On exponential sums with multiplicative coefficients
Haozhe Gou
Abstract
Let f:Nz∈C:|z|≤1 be multiplicative, and suppose that α has a rational approximation a/q with |α-a/q|≤ 1/(qx) and q≤ x/(( x)2( x)3). We prove that |Σn≤ xf(n)e(nα)|≤ (46+o(1))x q +O(x x), whereas complete multiplicativity improves 4/6 to 1. Both constants are optimal. In particular, this confirms a folklore conjecture on exponential sums with multiplicative coefficients motivated by the work of Montgomery and Vaughan (1977). We also obtain logarithmically weighted estimates and give counterexamples to related conjectures.
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