Ramanujan-Lodge's Harmonic Number approximation

[Martin Fuller]

It looks like a typo in the paper.  It should be:
((12 * gamma) - 11 + (12 * ln(2^0.5))) / (1 - gamma - ln(2^0.5))

or more simply:
1 / (1 - gamma - ln(2^0.5)) - 12

The second constant is more simply written as:
1 / (1 - gamma - ln(3/2)) - 54

--- Jonathan Post <jvospost3 at gmail.com> wrote:

> While preparing to submit to OEIS the decimal digits of two constant
> used in the Ramanujan-Lodge Harmonic Number approximation and the
> DeTemple-Wang Harmonic Number approximation, I ran into trouble.
> 
> I must be doing something really stupid.
> 
> Both the low-res Google calculator and the high-res WIMS calculator
> tell me that
> 	
> ((12 * gamma) - 11 - (12 * ln(2))) / (1 - gamma - ln(2^0.5)) =
> -162.5909
> 
> where gamma is Euler's constant.
> 
> But Theorem 6 of the arXiv citation, formula (1.13), page 6, insists
> that this constant is roughly 1.12150934.
> 
> http://arxiv.org/pdf/0707.3950
>     Title: Ramanujan's Harmonic Number Expansion into NegativePowers
> of a Triangular Number
>     Authors: Mark B. Villarino
>     Comments: sharp error estimates and general formulas for
> Ramanujan's harmonic number expansion
>     Subjects: Classical Analysis and ODEs (math.CA); General
> Mathematics (math.GM)
> 
>     An algebraic transformation of the DeTemple-Wang half-integer
> approximation to the harmonic series produces the general formula and
> error estimate for the Ramanujan expansion for the nth harmonic
> number
> into negative powers of the nth triangular number. We also discuss
> the
> history of the Ramanujan expansion for the nth harmonic number as
> well
> as sharp estimates of its accuracy, with complete proofs, and we
> compare it with other approximative formulas.
> 
> As Harminic numbers and Ramanujan appear to popular on EIS, may I ask
> for someone to help me in my befuddlement?
> 




I'm curious how a sieve, similar to the sieve of Erosthanese, would
"perform".  The basic idea is this: suppose that n is some number that you
want to factor. Assume that you can map, for x = 0 to some upper bound less
than the smallest factor of n, n == a mod x --> f == b mod x, where f is a
factor of n. Say, for instance:

n == 2 mod 3, therefore f == 1 mod 3
n == 1 mod 4, therefore f == 3 mod 4
n == 4 mod 5, therefore f == 2 mod 5

& etc.

Thus we could establish a "profile" for f, which can be turned into a sieve
by finding all integers congruent to 1 mod 3, then a subset congruent to 3
mod 4, then a subset of that congruent to 2 mod 5, etc. My question is how
"quickly" can we narrow in on possible values for f by using this sieve? I
think something like this may be possible for values of n with certain
properties, but I'm not sure how well it'd perform.