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riemann         hypothesis

Some numbers have the special property that they cannot be expressed as the product of two smaller numbers, e.g., 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. The distribution of such prime numbers among all natural numbers does not follow any regular pattern.  However, the German mathematician G.F.B. Riemann (1826 - 1866) observed that the frequency of prime numbers is very closely related to the behavior of an elaborate function

    ζ(s) = 1 + 1/2s + 1/3s + 1/4s + ...

called the Riemann Zeta function. The Riemann hypothesis asserts that all interesting solutions of the equation
    ζ(s) = 0
lie on a certain vertical straight line. This has been checked for the first 10,000,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers. This problem is:  Unsolved
Other Millennium Problems


Riemann 1859: courtesy of the State and University Library at Göttingen
Picture

keentween

  • Ask
  • FAQ
  • R Winter Series
    • Create & Manipulate Matrices
    • Data Cleaning, Data Frames & Lists
    • Tidyverse
    • Creating Graphs with ggplot
  • AP CS A
    • Java July Series >
      • :: Classes & OOPs
      • :: Interface
      • :: Inheritance
      • :: Intro To Arrays Java
      • :: Arrays Continued
    • :: CS-A or CS-P?
    • :: Control Structures
    • :: Recursion
    • :: While Do While Loops in Java
    • :: Interface in Java
    • :: ArrayLists
    • :: Java Naming Conventions
    • :: Logic Circuits
    • :: Getters and Setters
    • :: Binary & Hexadecimal
  • Summer 2020 Tutoring
    • :: REPRESENT IT!
    • Pre-Algebra Sessions >
      • :: Basic Division
      • :: Complex Division
      • :: Estimation Division
      • :: Division Practice Problems
    • Algebra II >
      • :: Cubic Equations
      • :: Complex Numbers
    • Chemistry >
      • Molarity Basics
    • C++ Sessions >
      • :: Introduction
      • :: Style v Syntax
      • :: Variables & Data Types
      • :: Intialize/Declare Variables
      • :: Types of Operators
      • :: Strings and Input - Output
      • :: How to Construct Arrays
  • AP Bio
    • :: Sketch Notes >
      • :: Part 1
      • :: Part 2
    • :: epigenetics
    • :: Chi-Squared Tests
    • :: Cancer
    • :: Hox Genes
    • :: Hardy-Weinberg Principle
    • :: Rule of Multiplication + Addition for Punnett Squares
    • :: CRISPR
    • :: Amino Acid
    • :: Peptide
    • :: Why study Peptides
    • :: Aquaporins
    • :: Gram Stains
    • :: Graph on Excel for Bio Lab
  • AP Chem
    • Organic Chemistry
    • I. Properties of Matter >
      • Neutralization
    • II. Periodic table >
      • Org of Periodic Table
      • :: Groups
    • III. Chemical bonding >
      • :: Mass to Mass conversion
      • :: Naming Acids
      • :: Cross Drop Charge
      • :: Predicting Products
      • :: Balance Equation Question
      • :: Learn to Balance Equation
      • :: VSEPR Simulation
    • IV. Molar Mass >
      • ::LR ER and excess reatant
      • :: Molecular/Formula Mass
      • :: Empirical Formula & Molar Mass
      • :: Percentages & Empirical Formula
      • :: Empirical formula
    • IV. Solutions and Solubility >
      • :: Types of Solutions with Solubility Curves
      • :: Solubility Curve
    • V. Easy Tricks and Tips >
      • :: Tip to Molecular Shapes
      • Memorizing Bond Angles and Polarity
      • :: Chemistry Formulas
      • :: Trick Polyatomic ions
    • VI. General AP Concepts >
      • :: Potential Energy Diagrams
      • :: Haber-Bosch
      • :: Le Chatelier
      • :: Pressure & Moles
      • ::Rydberg's Constant vs Unit of Energy
      • :: Equilibrium and RICE Tables
      • :: Kinetics
      • Galvanic Cells
    • :: Flash cards
    • :: VSEPR
  • AP Stats
    • Chi-Squared Tests
    • Solving Chi-Sqd Test Using Sheets
    • Applications of Statistics
    • Standardized Scores
    • Distributions Transformations
  • AP Calc
    • DI Method - Tabular Integration
    • Polar Curves: Tangent Line and Slope
    • Riemann Sums: Left and Right Approximations
    • :: Conic Sections Flash cards
    • :: Parent Functions Flash cards
    • Worked Out Problems >
      • :: Worked Out Problems I
      • :: Worked Out Problems II
      • :: Worked Out Problems III
      • :: Worked Out Problems IV
      • :: Worked Out Problems V
      • :: Worked Out Problems VI
      • :: Worked Out Problems VII
      • :: Worked Out Problems VIII
      • :: Worked Out Problems IX
      • :: Worked Out Problems X
      • :: Worked Out Problems XI
      • :: Worked Out Problems XII
      • :: Worked Out Problems XIII
    • Applying Trig Identities
    • L'Hopital's Rule
    • Differences Between Conic Sections
    • Graphing Conic Sections
    • :: Pre-Calc - Trig Identities
    • Tangent & Normal Lines
    • Indefinite integrals: U Sub
    • Calculus Derivatives >
      • Product Rule
      • Quotient Rule
      • Chain Rule
  • Arduino
    • Quick Look
    • Project #1: Blinking LED
    • Project #2: Button LED
    • Project #3: Flowing LED
    • Project #4: LCD Display
    • Project #5: Serial Monitor
  • AP Español
    • AP Español Salsa
  • App
    • AP Go Pow How?
    • AP Go Pow APP Page
  • Musings
    • :: Bayesian Example
    • :: Nash equilibria
    • :: Bayesian Nash Equilibrium
    • :: Backward induction
    • :: what is ISS
    • :: Rotational Matrices
    • :: Primary v Secondary Pollutants
    • :: Black Hole
    • :: Covid-19 Hackathon
    • :: Evolution of Immunizations
    • :: Predictions of Diseases
    • :: Book List
    • :: Patterncount
    • :: Binary Classification
    • :: Cybersecurity
    • :: What is CIA Triad
    • :: What is Networking
    • :: Self Similarity
    • :: Trig Identities
    • :: UIL Number S
    • :: Box Offensive Play
    • :: Why Card Trick Works
    • :: Easy Multiplication
  • AP CREDIT
  • About