A note on the exceptional set for Goldbach's problem in short intervals

Assume the Generalized Riemann Hypothesis and suppose thatHlog^–6 xt8. Then we prove that all even integers in any interval of the form (x, x, +H) butO(H^1/2log³x) exceptions are a sum of two primes.Partially supported by Consiglio Nazionale delle Ricerche, Visiting Professor Program.Partially supported by Hungarian National Foundation for Scientific Research, Grant n. 1901.     

Quadratic residues and the distribution of prime numbers

D. Shanks [11] has given a heuristical argument for the fact that there are “more” primes in the non-quadratic residue classes modq than in the quadratic ones. In this paper we confirmShanks' conjecture in all casesq<25 in the following sense. Ifl ₁ is a quadratic residue,l ₂ a non-residue modq, ε(n, q, l ₁,l ₂) takes the values +1 or ?1 according ton?l ₁ orl ₂ modq, then $$\mathop {\lim }\limits_{x \to \infty } \sum\limits_p {\varepsilon (p,q,l_1 ,l_2 )} \log pp^{ - \alpha } \exp ( - (\log p)^2 /x) = - \infty$$ for 0≤α<1/2. In the general case the same holds, if all zeros ?=β+yγ of allL(s, χ modq),q fix, satisfy the inequality β²?γ²<1/4.     

Bemerkungen zur Arbeit von S. Knapowski und P. Turán

The present paper shows that by an easy modification of the ideas of S. Knapowski and P. Turán [2] one can prove the following Theorem 1: LetV ₁ (Y) denote the number of sign changes of π(x)?lix in the interval [2,Y. Then forY>C ₁ the inequality $$V_1 (Y) > C_2 (\log \log Y)C_3 $$ holds with positive effectively calculable constantsC ₁, C₂ andC ₃.     

The Difference Between Consecutive Primes, II

The authors sharpen a result of Baker and Harman (1995), showingthat [x, x + x0.525] contains prime numbers for large x. Animportant step in the proof is the application of a theoremof Watt (1995) on a mean value containing the fourth power ofthe zeta function. 2000 Mathematical Subject Classification:11N05.     

Über eine Verallgemeinerung des Tschebyschef-Problems

Ohne Zusammenfassung     

Primes in short intervals

Let (x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/421,yx andx>x() then (x)–(x–y)>y/(100 logx). This implies for the difference between consecutive primes the inequalityp _n+1–p _n p _n ^23/42 .