Application to Euler’s Constant

Put f(x) := ∑_{n=0}^{∞} \frac{x^{n}}{n!^{2}}

g(x) := ∑_{n=0}^{∞} (\frac{1}{1} + ⋯ + \frac{1}{n}) \frac{x^{n}}{n!^{2}}

Then \frac{g(x)}{f(x)} = \frac{1}{2} log x + γ + O (e^{−4\sqrt{x}}).

Choose x = ⌈\frac{N log 2}{4}⌉^{2}.