Logarithmic exponential sums with multiplicative coefficients
Haozhe Gou
Abstract
Let f be a 1-bounded multiplicative function, and suppose that |α-a/q|≤ 1/(qx), where (a,q)=1 and q≤ x. We prove that |Σn≤ xf(n)ne( nα)| ≤ (46+o(1)) xq+O( q). Here the constant 4/6 is optimal. This settles two conjectures of Granville and Lamzouri recently. If f is completely multiplicative, the constant improves to 1. The main new ingredient is a two-prime Ramaré decomposition identity.
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