Problems on Dirichlet Series

John E. McCarthy ^{∗} St. Louis, Missouri 63130

July 21 2003

s := σ + it Ω_{ρ} := {s : ℜ(s) > ρ}.

If f(s) ∼ ∑ a_{n}n^{−s}, then

σ_{a}(f) := inf{ℜ(s) : ∑ a_{n}n^{−s} converges absolutely}
σ_{b}(f) := inf{ρ : ∑ a_{n}n^{−s} converges to a bounded function in Ω_{ρ}}
σ_{u}(f) := inf{ρ : ∑ a_{n}n^{−s} converges uniformly in Ω_{ρ}}
σ_{c}(f) := inf{ℜ(s) : ∑ a_{n}n^{−s} converges}.

Have [1]: σ_{c} ≤ σ_{b} = σ_{u} ≤ σ_{a} ≤ σ_{c} + 1.

Let D denote the set of functions that can be represented in some halfplane by a Dirichlet series. Let P denote the set of Dirichlet series with a finite number of terms, and P_{0} those with no constant term.

For α < 0, define H_{α}^{p} to be the closure of P_{0} in the norm

lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt ∫_{0}^{∞} dσ σ^{−1−α}|f(s + c)|^{p};

∗Partially supported by National Science Foundation Grant DMS 0070639

for α = 0 define H_{0}^{p} to be the closure of P in the norm

lim_{c→0^{+}} lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt|f(c + it)|^{p}.  (1)

Define H_{∞}^{p}(Ω_{\frac{1}{2}} ) to be the holomorphic functions g in Ω_{\frac{1}{2}} for which

sup_{y∈ℝ} sup_{σ> \frac{1}{2}} ∫_{y}^{y+1} |g(σ + it)|^{p}

is finite.

Theorem 1 (Carlson [2]). Let f(s) = ∑_{n=1}^{∞} a_{n}n^{−s} be a function in D that has σ_{b} = 0. Then

∑_{n=1}^{∞} |a_{n}|^{2} = lim_{c→0^{+}} lim_{T →∞} \frac{1}{2T} ∫_{−T}^{T} dt|f(c + it)|^{2}.  (2)

Problem 1 Prove Theorem 1 with the hypothesis σ_{b} = 0 replaced by (i) σ_{c}(f) < 0. (ii) ∑_{n=1}^{∞} |a_{n}|^{2} < ∞. (iii) log |f(s)| = O(log |t|).

Problem 2 Develop a “BMOA” type theory so that

(i) every function in “BMOA” with σ_{c} < 0 has the integrals (1) finite for every p < ∞.

(ii) The alternating zeta function ζ_{1}(s) := ∑_{n=1}^{∞}(−1)^{n}n^{−s} is in BMOA. (Every even power of ζ_{1} is in H_{0}^{2} [4]).

Problem 3 Does H_{0}^{p} embed in H_{∞}^{p}(Ω_{\frac{1}{2}} ) for every p < ∞? (It does if p is an even integer [3]).

Problem 4 How can the zero sets for H_{α}^{p} and Mult(H_{α}^{p}) be classified?

Problem 5 What are the interpolating sequences for H_{0}^{2}? What are the interpolating sequences that lie on the real axis?

Problem 6 Is H_{0}^{1} = H_{0}^{2} ⊗ H_{0}^{2}, i.e. is every function in H_{0}^{1} writable as a sum of products of functions in H_{0}^{2} with norm control?

References

[1] H. Bohr. Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen Σa_{n}/n^{s}. Nachr. Akad. Wiss. Göttingen math.-Phys. Kl., pages 441–448, 1913.

[2] F. Carlson. Contributions á la théorie des séries de Dirichlet, Note I. Ark. Mat., 16(18):1–19, 1922.

[3] H. Hedenmalm, P. Lindqvist, and K. Seip. A Hilbert space of Dirichlet series and systems of dilated functions in L^{2}(0, 1). Duke Math. J., 86:1– 37, 1997.

[4] H. Helson. Compact groups and Dirichlet series. Ark. Mat., 8:139–143, 1969.