  A long-lived scalar field ($\Phi$) which couples weakly to the right-handed
(RH) neutrinos ($N_{Ri}$), generates small RH neutrino masses ($M_i$) in
Low-Scale-Leptogenesis (LSL) mechanisms, despite having a large vacuum
expectation value $v_\Phi$. In this case, the correlation shared by the $M_i$s
and the duration of the non-standard cosmic history driven by the $\Phi$
provides an excellent opportunity to study LSL signatures on primordial
gravitational waves (GWs). We find it engaging, specifically for the
gravitational waves that originate due to the inflationary blue-tilted tensor
power spectrum and propagate through the non-standard cosmic epoch. Depending
on $M_i$, broadly, the scenario has two significant consequences. First, if LSL
is at play, GWs with a sizeable blue tilt do not contradict the
Big-Bang-Nucleosynthesis (BBN) bound even for the post-inflationary models with
very high-scale reheating. Second, it opens up a possibility to probe LSLs via
a low-frequency and a complementary high-frequency measurement of GW-spectral
shapes which are typically double-peaked. For a case study, we consider the
recent results on GWs from the Pulsar-Timing-Arrays (PTAs) as a `measurement'
at the low frequencies and forecast the signatures of LSL mechanisms at the
higher frequencies.
