For the purposes of this question, the Harmonic Number, $H_n$ is defined by the finite series sum $H_n=\sum_{k=1}^n\frac{1}{k}$ (n and k being positive integers) and the Harmonic Logarithm, $hlog(n)$ is here defined as $$hlog(n)=\left( H_{n^2}-H_{n} \right)=\sum_{k=n+1}^{n^2}\frac{1}{k}$$
The standard definition for $\gamma$ the Euler–Mascheroni constant is
$$\gamma=\lim_{n \rightarrow \infty} \gamma_n$$ where $\gamma_n=H_n-\log(n)$
Following Scott [REF2], let $a_n=2\gamma_n-\gamma_{n^2}$, giving $\gamma=\lim_{n \rightarrow \infty} a_n=\lim_{n \rightarrow \infty} \left(2\gamma_n-\gamma_{n^2} \right)$
$$\begin{align}a_n&=2\gamma_n-\gamma_{n^2}\\ &=2\left(H_n-\log(n) \right)-\left(H_{n^2}-\log(n^2) \right)\\ &=2H_n-2\log(n)-H_{n^2}-2\log(n)\\ &=2H_n-H_{n^2}\\ &=H_n-\left(H_{n^2}-H_n \right)\\ &=H_{n}-hlog(n)\\ \end{align}$$
Therefore in the $\lim_{n \rightarrow \infty}$ we have $$\gamma=\lim_{n \rightarrow \infty} \left( H_n-hlog(n)\right)$$ or coverging from below using the alternative definition $\gamma_n=H_n-\log(n+1)$ $$\gamma=\lim_{n \rightarrow \infty} \left( H_n-hlog(n+1)\right)\tag1$$
where $\gamma$ is the Euler–Mascheroni constant.
(1) gives a slowly converging rational series for $\gamma$ $$\gamma=\sum_{n=1}^{\infty} \frac{1}{n}+\frac{1}{n+1}-\sum_{k=n^2+1}^{(n+1)^2}\frac{1}{k}$$
$$\gamma=\frac{5}{12}+\frac{221}{2520}+\frac{1517}{2520}+...\tag2$$
For the theoretical proofs and background see for example:
[REF1] J. Lambek and L. Moser, (Feb., 1956), Rational Analogues of the Logarithm Function, The Mathematical Gazette, Vol. 40, No. 331, pp. 5-7.
[REF2] J. A. Scott, (Nov., 1996), The Euler Constant γ without Logarithms, , The Mathematical Gazette, Vol. 80, No. 489, pp. 585-586.
(Aside: The term "Harmonic Logarithm" appears to have been first coined by M.F. Egan in the same issue of the Mathematical Gazette as the Lambek and Moser paper, using a slightly different definition to mine above (see pages 8-10). If readers know of any other relevant papers please let me know. The term "Harmonic Logarithm" has more recently been applied to a definition involving both rational and transcendental numbers http://mathworld.wolfram.com/HarmonicLogarithm.html.)
Relation of the Harmonic Logarithm to the Prime Number Theorem
One way of stating the prime number theorem is
$$\pi(N) \thicksim \frac{N}{log(N)}$$
where $\pi(N)$ is the prime counting function, being the number of primes less than or equal to N.
A closer approximation to the prime counting function can be found using the $Li(x)=\int_2^x \frac{dt}{log t}$ function. However if we calculate $Li((N+1)^2)-Li((N)^2)$ we find that the number of primes between two consecutive squares, $N^2$ and $(N+1)^2$, is also approximately given by $\frac{N}{log(N)}$
Now one conjecture here involving series (2) is
$$\gamma_n \;(n^2)!= \left( \frac{1}{n}+\frac{1}{n+1}-\sum_{k=n^2+1}^{(n+1)^2}\frac{1}{k} \right) (n^2)! = \frac{C}{p_1 p_2 ... p_n}$$ where C is some arbitrary positive integer and $p_1 p_2 ... p_n$ is a product of all the primes between $n^2$ and $(n+1)^2$, none of which being divisors of C.
However perhaps of greater interest is how to tackle the proof of the prime number theorem involving $hlog(N)$ instead of $log(N)$ $$\pi(N) \thicksim \frac{N}{hlog(N)}$$
without involving transcendental functions in the proof at all.
I have seen proofs using $H_n$ that generate Chebyshev bounds for the prime counting function, as referenced in the J. Lambek and L. Moser paper above, but surely this can be improved upon using $hlog(N)$ instead.