Sum of number of divisors of first N natural numbers

suchith

If σ(N) is the sum of all the divisors of N and τ(N) is the number of divisors of N then what is the sum of sum of all the divisors of first N natural numbers and the sum of the number of divisors of first N natural numbers?

Is there any relation between σ(N) and τ(N) functions?

Can I do that without factorizing any of the number in the sequence?

HallsofIvy

Have you tried calculating those yourself up to, say, N= 100?

suchith

Of course I have using my computer. I don't remember the result now. I have tried for both. Well!! Using computer it is not a big job.

But can I find it out without listing the factors or factorizing individual numbers?

Avoiding factorization is the goal. I wrote these functions in the form of series, for sum of tau function in the form of infinite series without factorization, and for sum of sigma function it is a finite series. I have written the proof by myself, but don't know about the correctness. Is there anything to find the sum in the mathematical literature?