Shifting the ordinates of zeros of the Riemann zeta function

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>y</m:mi> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> y\neq 0 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> C&gt;0 . Under the Riemann Hypothesis, there is a number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mo>*</m:mo> </m:msub> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> T_{*}&gt;0 (depending on y and C ) such that for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>T</m:mi> <m:mo>⩾</m:mo> <m:msub> <m:mi>T</m:mi> <m:mo>*</m:mo> </m:msub> </m:mrow> </m:math> T\geqslant T_{*} , both <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mi>ζ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>+</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>γ</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mtext>and</m:mtext> </m:mrow> </m:mrow> <m:mo mathvariant="italic" separator="true"> </m:mo> <m:mrow> <m:mrow> <m:mi>ζ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>+</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>γ</m:mi> <m:mo>+</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mrow> </m:math> \zeta(\frac{1}{2}+i\gamma)=0\quad\mbox{and}\quad\zeta(\frac{1}{2}+i(\gamma+y))\neq 0 hold for at least one γ in the interval <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>T</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>T</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>ε</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> [T,T(1+\varepsilon)] , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ε</m:mi> <m:mo>:=</m:mo> <m:msup> <m:mi>T</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mfrac> <m:mi>C</m:mi> <m:mrow> <m:mi>log</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mo>⁡</m:mo> <m:mi>T</m:mi> </m:mrow> </m:mrow> </