Discrepancies in the distribution of prime numbers

AbstractChebyshev has noticed a certain predominance of primes of the form 4n + 3 over those of the form 4n + 1. He asserted that limx→∞∑p > 2 (−1)(p − 1)2e−px = −∞. This was unproven until today. G. H. Hardy, J. E. Littlewood and E. Landau have shown its equivalence with an analogue to the famous Riemann hypothesis, namely, L(s, χ1mod 4) ≠ 0, Re(s) >12. S. Knapowski and P. Turán have given some similar (unproven) relations, e.g., limx→∞∑p > 2(−1)(p − 1)2logpe−log2(px) = −∞, which are also equivalent to the above. Using Explixit Formulas the author shows that limx→∞∑p > 2(−1)(p − 1)2logpp−12e−(log2p)x = −∞ (∗) holds without any conjecture. (In addition, the order of magnitude of divergence is calculated.) It turns out that (∗) is only a special case (in several respects). At first, it may be enlarged into limx→∞∑p > 2(−1)(p − 1)2logpp−αe−(log2p)x) = −∞, 0⩽α⩽12. Then, it can be generalised to a wider class of progressions. For example, the same is true if one sums over the primes in the classes 3n + 2 and 3n + 1, with a “−” and a “+” sign, respectively. All results of this type depend on the location of the first nontrivial zero of the corresponding L-series. D. Shanks has given some arguments for the predominance of primes in residue classes of nonquadratic type. He conjectured “If m1 mod k is a quadratic residue and m2 mod k a non-residue, then there are “more” primes congruent m2 than congruent m1 mod k.” This indeed turns out to be true in the sense of (∗), not only for k = 3, 4, but for some higher moduli as well. Finally, numerical calculations were made to investigate the behaviour of Δ3(X) ≔ π(X, 2 mod 3) − π(X, 1 mod 3) in the interval 2 ≤ X ≤ 18, 633, 261. No zero was found in this range. In the analogue case of Δ4(X) ≔ π(X, 3 mod 4) − π(X, 1 mod 4) the first sign change occurs at X = 26, 861