The total number of divisors of those integers with the most divisors

The total number of divisors of those integers with the most divisors

Im interested in summing taum, the number of positive divisors of m, not over all integers in an interval but rather over only the integers with the most divisors. More specifically: Given a large positive integer x, let n_1, n_2, dots, n_x be the integers 1, 2, dots, x, arranged so that taun_1 ge taun_2 ge cdots ge taun_x. Im interested in the behavior of the function fx,y sum_j1y taun_j. For example, when y1 then fx,1 is just the maximal order of the divisor function tau; when yx we have fx,x sim xlog x by Dirichlet. I believe that we have fx,y sim xlog x even when y is as small as xlog xbeta for betalog4-1. Im most interested in getting values of fx,y that look like xlog xdelta. For example, I believe we can show using the SelbergSathe formula that fbigx, xlog xalpha1-logalpha-1 big sim xlog xalpha1-logalpha-1alphalog 2o1 for alphage 2. Soundararajan mentions a similar result in his paper Omega results for the divisor and circle problems, but for sum taun_j n_j-34 rather than sum taun_j. Has the behavior of fx,y already been worked out in the literature I would very much like to cite existing results, assuming they exist, rather than prove this from scratch.