The motivation for the following problem stems from the recent preprint by James Maynard, see also Proposition 5 of the recent blogpost by Terrence Tao. The solution of this problem could give better bounds on the length of intervals whose translates contain tuples of primes infinitely often.
Let $\Delta_k$ be the standard simplex in $\mathbb{R}^k$, that is $$ \Delta_k = \left\{ (x_1,\ldots,x_k) \in \mathbb{R}^k \, | \, 0 \leq x_i \leq 1 \,\&\, \sum_{i=1}^k x_i \leq 1 \right\}. $$ amd for a smooth function $F$ on $\Delta_k$ define its $m$th integral slice by $$ F_{(m)}(u_1,\ldots,\widehat{u_m},\ldots,u_k) = \int_0^{1-\sum_{i\neq m} u_i} F(u_1,\ldots,u_{m-1},t_m,u_{m+1},\ldots,u_k) \,\mathrm{d}t_m $$
Consider two functionals on the space of smooth functions on $\Delta_k$ $$ I_k(F) = \int_{\Delta_k} F(x)^2\, \mathrm{d}x $$ and $$ J^{(m)}_k (F) = \int_{\Delta_k^{(m)}} F_{(m)}(x_1,\ldots,\widehat{x_m},\ldots,x_k)^2 \, \mathrm{d}x_1\cdots \mathrm{d}x_{m-1} \mathrm{d}x_{m+1} \cdots \mathrm{d}x_k $$ where $\Delta_k^{(m)}$ is the standard simplex in coordinates $\{ x_1, \ldots, \widehat{x_m}, \ldots, x_k \}$.
Problem: The ultimate goal is to find $$ M_k = \sup_{F\in\mathcal{C}^\infty(\Delta_k)} \frac{\sum_{m=1}^k J^{(m)}_k(F)}{I_k(F)}. $$ but even a nice basis of orthogonal symmetric polynomials on $\Delta_k$ would be useful for numerical experiments.
Notes:
- Since the numerator in the definition of $M_k$ is symmetric, we can restrict to smooth symmetric functions on $\Delta_k$. Maynard's preprint contains some results with certain symmetric polynomials as well as some bounds for large $k$.
- The maximizer $F$ of the ratio $\sum_{m=1}^k J^{(m)}_k(F)/I_k(F)$ is an eigenfunction for $\mathcal{L}_k$ with eigenvalue equal to the vaule of ratio at $F$ where $$\mathcal{L}_kF(u_1,\ldots,u_k) = \sum_{m=1}^k F_{(m)}(u_1,\ldots,\widehat{u_m},\ldots,u_k). $$
- If one considers only functions of the form $F(u_1,\ldots,u_k) = g(u_1+\cdots+u_k)$, then the solution is given in terms of Bessel functions $J_{k-2}$, see [Tao] and [FPR] for details.
