Effective uniform non-vanishing bounds

Determine a computable function η_{k,s}(X) tending to zero that bounds the left-hand sides of the uniform non-vanishing estimates for standard and twisted symmetric-square L-values by η_{k,s}(X)P_b(X), and provide an explicit X_0(ε) such that η_{k,s}(X)<ε for every X≥X_0(ε).

Background

Theorems 1 and 2 prove that the proportion of primes p≤X in a fixed residue class modulo 4 for which a standard twisted L-value, or a nontrivial linear combination of twisted symmetric-square L-values, vanishes tends to zero uniformly in the nonzero coefficient vector.

These results are qualitative: they provide an o(P_b(X)) estimate but no computable rate of convergence or explicit threshold. The first question in the future-work section asks for effective versions supplying both.

References

For fixed $b\in\{1,3\}$, can one give a computable function $\eta_{k,s}(X)\to0$ that bounds the left sides of `eq:standarduniformand \eq:symuniform` by $\eta_{k,s}(X)P_b(X)$ in their respective ranges, together with an explicit $X_0(\epsilon)$ such that $\eta_{k,s}(X)<\epsilon$ for every $X\geq X_0(\epsilon)$?

— Uniform non-vanishing of prime quadratic twists of standard and symmetric square $L$-functions  (2610.11974 - Ni, 8 Oct 2026) in Section 8, Questions for future work, Question 1