Asymptotic evaluation of the diagonal main-term expression in the transition range

Obtain an asymptotic formula for the diagonal main-term expression for the averaged twisted second moment of symmetric-square L-functions displayed in equation (2mom MT2 eq3), particularly in the transitional regime e_1e_2 approximately k when the twist satisfies r much larger than k.

Background

The paper studies the averaged twisted second moment M_2^a(r,K) of symmetric-square L-functions attached to holomorphic Hecke cusp forms, decomposing it into a diagonal main term and off-diagonal contributions. The diagonal component MT_2(r,K) is represented by a contour integral involving a sum over divisors e=e_1e_2^2 of r^2 and the archimedean factor of the symmetric-square L-function.

The authors explain that the archimedean quotient behaves like k^z, producing a factor (k/(e_1e_2))^z in the contour integrand. When e_1e_2 is much larger or much smaller than k, contour shifting yields an asymptotic formula. The unresolved case is the transition range e_1e_2 approximately k, which can arise when r is larger than k. The paper resolves the overall moment problem by combining the diagonal and off-diagonal terms, but it explicitly states that it does not obtain an asymptotic formula for this diagonal expression on its own.