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3. Distribution of the Zeros of L-functions and their Derivatives.

    1. Zeros of $\zeta'(s)$ Close to the Half Line.


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      Problem 3.1.

      [Brian Conrey] Prove that there are zeros close to the half line and hence, improve the proportion of zeros of $\zeta$ on the half line.

        • Algorithms for Computing Zeros of $\zeta'(s)$


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          Problem 3.2.

          [Chris Hughes] Can one find practical algorithms for finding zeros of the derivative of $\zeta(s)$?

            • The Double Hump Problem

                  It has been observed with numerical simulations that the (normalized) radial distribution of the zeros of $\Lambda'_X(z)$ has a bimodal distribution. David Farmer has a paper on the arxiv with numerical evidence of this. Analogously, a similar phenomenon has been observed for the horizontal distribution of zeros of $\zeta'(\frac{1}{2 }+it)$.

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              Problem 3.3.

              [Brian Conrey] Can we identify the source of the bimodal distribution?

                • Correlation Statistics for Consecutive Pairs of Zeros of $\zeta(\frac{1}{2}+it)$.


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                  Problem 3.4.

                  [Hugh Montgomery] If one chooses four consecutive zeros of $\zeta(\frac{1}{2}+it)$ at random high up the critical line, can you quantify the probability that the spacings of the two consecutive pairs differ from the mean? Some similar questions to consider are:
                  • 1. Is the ‘Dot-Dash Conjecture’ true? This says that if two zeros are closer to each other on average, then this biases the following gap to be larger than average.
                  • 2. Soundararajan has asked, do $50$ percent of pairs have gaps smaller than the mean spacing?
                  • 3. Is the mean spacing also the median spacing? If not, then what is it?

                    • Gaps Between Low Lying Zeros of Dirichlet L-functions


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                      Problem 3.5.

                      [Henryk Iwaniec] Is it possible to quantify the gaps between low lying zeros of $L(s,\chi)$ and $L(s,\overline{\chi})$? For instance, can one estimate the size, as $\chi$ varies, of $|\gamma_1-\gamma_2|$ where $\gamma_1,\gamma_2$ are zeros of $L(s,\chi),L(s,\overline{\chi})$, respectively.

                        • Long Mollifiers


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                          Problem 3.6.

                          [Andrew Pearce-Crump] Can we improve the percentage of simple zeros by taking a longer mollifier?
                              David Farmer has remarked that if one is using the simple zero method, then taking $\theta=4/7$ instead of $1/2$ gives a $4$ percent increase in the proportion of zeros.

                            • Short Mollifiers

                                  In a 2025 paper of Conrey, Farmer, Kwan, Lin and Turnage-Butterbaugh, it was shown that Levinson’s method will give a positive proportion of zeros on the critical line no matter how short the mollifier is.

                              Levinson’s method with twisted 4th vs 2nd moments can be found in a 2014 preprint of Hung Bui.

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                              Problem 3.7.

                              [Andrew Pearce-Crump] Prove that a positive proportion of zeros lie on the critical line by using Levinson’s method with a twisted 4th moment, no matter how short the mollifier is.

                                • The Distribution of Zeros of $\zeta'(s)$.


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                                  Problem 3.8.

                                  [Brian Conrey] What is the horizontal distribution of $\rho-\frac{1}{2}$, where $\rho$ are the zeros of $\zeta'(s)$, on the scale of the mean spacing?

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