Large differences between consecutive primes.

*(English)*Zbl 1141.11042Denote by \(p_n\) the \(n\)-th prime number. In the article under review the author shows that

\[ \sum_{\substack{ p_{n+1}-p_n>x^{1/2}\\ x\leq p_{n\leq 2x}}}\ll x^{2/3}, \]

improving on work of D. Wolke [Math. Ann. 218, 269–271 (1975; Zbl 0301.10038)], D. R. Heath-Brown [J. Lond. Math. Soc. (2) 20, 177–178 (1979; Zbl 0407.10032)] and A. S. Peck [Proc. Lond. Math. Soc. (3) 76, No.1, 33–69 (1998; Zbl 0891.11046)]. To do so, the problem is converted into a problem on primes in short intervals. More precisely, the author shows that there exists a function \(A(x, y)\) and a constant \(c>0\), such that

\[ \pi(y+yx^{-1/2}) - \pi(y) \geq \frac{cyx^{-1/2}}{\log y}\big(1+o(1)+A(x, y)\big), \]

and such that \(A\) satisfies a certain mean value estimate as \(y\) ranges over the interval \([x, 2x]\). The proof is related to the proof of the fact that \(p_{n+1}-p_n\ll p_n^{21/40}\) by Baker, Harman and Pintz.

For the proof the quantity \(\pi(y+yx^{-1/2}) - \pi(y)\) is written as \(S(\mathcal{A}, 2x^{1/2})\), where \(\mathcal{A}=\)

\([y, y+x^{-1/2}]\cap\mathbb{N}\), and, as usual in sieve theory, \(S(\mathcal{A}, z)\) denotes the number of elements of \(\mathcal{A}\)

without prime divisors \(\leq z\). Then \(S(\mathcal{A}, 2x^{1/2})\) is decomposed via the Buchstab identity. To estimate the resulting terms, one needs asymptotic estimates for \(\sum a_m S(\mathcal{A}_m, z)\), where the coefficients \(a_m\) are certain restricted divisor sums, and the sets \({\mathcal A}_m\) are defined as \(\mathcal{A}_m = \{d: md\in\mathcal{A}\}\). Most of the paper now deals with the derivation of these estimates under various restrictions on the sequence \(a_m\) and the range of summation.

The estimates are performed by transforming the sum in question into a complex integral and applying mean value theorems and large value estimates to the resulting Dirichlet polynomials. The computations required to do so are very involved and need distinctions into several cases.

Slight improvements are possible, however, these would require even more work, and, as the author remarks, \(\frac{21}{32}=0.656\) would be the limit of the method.

The technical main result (Lemma 2.4) is stated in a rather general way, which should allow for other applications as well.

\[ \sum_{\substack{ p_{n+1}-p_n>x^{1/2}\\ x\leq p_{n\leq 2x}}}\ll x^{2/3}, \]

improving on work of D. Wolke [Math. Ann. 218, 269–271 (1975; Zbl 0301.10038)], D. R. Heath-Brown [J. Lond. Math. Soc. (2) 20, 177–178 (1979; Zbl 0407.10032)] and A. S. Peck [Proc. Lond. Math. Soc. (3) 76, No.1, 33–69 (1998; Zbl 0891.11046)]. To do so, the problem is converted into a problem on primes in short intervals. More precisely, the author shows that there exists a function \(A(x, y)\) and a constant \(c>0\), such that

\[ \pi(y+yx^{-1/2}) - \pi(y) \geq \frac{cyx^{-1/2}}{\log y}\big(1+o(1)+A(x, y)\big), \]

and such that \(A\) satisfies a certain mean value estimate as \(y\) ranges over the interval \([x, 2x]\). The proof is related to the proof of the fact that \(p_{n+1}-p_n\ll p_n^{21/40}\) by Baker, Harman and Pintz.

For the proof the quantity \(\pi(y+yx^{-1/2}) - \pi(y)\) is written as \(S(\mathcal{A}, 2x^{1/2})\), where \(\mathcal{A}=\)

\([y, y+x^{-1/2}]\cap\mathbb{N}\), and, as usual in sieve theory, \(S(\mathcal{A}, z)\) denotes the number of elements of \(\mathcal{A}\)

without prime divisors \(\leq z\). Then \(S(\mathcal{A}, 2x^{1/2})\) is decomposed via the Buchstab identity. To estimate the resulting terms, one needs asymptotic estimates for \(\sum a_m S(\mathcal{A}_m, z)\), where the coefficients \(a_m\) are certain restricted divisor sums, and the sets \({\mathcal A}_m\) are defined as \(\mathcal{A}_m = \{d: md\in\mathcal{A}\}\). Most of the paper now deals with the derivation of these estimates under various restrictions on the sequence \(a_m\) and the range of summation.

The estimates are performed by transforming the sum in question into a complex integral and applying mean value theorems and large value estimates to the resulting Dirichlet polynomials. The computations required to do so are very involved and need distinctions into several cases.

Slight improvements are possible, however, these would require even more work, and, as the author remarks, \(\frac{21}{32}=0.656\) would be the limit of the method.

The technical main result (Lemma 2.4) is stated in a rather general way, which should allow for other applications as well.