Zero-Free Regions of the Riemann Zeta Function and Approximation in Weighted Dirichlet Spaces

Abstract We study zero-free regions of the Riemann zeta function $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> related to an approximation problem in the weighted Dirichlet space $$D_{-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> which is known to be equivalent to the Riemann Hypothesis since the work of Báez-Duarte. We prove, indeed, that analogous approximation problems for the standard weighted Dirichlet spaces $$D_{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> when $$\alpha \in (-3,-2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> give conditions so that the half-plane $$\{s \in \mathbb {C}: \Re (s) &gt; -\frac{\alpha +1}{2}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>:</mml:mo> <mml:mi>ℜ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is also zero-free for $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> . Moreover, we extend such results to a large family of weighted spaces of analytic functions $$\ell ^p_{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ℓ</mml:mi> <mml:mi>α</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:math> . As a particular instance, in the limit case $$p=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$\alpha =-2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we provide a new equivalent formulation of the Prime Number Theorem.