The maximum over characters of |Σ_{n≤N} f(n)χ(n)|, for any multiplicative f with |f(n)| = 1, is at least √N exp((1+o(1))√(log(q/N)/log₂(q/N))) in the range exp((log q)^{1/2+δ}) ≤ N ≤ √q.
(French) [G´ al-type sums and applications], Proc
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Large values of character sums with multiplicative coefficients
The maximum over characters of |Σ_{n≤N} f(n)χ(n)|, for any multiplicative f with |f(n)| = 1, is at least √N exp((1+o(1))√(log(q/N)/log₂(q/N))) in the range exp((log q)^{1/2+δ}) ≤ N ≤ √q.