Estimating 𝜋(𝑥) and related functions under partial RH assumptions

Abstract

We give a direct interpretation of the validity of the Riemann hypothesis for all zeros with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal fraktur upper I left-parenthesis rho right-parenthesis element-of left-parenthesis 0 comma upper T right-bracket"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> ℑ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Im (\rho )\in (0,T]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the prime-counting function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> π </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by proving that Schoenfeld’s explicit estimates for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> π </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Chebyshov functions hold as long as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4.92 StartRoot x slash log left-parenthesis x right-parenthesis EndRoot less-than-or-equal-to upper T"> <mml:semantics> <mml:mrow> <mml:mn>4.92</mml:mn> <mml:msqrt> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:msqrt> <mml:mo> ≤ </mml:mo> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4.92\sqrt {x/\log (x)} \leq T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also improve some of the existing bounds of Chebyshov type for the function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ψ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\psi (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .