On a hybrid mean value of certain Hardy sums and Ramanujan sum (Q1412388)

For odd \(c\geq 3\), the author provides an asymptotic formula for sums of the form \[ \sum^c_{h=1} R_c(2h+ 1) S_1(2h, c), \] taken over \(h\) relatively prime to \(c\), where \(R_c(d)= \sum^c_{h=1} \exp(2\pi idh/c)\) is a Ramanujan sum, and \(S_1(d, c)= \sum^{c-1}_{j=1} \exp(\pi i(j+ 1+ [jd/c]))\) is a Hardy sum.