Goldbach numbers in short interval. I
Soon after the proof of the ternary Goldbach conjecture, it has known that almost all even numbers are Goldbach numbers, which are sums of two primes. \textit{K. Ramachandra} [J. Indian Math. Soc., New Ser. 37 (1973), 157-170 (1974; Zbl 0326.10041)]\ first established a short interval result for such numbers in the following form. Let \(A= N^{\varphi+ \varepsilon}\), where \(\varphi= 3/5\), \(\varepsilon>0\), and \(B\) be a sufficiently large constant. Then, for all large values of \(N\), the even numbers in the interval \(N< n\leq N+A\), with the possible exception of \(O(A/ \log^B N)\) values, are all Goldbach numbers. The admissible value for \(\varphi\) has since been lowered successively by mathematicians, incuding the author, using a combination of the sieve method, the circle method and various new techniques. In Part I of the paper the author gives the first part of the proof that \(\varphi= 7/78\) is an admissible value. He also shows that if \(\psi= 1/13\) then, for almost all \(n\), the interval \(n< m<n+ n^{\psi+ \varepsilon}\) contains a prime number. This improves on a result of \textit{G. Harman} [Math. Z. 180, 335-348 (1982; Zbl 0482.10040)], who had \(\psi= 1/10\). For a review of Part II see below.
scientific article; zbMATH DE number 783477