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4. Distribution of Values of L-functions


    1. Add remark

      Problem 4.1.

      [Kannan Soundararajan] Suppose $T>1000$, say. Show that \[\int_0^T |\zeta(\frac{1}{2}+it)|^{6}dt< T(\log T)^9.\] More generally, for $k\geq 2$, prove that

      \[\int_0^T |\zeta(\frac{1}{2}+it)|^{2k}dt\leq T(\log T)^{k^2}.\]
          Remarks. i) This would capture sub-Gaussianity.

      ii) We believe that \[\int_0^T |\zeta(\frac{1}{2}+it)|^{2k}dt\sim c_k T(\log T)^{k^2},\] where $c_k\sim \exp(-k^2\log k -k^2 \log\log k),$ as $k\to \infty$.

      iii) From the works of Soundararajan and Harper we know, conditional on RH, that \[\int_0^T |\zeta(\frac{1}{2}+it)|^{2k}dt\leq f_k T(\log T)^{k^2},\] where $f_k=\exp(\exp(k^{100}))$.

        • Lower Order Terms of Moments


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          Problem 4.2.

          [Henryk Iwaniec] Is it possible to determine subleading terms for moments of L-functions from RMT heuristics? In particular, can this be understood for low moments?

            • Weighted Central Limit Theorems


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              Problem 4.3.

              [Anurag Sahay] In Fazzari’s thesis, he proved, under RH, a central limit theorem for $\log|\zeta(\frac{1}{2}+it)|$ where the measure is twisted by $|\zeta|^2$ and $|\zeta|^4$. Can you do the analogous problem for $\log|\Lambda_X(z)|$?

                • Lower Order Terms of Moments of the Derivative


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                  Problem 4.4.

                  [Michael Rubinstein] In RMT we have finite $N$ formulae for moments of $\Lambda'_X(1)$. From this, can we predict lower order terms for the moments of $\zeta'(\frac{1}{2}+it)$?

                    • Twisted Moments in Random Matrix Theory

                          In a series of papers in 2015, Conrey and Keating established a heuristic for predicting asymptotic formulas for the moments of $\zeta(s)$. This involved using lower twisted moments of $\zeta(s)$ to calculate the higher moments. In a recent paper, Baluyot and Conrey proved that this heuristic, the ‘BK splitting’, can be rigorously understood in random matrix theory, where one looks at ‘twisted moments’ over the unitary group.

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                      Problem 4.5.

                      [Anurag Sahay] Can you prove an analogous BK-splitting for the other classical compact groups, such as $SO(2N)$ and $Sp(2N)$?

                          Cite this as: AimPL: Moments of the derivative of characteristic polynomials and L-functions, available at http://aimpl.org/lprime.