The least primitive roots mod <i>p</i>

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> p\gt 1 be a large prime number, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ε</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:math> \varepsilon \gt 0 be a small number. The established unconditional upper bounds of the least primitive root <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo>≠</m:mo> <m:mo>±</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> u\ne \pm 1,{v}^{2} in the prime finite field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {{\mathbb{F}}}_{p} have exponential magnitudes <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo>≪</m:mo> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>⁄</m:mo> <m:mn>4</m:mn> <m:mo>+</m:mo> <m:mi>ε</m:mi> </m:mrow> </m:msup> </m:math> u\ll {p}^{1/4+\varepsilon } . This note contributes a new result to the literature. It proves that the upper bound of the least primitive root has polynomial magnitude <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo>≤</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mi>p</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>ε</m:mi> </m:mrow> </m:msup> </m:math> u\le {\left(\log p)}^{1+\varepsilon } unconditionally.