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On naturally occuring sums
I was thinking about naturally accoring sums. Though "naturally" might be a bit subjective I came to this as follows.

Most people are intrested in sums and integrals of expressions of modest lenght.

Also expressions of modest lenght are good asymptotics in general.

So the idea is to consider elementary functions because they are good asymptotics to most functions.

In fact I consider asymptotics to elementary functions that are themselves elementary.

For instance the number of primes between 2 and large x is about li(x).

li(x) is asymptotic to x/(ln(x)-1). And x/ln(x) is again asymptotic to x/(ln(x)-1).

Lets consider asymptotic exp-free elementary functions (AEFE).

So exp(x) * something or x^x * something or exp(x)/(exp(x)+1) * something is not allowed ; it is not a AEFE.

Now all elementary functions are compositions of powers , exponentials and logaritms. ( arctan , sin , cos , tan , arcsin are not allowed : sin is of exponential type )

We also want to stay in the real domain.

Also note that x^4 + x^2 + 17 is simply asymptotic to x^4.

This thread is about sums of AEFE.

THUS sum (x^4 + x^2)/(x^3 + 4) reduces to sum x^4/x^3 = sum x

Hence we can conclude that since sum a + sum b = sum (a+b) we only consider expressions involving

1) powers
2) logaritms
3) no sums or differences in the denumerator or numerator.

These are thus the AEFE functions by the arguments above.

The simplest occuring AEFE are the ones without many compositions.

Lets consider these simple AEFE.

Thus we end up with a small amount of products and compositions of powers and logaritms.

Again short expression are preferred.

Hence we naturally start with 1 or 2 products and compositions at most.

The first examples are

sum x^a

this is well studied.

sum ln(x^a)/x^b

this is also well studied : derivatives of the riemann zeta and related.

sum ln(x^c)

this reduces to c ln(gamma(x)

So all that remains for the simplest cases is

sum ln^A (x)

In particular Im intrested in

sum ln^2(x)

We have continuum sum and matrix methods.

Notice sum ln^2(x) - integral ln^2(x) - ln^2(x)/2 grows slowly and with some modifications we can get an euler like constant.
( the integral is trivial to compute )

I am of course referring to the trapezoidal rule to show the connection between sums and integrals involving ln^2.

Also intresting imho is sum ln^2((m+n)/m) where the sum is over n and n goes from 1 to m.

If m gets large we get lim (sum (ln(m+n) - ln(m))^2)/m approximates the integral from 1 to 2 of ln^2.

Since that last expression can be simplied by working out the square and using lngamma , we get an imho intresting property.

But there is another way of getting approximations.

Notice the derivative of ln^2(x) is 2 ln(x)/x.

It suggests that the closed form is something like integral ln(gamma(x)) + digamma(x) + trigamma(x) + ... dx

Thus relating this to sums and integrals of the polygamma functions !!

I wonder about simplifying such expressions too.

Since ln^2(x) -> ln^2(x+1) is mapped by ln^2 ( (exp(sqrt(x))+1) ) this means the superfunction of ln^2 ( (exp(sqrt(x))+1) ) is given by something like

integral ln(gamma(x)) - ln(gamma(x-1)) + digamma(x) - digamma(x-1) + trigamma(x) - trigamma(x-1) + ... dx

That seems intresting to me.

Notice that the analogue post about exp and powers instead of logarithms and powers would have led us to the polylogaritms.

Maybe this post is just a good defense for the introductions of the polygamma and polylog functions.

One of the reasons I posted this is because I am intrested in euler like constants and maybe closed forms for them ... perhaps with continuum sum techniques ?

Also a continuum sum of the form entire(exp(x)) is simple to compute by the very algorithm we use.

But logarithms are not entire and thus not so easily summable.

A deeper understanding of continuum sums and perhaps a closed form for integral ln(gamma(x)) - ln(gamma(x-1)) + digamma(x) - digamma(x-1) + trigamma(x) - trigamma(x-1) + ... dx or similar is a goal of this thread.

Also a deeper understanding of gamma like constants would be nice.

Notice I avoided things like writing ln^2 as a double sum ( not hard with Taylor series , nice combinatorics ) or working with sums or double sums and bernouilli numbers. I deliberately avoided that.

Probably the ultimate solution is writing sum ln^2 as an integral transform.

That integral transform should follow from the fact that the polygamma has an integral transform representation.

Then take the sums and integrals under the integral sign and your result should follow.

Analoge for the superfunction of ln^2 ( (exp(sqrt(x))+1) ).

I am not aware of integral transforms as solutions to superfunctions unless in case where they can be expressed in standard functions or are derived in a similar way as above.

It would look nice in a textbook to have an integral transform of a superfunction or for the sum ln^2(x) , not ?

Just to show that integral transforms do not " stop " at gamma , polylog , bessel and zeta.

Of course this is somewhat sketchy ; I did not consider constants of integration and such so this all needs to be made formal too.

Another reason to look into this is that functions like ln^2 occur frequently in number theory.

Maybe this will inspire you and you are intrested in giving this all a second look.



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