On the Li coefficients for the Dirichlet L-functions

In this paper, we prove under the Riemann hypothesis that the Li coefficients for the Dirichlet $L$ -functions $\lambda _{\chi }(n)$ are increasing in $n$ . We also prove unconditionally that the first Li coefficients are increasing using the Bell polynomials. Furthermore, we give a probabilistic interpretation and describe another method differently as stated in Omar et al. (LMS J Comput Math 14:140–154, 2011) to compute them.