The fourth moment of the Hurwitz zeta function

Abstract We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ζ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \zeta(s,\alpha) on the critical line when the shift parameter 𝛼 is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> </m:math> 2k th moment for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>⩽</m:mo> <m:mi>k</m:mi> <m:mo>⩽</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> 0\leqslant k\leqslant 2 in this case. In contrast to the Riemann zeta function and other 𝐿-functions from arithmetic, these grow like <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>T</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mo lspace="0.167em">⁡</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> T(\log T)^{k} . This suggests, and we conjecture, that the value distribution of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ζ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \zeta(s,\alpha) on the critical line is Gaussian.