Hardy's function $Z(t)$ - results and problems

Abstract

This is primarily an overview article on some results and problems involving the classical Hardy function $$Z(t):=\zeta (1/2+it){(\chi (1/2+it))}^{-1/2},\zeta (s)=\chi (s)\zeta (1-s).$$ In particular, we discuss the first and third moment of $Z(t)$ (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution of its values is presented.

15 pages. arXiv admin note: text overlap with arXiv:1511.07140

References in corpus (3)

• Continuous lower bounds for moments of zeta and L-functions
• Sharp conditional bounds for moments of the Riemann zeta function
• On the value-distribution of the Riemann zeta-function on the critical line