Central-point partial Euler-product conjecture

Establish the stated limiting formula for the partial Euler product at the central point s=1/2 of an isobaric automorphic representation whose cuspidal constituents have entire standard L-functions, including the logarithmic factor determined by the central-point vanishing order and the constant involving the second-moment order and the central derivative.

Background

This is the Kaneko–Koyama–Kurokawa conjecture used as the key analytic input in the paper. It predicts convergence, or a precise logarithmic decay rate, for Euler products on the critical line, extending the range in which Euler products of automorphic L-functions are expected to behave regularly.

References

Conjecture \ref{DRH} predicts that the partial Euler product defining $L(s, \pi)$ should converge at the central point $s=\frac{1}{2}$ to $\sqrt{2}{\nu (\pi)} \cdot L \left( \frac{1}{2}, \pi \right)$ if $m =0$, and that it goes to zero at the rate of $(\log x){-m}$ if $m 1$.

— Products of point counts of higher genus curves over finite fields  (2608.18014 - Bucur et al., 18 Aug 2026) in Conjecture \ref{DRH}, Section 2.3, subsection “Euler products at the central point”