Determine the correct admissible exponent at the critical moment

Determine the correct order of magnitude of the critical moment \(\int_0^1 |S_k(\alpha;K)|^{1+1/k}\,d\alpha\) for short intervals and thereby determine whether the exponent \(((k+1)/2)\delta_k\) or its \(\Delta_k\)-analogue should be regarded as \((k,1+1/k)\)-admissible.

Background

The paper defines $\theta_{k,s}$ using the expected moment asymptotics away from the critical exponent $s=1+1/k$. At the critical exponent, however, the known complete-interval bounds differ by a substantial logarithmic factor, and the paper proves only short-interval upper and lower bounds of different logarithmic strengths.

Because the correct order of the critical moment is not known, the authors deliberately avoid assigning the resulting interval-length threshold the formal status of a $(k,1+1/k)$-admissible exponent. Thus, resolving the critical moment’s order would also settle the associated admissibility question.