Given a real $t > 0$, we set
$$m _ { t } : = S _ { 1,1 } ( t ) \quad \text { and } \quad \sigma _ { t } : = \sqrt { S _ { 2,1 } ( t ) }$$
Given reals $t > 0$ and $u$, we set
$$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$
Show that $\sigma _ { t } \sim \frac { \pi } { \sqrt { 3 } t ^ { 3 / 2 } }$ as $t$ tends to $0 ^ { + }$. Deduce from this that, for all real $u$,
$$j ( t , u ) \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } e ^ { - u ^ { 2 } / 2 }$$