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Some Questions About Spaces of Dirichlet Series Some questions about spaces of Dirichlet series. Jos´eBonet Instituto Universitario de Matem´aticaPura y Aplicada Universitat Polit`ecnicade Val`encia Eichstaett, June 2017 Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet Series P1 an A Dirichlet Series is an expression of the form n=1 ns with s 2 C. Riemann's zeta function: If s 2 C, 1 1 1 X 1 ζ(s) := 1 + + + ::: = : 2s 3s ns n=1 Euler 1737 considered it for s real: He proved ζ(2) = π2=6, and he relation with the prime numbers: −1 Y 1 ζ(s) = 1 − ; ps p the product extended to all the prime numbers p. Jos´eBonet Some questions about spaces of Dirichlet series Riemann Riemann (1826-1866) and his Memoir \Uber¨ die Anzahl der Primzahlen unter einer gegebenen Gr¨osse", in which he formulated the \Riemann Hypothesis". Jos´eBonet Some questions about spaces of Dirichlet series Riemann The series ζ(s) converges if the real part of s is greater than 1, and ζ(s) has an analytic extension to all the complex plane except s = 1, where it has a simple pole. Obtained the functional equation. s s−1 ζ(s) = 2 π sen(πs=2)Γ(1 − s)ζ(1 − s); s 2 C: Proved that there are no zeros of ζ(s) outside 0 ≤ <(s) ≤ 1, except the trivial zeros s = −2; −4; −6; :::. Jos´eBonet Some questions about spaces of Dirichlet series Riemann's Hypothesis Riemann conjectured that all the non-trivial zeros of ζ(s) are in the line <(s) = 1=2. His work was essential for the prime number theorem of Hadamard and de la Vall´eePoussin (1896): π(n) log n l´ım = 1; n!1 n π(n) = number of primes smaller or equal than n 2 N. Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet Series P1 an Given a Dirichlet Series D(s) = n=1 ns in C we define, with Cr = [Re s > r], 1 X an σ (D) := ´ınffr 2 : is abs. conv. in the half plane g a R ns Cr n=1 1 X an σ (D) := ´ınffr 2 : is unif. conv. in the half plane g u R ns Cr n=1 1 X an σ (D) := ´ınffr 2 : is conv. in the half plane g c R ns Cr n=1 Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet Series 6 conv. - unif. conv. - abs. conv. - - σc σu σa Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet Series σc (D) ≤ σu(D) ≤ σa(D) for each Dirichlet series D. Moreover D(s) defines a holomorphic function on Cσc (D) = [Re s > σc (D)]. Proposition P1 an Given a Dirichlet Series D = n=1 ns in C one has 0 ≤ σa(D) − σc (D) ≤ 1; but 1 0 ≤ σ (D) − σ (D) ≤ : a u 2 The second one is not completely trivial. Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet Series Proposition (i) sup(σa(D) − σc (D)) = 1: D (ii) sup(σu(D) − σc (D)) = 1: D P1 (−1)n (i) D = n=1 ns satisfies σa(D) = 1 and σc (D) = 0. P1 (−1)n (ii) (pn)n prime numbers. D = s satisfies σu(D) = 1 (the proof n=1 (pn) requires Kronecker's theorem and the prime number theorem) and σc (D) = 0. Jos´eBonet Some questions about spaces of Dirichlet series Bohr's absolute convergence problem Definition ( ) X 1 S := sup σ (D) − σ (D) j D = a Dirichlet series : a u n ns n Problem of absolute convergence of Bohr S = ? Theorem, Bohr (1913), Bohnenblust, Hille (1931) 1 S = 2 Jos´eBonet Some questions about spaces of Dirichlet series Bohr's absolute convergence problem P1 an Given a Dirichlet Series D(s) = n=1 ns in C we define the bounded abscissa σb(D) as the infimum of r 2 R such that D(s) defines a bounded holomorphic function in the half plane Cr = [Re s > r]. σc (D) ≤ σb(D) ≤ σu(D) ≤ σa(D) for each Dirichlet series D. Theorem. Bohr, 1913 σb(D) = σu(D) for each Dirichlet series D. Jos´eBonet Some questions about spaces of Dirichlet series Bohr's absolute convergence problem Jos´eBonet Some questions about spaces of Dirichlet series Harald Bohr (1887-1951) Younger brother of Niels Bohr, Nobel Prize in Physics 1922. Football player. He played for Denmark national team. Silver medal in Olympic Games in London in 1908. They defeated France 17-1. England won the tournament. He presented his Ph.D. thesis \Contributions to the theory of Dirichlet series" in 1910. The audience consisted mainly of team mates and soccer fans. He had a long fruitful collaboration with Landau. Perhaps his most famous contribution is the theory of almost periodic functions. Jos´eBonet Some questions about spaces of Dirichlet series Niels and Harald Bohr Jos´eBonet Some questions about spaces of Dirichlet series Denmark team Olympics London 1908 Jos´eBonet Some questions about spaces of Dirichlet series Landau and Bohr Landau Bohr Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet series and complex analysis on polydiscs. Bohr's vision Definition p = the sequence of prime numbers: p1 < p2 < p3 < : : : α α1 αn (N) p = p1 × · · · × pn where α = (α1; : : : ; αn; 0 ;::: ) 2 N0 The Bohr transform X 1 X B : a 2 D ! c zα 2 P; n ns α n α with an = apα = cα: It is a one to one correspondence between the formal Dirichlet series and the formal power series by the fundamental theorem of prime numbers. Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet series and complex analysis on polydiscs The space H1 H1 is the space of all those Dirichlet series D which converge on C0 := [Re > 0] such that the holomorphic function D(s) is bounded on this halfplane. It is a Banach algebra under the norm jjDjj := sup jD(s)j. 1 s2C0 Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet series and complex analysis on polydiscs The space H1(Bc0 ) P α H1(Bc0 ) is the space of all formal series (N) cαz such that for each α2N0 P α N the series, N cαz defines a holomorphic and bounded function α2N0 N fN in the polydisk D and, moreover, X α sup kfN k1 = sup sup f cαz g < 1: N N N z2D N α2N0 It coincides with the space of bounded holomorphic functions f on the unit ball of the Banach space c0 of complex null sequences, and jjf jj1 = supN kfN k1. It is also a Banach algebra. Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet series and complex analysis on polydiscs Theorem. Bohr (2013), Hedenmalm, Lindqvist, Seip, Duke Math. J. (1997)) The Bohr transform B is a Banach algebra isometry between H1 and H1(Bc0 ). Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet series and complex analysis on polydiscs Hedenmalm, Lindqvist, Seip solved a problem of Beurling from 1945. Theorem. Hedenmalm, Lindqvist, Seip, Duke Math. J. (1997)) p P1 The function '(x) = n=1 an 2 sin(nπx) 2 L2(0; 1) satisfies that ('(n:))n is a Riesz basis of L2(0; 1) (i.e. orthonormal with respect to an P1 −s equivalent norm) if and only if S'(x) := n=1 ann 2 H1 and it is bounded way from 0 in C0. P1 −s Important tool: The Hilbert space H2 of Dirichlet series n=1 ann P1 2 with n=1 janj < 1. Jos´eBonet Some questions about spaces of Dirichlet series Solution of Bohr's problem. Reformulation of Bohr's problem S = supD (σa(D) − σu(D)) = supfσa(D) j D 2 H1g: An example of Toeplitz (1913) proved 1=4 ≤ S ≤ 1=2. The final solution was obtained by Bohnenblust and Hille (Annals Math. 1931). They presented the following inequality that implies S = 1=2. Jos´eBonet Some questions about spaces of Dirichlet series Bohnenblust and Hille inequality Theorem. There is C > 1 such that for each N and for each m homogeneous P α polynomial P(z) = N cα(P)z the following holds: α2N0 ;jαj=m 2m m+1 X m+1 2m m cα(P) ≤ C sup jP(z)j: N z2Bc0 α2N0 ;jαj=m The exponent 2m=(m + 1) is optimal. The constant C can be taken as close to 1 as necessary. This is an extension of Littlewood's 4=3 inequality for 2-homogeneous polynomials in CN . It was improved by Defant, Frerick, Ortega-Cerd`a, Ouna¨ıesand Seip in 2011, and again by Bayart, Pellegrino and Seoane in 2014. Jos´eBonet Some questions about spaces of Dirichlet series Solution of Bohr's problem. Bohr's transform maps the space Pm(c0) of m-homogeneous m continuous polynomials on c0 bijectively onto H1, the space of all P −s n ann 2 H1 such that an 6= 0 implies that the number of prime divisor of n counting multiplicity is m. Bohnenblust and Hille inequality implies that m m−1 Sm := supfσa(D) j D 2 H1g ≤ 2m . m−1 The proof that Sm ≥ 2m requires the construction of certain continuous m-homogeneous polynomials on c0. m−1 Since Sm = 2m ≤ S ≤ 1=2 for each m, one concludes S = 1=2. Jos´eBonet Some questions about spaces of Dirichlet series Dirichlet series and complex analysis on polydiscs Bohr's power series theorem, Math. Ann. 1914 P n For all z 2 D; jzj < 1=3 and for all f (z) = n cn(f )z 2 H1(D) we have X n jcn(f )z j ≤ kf k1: n 1 Moreover, is optimal.
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