Refer to the smaller prime in a twin prime pair as a lesser twin prime. As an odd number, a lesser twin prime is congruent to either 1 or 3 mod 4: is anyone aware of existing heuristics which predict the percentage of lessser twin primes congruent to 1 mod 4? I checked using Python that 50,013 of the first 100,000 lesser twin primes are congruent to 1 modulo 4, which hints at the rather satisfying answer of 50%.
In case it is helpful, I'll say a bit about what inspired this question. I am currently reading Washington's Introduction to Cyclotomic Fields which involves a lot of expressions involving $(p-1)/2$ and $(p+1)/2$ for an odd prime $p$ (for example, as subscripts of Bernoulli numbers). I started thinking about how a twin prime pair $(p,q)$ has $(p+1)/2 = (q-1)/2$ and thus that it could be interesting to compare expressions involving $(p+1)/2$ for $p \equiv 1 \mod 4$ with expressions involving $(p-1)/2$ for $p \equiv 3 \mod 4$.