L-functions and Random Matrix Theory

This web page highlights some of the conjectures and open problems concerning L-functions and Random Matrix Theory.

If you would like to print a hard copy of the whole outline, you can download a dvi, postscript or pdf version.

  1. Distribution of zeros of L-functions
    1. The GUE hypothesis
    2. Correlations of zeros
    3. Neighbor spacing
    4. Zeros of derivatives
      1. Horizontal distribution of zeros of zeta prime
      2. Statistics of the zeros of the derivative of xi
      3. Relationship between symplectic and odd orthogonal
    5. The Alternative Hypothesis
    6. p-adic L-functions
  2. Zeros and primes
    1. The distribution of primes
    2. Gaps between primes
    3. Primes of a special form
    4. An exponential sum involving primes
  3. Mean-values
    1. The mean-value conjectures
    2. Shifted mean values
    3. Multiple Dirichlet series
    4. Ratios of zeta-functions
    5. Lower order terms
      1. Fractional moments
    6. Mollified mean values
      1. Mollifying a Family
      2. Long Mollifiers
    7. Moments of S(T)
    8. The integral of exp(i lambda S(t))
    9. High moments are not natural
  4. Critical values
    1. Ranks of elliptic curves
      1. The frequency of rank 2 curves in a family of quadratic twists
      2. The frequency of higher rank in a family of quadratic twists
      3. Connection with the class number problem
  5. Extremal problems
    1. Critical values
      1. The maximal order of S(T)
      2. The maximal order of the zeta-function on the critical line
      3. The maximal rank of an elliptic curve as a function of its conductor
      4. The distribution of Fourier coefficients of half-integral weight forms
      5. Omega results for twists
    2. Non-critical values
      1. The zeta-function on the 1-line
      2. The order of zeta'/zeta
      3. Class numbers of quadratic fields
    3. Difficult to classify
      1. Large gaps between primes
      2. Extreme gaps between consecutive zeros of the zeta-function
      3. Maximal clusters of zeros of the zeta-function
  6. Function field zeta-functions
    1. Fixed q
    2. Corrections for small degree
    3. Effective computation of zeta-functions
      1. Elliptic curves
      2. Higher genus
    4. More function field analogues
    5. Distribution of sha
  7. Random matrix questions
    1. Derivatives of characteristic polynomials
    2. Eigenvalues of high powers of a matrix
    3. p-adic random matrix theory
  8. Miscellaneous topics
    1. Explicit Formula
    2. Value distribution
    3. GOE and Graphs

The individual contributions may have problems because converting complicated TeX into a web page is not an exact science. The dvi, ps, or pdf versions are your best bet.