  Improved lower bounds on the average and the worst-case rate-memory tradeoffs
for the Maddah-Ali&Niesen coded caching scenario are presented. For any number
of users and files and for arbitrary cache sizes, the multiplicative gap
between the exact rate-memory tradeoff and the new lower bound is less than
2.315 in the worst-case scenario and less than 2.507 in the average-case
scenario.
