Optimal totient dependence for multiplicative coefficients

Determine the optimal totient dependence in exponential-sum estimates for multiplicative coefficients, beyond the \(\varphi(r)^{-1/2}\) dependence obtained in the main theorem.

Background

The paper shows that an O(E(c))O(E(c)) replacement for the reduced-residue mean is impossible for arbitrary coefficient vectors, by an explicit orthogonality example. However, that example need not arise from a multiplicative function, so it does not settle the arithmetic factor in the multiplicative-function theorem. The conclusion explicitly identifies determining this factor as unresolved and notes that no improvement is proved.

References

The remaining arithmetic dependence presents a different question. Remark~\ref{rem:totient-obstruction} rules out replacing the right-hand side of eq:short-residue by $O(E(c))$ uniformly for arbitrary coefficient vectors. This does not establish the optimal arithmetic factor in the multiplicative-function theorem. A possible direction is to retain multiplicativity or other structure before passing to the unrestricted coefficient estimate, or to change that reduction; this will be the subject of future research. No improvement of the totient factor is proved here.

eq:short-residue:

1φ(r)∑b∈R(r)∣∑n≤Ucne(bn/r)∣2≪log⁡log⁡(3U)∑n≤U∣cn∣2.\frac1{\varphi(r)}\sum_{b\in\mathcal R(r)} \bigg|\sum_{n\le U}c_ne(bn/r)\bigg|^2 \ll \log\log(3U)\sum_{n\le U}|c_n|^2.

— Effective estimates for exponential sums with multiplicative coefficients  (2609.11070 - Robles, 10 Sep 2026) in Remark 2.14, Section "The displacement exponent and a Fourier obstruction"; Section "Conclusion and future work"