Is there a general formula to calculate the sum of the squares logarithms of first n natural numbers?

Is there a general formula for the following sequence?

S(n) = [log(1)]^2 + [log(2)]^2 + ......... + [log(n-1)]^2 + [log(n)]^2

and similarly for the sum of cubes of logarithms of first n natural numbers and if there is one please let me know the procedure you have taken to arrive at the solution so that I can extend that to higher orders.

[Stephen23]

@Yaswanth Sai: this is not a MATLAB question. You should ask this on a math forum.

Answer 1

[Walter Roberson]

No there is no formulas for that. There are some inequalities known and there are some conjectures. One approximation is discussed in the link

https://www.physicsforums.com/threads/sum-of-log-squared-terms.627139/

Answer 2

[Birdman]

Symbolic Toolbox and its features are best for you to get what you want. For instance:

syms S(n,x) m
S(n,x)=symsum(log(n).^x,n,1,n)

This code simply defines symbolic variables m,n,x and symbolic function S which is a function of n and x. Then, we define a series which sums log(n)^x starting from 1 to n and also lets you define the power of logx depending on your input. A numeric example:

>> S(5,3)

ans =

log(2)^3 + log(3)^3 + log(4)^3 + log(5)^3

[Yaswanth Sai]

Birdman, thank you for your solution, but I am specifically looking for a formula kind of thing(eg: n(n+1)/2 for the sum of first n natural numbers) so that I can use the result to further manipulate my expressions.

[Birdman]

Ok, doesn't my solution give you that? I did not understand. The thing you asked for in your question seems to differ from what you now want.

[Yaswanth Sai]

I actually asked for a general formula instead of a numerical sum, that is the reason I have specifically told general formula because numerical value can be easily obtained by a simple for loop or by the above method you have specified.

[Birdman]

Hmm, I do not know a general formula for that, Google will be your best friend in this case.