeta Explained #94: Alternate Proof of 1/6 Bound on Lindelöf Hypothesis (van der Corput's Method)
• Zeta Explained #95: Exponent Pairs and the Lindelöf Hypothesis
• Zeta Explained #96: The A and B Processes on Exponent Pairs
• Zeta Explained #97: An Overview of Two-Dimensional Exponential Sums
• Zeta Explained #98: The Dirichlet Divisor Problem
• Zeta Explained #99: The Gauss Circle Problem
• Zeta Explained #100: Riemann Hypothesis Implies Lindelöf Hypothesis