The Euler constant designated g is defined:
lim b->inf (å(1,b) 1/i - S(1,b) 1/x dx) =
lim b->inf (å(1,b) 1/i - In b)
We say that a constant is a member of the "gamma" set if there is a formula
lim b->inf (å(a,b) f(i) - S(a,b) f(x) dx) = c, with c non-zero.
A curious thing about this limit is that it differs from lim(b->inf)å(a,b)f(i) - lim(b->inf)S(a,b)f(x) dx.
That last formula always goes to zero in the limit. Visualize a series of bar graphs with h = 1. So, for i-1 we put each value side by side. The continuous curve x-1 intersects the bar graph at integer values. So what we have is the area under the bar graph curve minus the area under the smooth curve. The difference is a sequence of ever-smaller, by percentage, pieces of the bars. Their total area equals gamma.
There is an infinity of formulas whereby the difference between a pair of converging curves yields a constant.
For example, lim (b->inf) (å(0,b) 1/i2 - S(0,b) 1/x2 dx) = constant less than or equal to p2/6. And in general, for r > 1, lim (b->inf) (å(0,b) 1/ir - S(0,b) 1/xr dx = constant. That is, there is a set of left-over pieces of the bars. This area is less than or equal to z(r). We do not know whether any particular r, including r = 2s, corresponds to a rational number.
However, for divergent curves other than the euler constant curves, I have not found a formula which produces a nonzero constant.
The family of Euler constants -- also known as Stieltjes constants -- follows this pattern:
c = lim (n->inf) (Sum (1,n) (In m)r/r - Integral(1,n) (In x)m /x dx)
Other cases:
lim (b->inf) (å(0,b) i - S(0,b) x dx). Now at every value of b, we have the b(b-1)/2 - b2/2 = -b/2 -- and lim(0,b)-b/2 diverges.
Also:
lim(b->inf) (å(0,b) i2 - S(0,b) x2 dx) = -b2/4, which diverges. Negative divergence also holds for f(i) = i3 and f(x) = x3 and also for exponent 4.
We also have
lim(b->inf)(å(0,b) ei - S(0,b) ex)dx) =
lim(b->inf)(å ei - (ex - 1)) =
lim(b->inf)(å(0,b) e(i-1)) - 1 -- which diverges.