Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I have never been able to derive a "satisfying answer."

Set-up: Let $r\ge 1$ and let $f \in L^2[0,1]$ be a continuous real-valued function of bounded variation on $[0,1]$, normalized so that $$ \int_0^1(1-u)^{r^2-1}f(u)^2 du = 1. $$ Further define $M(c)=M(c,f,r)$ as $$ M(c):=c+\frac{2 r}{\pi}\int_0^1 (1-u)^{r^2-1}f(u) \int_0^u \frac{\sin(\pi c v)}{v} f(u-v) \ dv \ du.$$

Question: How does one choose $r$ and $f$ optimally so that $$ M(c) >1$$ for $c$ as small as possible?

An argument of Conrey, Ghosh, and Gonek [2] can be used to show that $M(c)<1$ if $c<\frac{1}{2}$ for any such $f$ and $r$. In [1], choosing $f$ to be a polynomial of low degree ($\le 6$) and using Mathematica to numerically optimize the $r$ and the coefficients, we were able to find $f$ and $r$ such that $M(.5155)>1$.

In the special case when $r=1$, Montgomery and Odlyzko [3] actually solved the optimization problem using prolate spheroidal wave functions. Here is the basic idea of their argument. In this case, we have $$ M(c) = c+ \int_0^1\int_0^1 f(u) f(v) \frac{\sin(\pi c(u-v))}{\pi(u-v)} \ dv \ du. $$ The double integral on the right-hand side is $$ \int_{-c/2}^{c/2} \left| \int_0^1 f(v) e^{2\pi i v} dv \right|^2 dt :=I(c),$$ say. They then observed that choosing $$ f(x) = aR_{00}^{(1)}[\pi c/2,2x-1]$$ maximizes $I(c)$, where $R_{mn}^{(1)}[c,x]$ is the radial prolate spheroidal wave function of the first kind of order m and degree n, and a is a constant to be chosen according to our above normalization.

These wave functions are very hard to study numerically, and Montgomery and Odlyzko approximated them using modified Bessel functions. In [1], when $r=1$, we recovered their results to four decimal places using polynomials of degree four. So in this case it seems that polynomials of small degree work (almost) as well as more sophisticated techniques.

References:

[1] H. M. Bui, M. B. Milinovich, and N. C. Ng, A note on the gaps between consecutive zeros of the Riemann zeta-function, Proc. Amer. Math. Soc. 138 (2010), no. 12, pp. 4167-4175.

[2] J. B. Conrey, A. Ghosh, and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull. London Math. Soc. 16 (1984), 421–424.

[3] H. L. Montgomery and A. M. Odlyzko, Gaps between zeros of the zeta function, Colloq. Math. Soc. Janos Bolyai, 34. Topics in Classical Number Theory (Budapest, 1981), North-Holland, Amsterdam, 1984.