The Riemann Hypothesis

This web page highlights some of the conjectures and open problems concerning The Riemann Hypothesis.

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  1. What is an $L$-function?
    1. Terminology and basic properties
      1. Functional equation
      2. Euler product
      3. $\xi$ and $Z$ functions
      4. Critical line and critical strip
      5. Trivial zeros
      6. Zero counting functions
    2. Arithmetic L-functions
      1. The Riemann zeta function
      2. Dirichlet L-functions
      3. Dedekind zeta functions
      4. GL(2) L-functions
        1. Dirichlet series associated with holomorphic cusp forms
          1. Examples
        2. Dirichlet series associated with Maass forms
      5. Higher rank L-functions
    3. The Selberg class
      1. Dirichlet series
      2. Analytic Continuation
      3. Functional Equation
      4. Euler Product
      5. Ramanujan Hypothesis
      6. Selberg Conjectures
    4. Analogues of zeta-functions
      1. Dynamical zeta-functions
      2. Spectral zeta functions
  2. Riemann Hypotheses
    1. Riemann Hypotheses for global L-functions
      1. The Riemann Hypothesis
      2. The Generalized Riemann Hypothesis
      3. The Extended Riemann Hypothesis
      4. The Grand Riemann Hypothesis
    2. Other statements about the zeros of L-functions
      1. Quasi Riemann Hypothesis
      2. The Density Hypothesis
      3. 100 percent hypothesis
      4. Zeros on the $\sigma=1$ line
      5. Landau-Siegel zeros
      6. The vertical distribution of zeros
    3. The Lindelof hypothesis and breaking convexity
    4. Perspectives on RH
      1. Analytic number theory
      2. Physics
      3. Probability
      4. Fractal geometry
  3. Equivalences to RH
    1. Primes
      1. The error term in the PNT
      2. More accurate estimates
    2. Arithmetic functions
      1. Averages
        1. The von Mangoldt function
        2. The M\"obius function
      2. Large values
        1. The sum of divisors of $n$
        2. The Euler function
        3. The maximal order of an element in the symmetric group
    3. The Farey series
      1. Mikolas functions
      2. Amoroso's criterion
    4. Weil's positivity criterion
      1. Li's criterion
      2. Bombieri's refinement
    5. Complex function theory
      1. Speiser's criterion
      2. Logarithmic integrals
      3. An inequality for the logarithmic derivative of xi
    6. Function spaces
      1. The Beurling-Nyman Criterion
        1. A mollification problem
      2. Salem's criterion
    7. Other analytic estimates
      1. M. Riesz series
      2. Hardy-Littlewood series
      3. Newman's criterion
      4. Polya's integral criterion
    8. Grommer inequalities
    9. Dynamical systems
    10. Redheffer's matrix
  4. Attacks on the Riemann Hypothesis
    1. Alain Connes's approach
      1. The dynamical system problem studied by Bost--Connes
      2. The C*-algebra of Bost--Connes
    2. Iwaniec' approach
      1. families of rank 2 elliptic curves
    3. Unsuccessful attacks on the Riemann Hypothesis
      1. Zeros of Dirichlet polynomials
      2. de Branges' positivity condition
  5. Zeta Gallery
  6. Anecdotes about the Riemann Hypothesis

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