Abstract
We prove effective results on when a function can be approximated by a Dirichlet polynomial with bounded coefficients. Assuming that Φ(n) is an increasing function we prove that the set of polynomials {\sum_{n=2}^N a_n n^{it-1}: N \geq 2, |a_n| \leq Φ(n)}, is dense in L^2(0,H) if and only if \sum_{n=2}^\infty \frac{\log Φ(n)} {n \log^2 n} = \infty. We also prove variants of this result for generalized Dirichlet polynomials. The main tools are theorems of Paley and Wiener related to quasianalytic
Results & Lemmas (20)
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Theorem 1.
Theorem 1. Suppose Φ(n) is an increasing positive function and H > 0. Then lim N→∞ min |an|≤Φ(n) Z H 0 1 + N X n=2 annit−1
Theorem 1. Suppose Φ(n) is an increasing positive function and H > 0. Then lim N→∞ min |an|≤Φ(n) Z H 0 1 + N X n=2 annit−1
Theorem 2.
Theorem 2. Suppose 0 = λ0 < λ1 < · · · satisfy the Dirichlet condition. Let A0 = 1 and An, n ≥2 be positive real numbers. Define Λ(x) = X…
Theorem 2. Suppose 0 = λ0 < λ1 < · · · satisfy the Dirichlet condition. Let A0 = 1 and An, n ≥2 be positive real numbers. Define Λ(x) = X λn≤x An, and suppose that Z ∞ 1 log Λ(x) x2 dx < ∞. Then we have for each H > 0 that lim N→∞
Lemma 1.
Lemma 1. Let λn fulfill the Dirichlet condition (5), and let Bn be a sequence of positive numbers such that ∞ X n=1 Bn < ∞. Then for any H >…
Lemma 1. Let λn fulfill the Dirichlet condition (5), and let Bn be a sequence of positive numbers such that ∞ X n=1 Bn < ∞. Then for any H > 0 we have inf |bn|≤Bn Z H 0 1 + ∞ X n=1
Lemma 2.
Lemma 2. (Paley-Wiener) Suppose S(x) is a positive increasing function such that Z ∞ 0 log S(x)dx 1 + x2 < ∞. Then given any ǫ > 0 there…
Lemma 2. (Paley-Wiener) Suppose S(x) is a positive increasing function such that Z ∞ 0 log S(x)dx 1 + x2 < ∞. Then given any ǫ > 0 there exists an entire function φ(x) of finite type ǫ such that φ(x) ≤ 1 S(|x|), x ∈R.
Lemma 3.
Lemma 3. (Paley-Wiener) Suppose φ(x) is an entire function of exponential type A such that Z ∞ −∞ |φ(x)|2dx < ∞. Then the Fourier-transform…
Lemma 3. (Paley-Wiener) Suppose φ(x) is an entire function of exponential type A such that Z ∞ −∞ |φ(x)|2dx < ∞. Then the Fourier-transform ˆφ will have support on [−A, A]. A direct consequence of Lemma 2 and Lemma 3 is the following:
Lemma 4.
Lemma 4. Let ǫ > 0 and suppose S(x) is a positive increasing function such that Z ∞ 0 log S(x)dx 1 + x2 < ∞. Then there exists a continuous…
Lemma 4. Let ǫ > 0 and suppose S(x) is a positive increasing function such that Z ∞ 0 log S(x)dx 1 + x2 < ∞. Then there exists a continuous function f with support on [0, ǫ] such that ˆf(0) ̸= 0, and such that | ˆf(t)| ≤ 1 S(|t|), t ∈R.
Theorem 3.
Theorem 3. Let An and Λ(n) be defined as in Theorem 2. Suppose Z ∞ 1 ε(x) x dx < ∞, for some positive decreasing function ε(x) and that Λ(X)…
Theorem 3. Let An and Λ(n) be defined as in Theorem 2. Suppose Z ∞ 1 ε(x) x dx < ∞, for some positive decreasing function ε(x) and that Λ(X) ≪Λ(X + Y ) −Λ(X), (ε(X) ≪Y ≪1) (11) for some δ > 1. Let H > 0. Then the set of Dirichlet polynomials ( N X n=2 ane−λnit, |an| ≤An )
Lemma 5.
Lemma 5. Let f(x) be an entire function of exponential type. Then Z ∞ 0 log+ |f(x)|dx 1 + x2 < ∞=⇒ Z ∞ 0 log−|f(x)| 1 + x2 dx < ∞. 3.5
Lemma 5. Let f(x) be an entire function of exponential type. Then Z ∞ 0 log+ |f(x)|dx 1 + x2 < ∞=⇒ Z ∞ 0 log−|f(x)| 1 + x2 dx < ∞. 3.5
Lemma 6.
Lemma 6. Suppose f(x) is a a continuous function with compact support and that ˆf(0) = 1. If inf ˆf(z)=0 |x −z| > δ > 2ε(x), then min…
Lemma 6. Suppose f(x) is a a continuous function with compact support and that ˆf(0) = 1. If inf ˆf(z)=0 |x −z| > δ > 2ε(x), then min t∈[x,x+ε(x)] ˆf(t) = log ˆf(x) + O δ−1ε(x)x . (14)
Theorem 4.
Theorem 4. Suppose An > 0 and that λn fulfill the Dirichlet condition (5). Let Λ(X) = X λn≤X An, and Z ∞ 1 ε(x) x dx < ∞ for some positive…
Theorem 4. Suppose An > 0 and that λn fulfill the Dirichlet condition (5). Let Λ(X) = X λn≤X An, and Z ∞ 1 ε(x) x dx < ∞ for some positive decreasing function ε(x) and suppose that Λ(X + Y ) −Λ(X) ≫Y Λ(X), (ε(X) ≪Y ≪1), for X ≥X0. Then we have that for any H > 0
Theorem 5.
Theorem 5. Suppose that An are positive numbers such that 1 M T+M X n=T An ≍Φ(T), T/(log log T)1+δ ≤M ≤T. for some δ > 0, T ≥T0 and some…
Theorem 5. Suppose that An are positive numbers such that 1 M T+M X n=T An ≍Φ(T), T/(log log T)1+δ ≤M ≤T. for some δ > 0, T ≥T0 and some positive increasing function Φ(n). Then lim N→∞ min |an|≤An Z H 0
Theorem 1
Theorem 1 follows from using An = Φ(n) in Theorem 5. 4.2.2 Classical Dirichlet polynomials with arithmetical coefficients We will mention two…
Theorem 1 follows from using An = Φ(n) in Theorem 5. 4.2.2 Classical Dirichlet polynomials with arithmetical coefficients We will mention two other simple applications which also follows directly from
Theorem 5.
Theorem 5. First for primes:
Theorem 5. First for primes:
Corollary 1.
Corollary 1. Suppose Φ(p) is an increasing positive function and H > 0. Then lim N→∞ min |ap|≤Φ(p) Z H 0
Corollary 1. Suppose Φ(p) is an increasing positive function and H > 0. Then lim N→∞ min |ap|≤Φ(p) Z H 0
Corollary 2.
Corollary 2. Suppose Φ(n) is an increasing positive function and H > 0. Then lim N→∞ min |an|≤Φ(n) Z H 0 1 + N X n=2 and(n)nit−1
Corollary 2. Suppose Φ(n) is an increasing positive function and H > 0. Then lim N→∞ min |an|≤Φ(n) Z H 0 1 + N X n=2 and(n)nit−1
Theorem 6.
Theorem 6. Suppose α > 0, and that Φ(n) is an increasing positive function and H > 0. Then lim N→∞ min |an|≤Φ(n) Z H 0 αit−1 + N X n=1 an(n…
Theorem 6. Suppose α > 0, and that Φ(n) is an increasing positive function and H > 0. Then lim N→∞ min |an|≤Φ(n) Z H 0 αit−1 + N X n=1 an(n + α)it−1
Theorem 7.
Theorem 7. Suppose ω(t) ≤1 is an increasing function such that Z ∞ 2 1 −ω(t) t log t dt < ∞. Then for each δ > 0 there exists a Cδ > 0 such…
Theorem 7. Suppose ω(t) ≤1 is an increasing function such that Z ∞ 2 1 −ω(t) t log t dt < ∞. Then for each δ > 0 there exists a Cδ > 0 such that Z T+δ T |ζ(σ + it)|dt ≥Cδ, ω(T) ≤σ. 18
Theorem 8.
Theorem 8. Suppose α > 0 and that ω(t) ≤1 is an increasing function such that Z ∞ 2 1 −ω(t) t log t dt < ∞. Then for each δ > 0 there…
Theorem 8. Suppose α > 0 and that ω(t) ≤1 is an increasing function such that Z ∞ 2 1 −ω(t) t log t dt < ∞. Then for each δ > 0 there exists a Cδ > 0 such that Z T+δ T |ζ(σ + it, α)|dt ≥Cδ, ω(T) ≤σ.
Theorem 1.
Theorem 1. 19
Theorem 1. 19
Corollary 3.
Corollary 3. Let α > 0. Then Z T+δ T |ζ(1 + it, α)|dt ≥Cδ > 0, This is the first result that shows a a lower bound in this problem for the…
Corollary 3. Let α > 0. Then Z T+δ T |ζ(1 + it, α)|dt ≥Cδ > 0, This is the first result that shows a a lower bound in this problem for the Hurwitz zeta-function that is independent of T, and thus that the Hurwitz zeta-function is not universal on the line Re(s) = 1. For the special case of the Riemann zeta-function and for an argument for why this implies non uni- versality and what universality on a line means, see the discussion in [5, pp. 5-6]. References [1] J. Andersson. Disproof of some con