The Spectrum of Multiplicative Functions

An uncertainty principle for arithmetic sequences

“…The connection between multiplicative functions and the integral equation (7.1) is given by the following result which is Proposition 1 in [6]. The converse to Proposition 7.1 also holds (see Proposition 1 (converse) of [6]) so that the study of these integral equations is entirely analogous to the study of mean-values of multiplicative functions.…”

“…In [6] we showed that there is a unique solution σ(u) to (6.1) and that σ(u) is continuous and |σ(u)| ≤ 1 for all u. In fact σ(u) is given by …”

“…In our case the problem boils down to proving oscillations in the mean-value of the more general class of multiplicative functions satisfying 0 ≤ f (n) ≤ 1 for all n (see Theorem 3.1). Along with our general formalism, this forms the main new ingredient of our paper and is partly motivated by our previous work [6] on multiplicative functions and integral equations. In Section 7 we present a simple analogue of such oscillation results for a wide class of integral equations which has the flavor of a classical "uncertainty principle" from Fourier analysis.…”

“…An uncertainty principle for integral equations E. Wirsing [18] observed that questions on mean-values of multiplicative functions can be reformulated in terms of solutions to a certain integral equation. We formalized this connection precisely in our paper [6] and we now recapitulate the salient details. Let χ : (0, ∞) → C be a measurable function with χ(t) = 1 for 0 ≤ t ≤ 1 and |χ(t)| ≤ 1 for all t ≥ 1.…”

Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

“…For the first assertion see the remark after Proposition 8.1 of [7] and note that if |z| ≤ 1, then 2 − |1 + z| ≥ 1 2 (1 − Re z). To see the second assertion note that if y ≤ x, then…”