A Montgomery-Hooley type theorem for prime \(k\)-tuplets
This paper proves an average mean square estimate for prime \(k\)-tuples in long arithmetic progressions. This result is a generalization of the Barban-Davenport-Halberstam theorem for primes in arithmetic progressions. Given \(2k\) integers \(a_j\) and \(b_j\), \(0\leq j\leq k-1\), then the \(k\)-tuplet \((a_0n+b_0, a_1n+b_1, \cdots , a_{k-1}n+b_{k-1})\) is called a prime \(k\)-tuplet for a given \(n\) if each component is prime. The case \(k=2\), \(a_0=a_1=1\), \(b_0=0\), \(b_1=2\) give the familiar twin primes. Letting \({\mathbf b}= (b_0, b_1, \cdots, b_{k-1})\), we count the prime \(k\)-tuplets in the arithmetic progression \(a(\text{mod } q)\) with the function \[ \psi(x; {\mathbf b}, a,q) = \sum_{\scriptstyle n \in N({\mathbf b}) \atop \scriptstyle n \equiv a(\text{mod } q)} \prod_{j=0}^{k-1} \Lambda( a_jn+b_j) \] where \(N({\mathbf b}) = \{t : 1\leq a_jt+b_j\leq x\) for all \(0\leq j\leq k-1 \} \), and \(\Lambda\) is the von Mangoldt function. It has been conjectured that \( \psi(x; {\mathbf b}, a,q) \sim \sigma({\mathbf b}; a,q) |N({\mathbf b})|\) when \( \sigma({\mathbf b}; a,q) >0\), where \(\sigma({\mathbf b}; a,q)\) is the singular series for this problem whose definition is to complicated to give here, and \(|N({\mathbf b})|\) is the length of the interval \(N({\mathbf b})\). Let \(Z=Z(x) = \{{\mathbf b} : |N({\mathbf b})|\neq 0\}\). Now define \[ E(x,Q) = \sum_{q\leq Q}\sum_{a=1}^q\sum_{{\mathbf b}\in Z}\left|\psi(x; {\mathbf b}, a,q) - \sigma({\mathbf b}; a,q) |N({\mathbf b})|\right|^2. \] The main result proved is that for \(Q\leq x/a_*\), where \(a_* = \max_{0\leq j\leq k-1} |a_j|\), \[ \begin{align*}{ E(x,Q) = {1\over \phi(|a_0|)}x^kQ(\log x-1)^k-x^kQ& \sum_{m=0}^k\beta_m (\log{x\over Qa_*})^m \cr &+O(x^{k- {1\over k+2}}Q^{k+3\over k+2}+ {x^{k+1}\over (\log x)^a}),\cr}\end{align*} \] where the numbers \(\beta_m\) depend only on the \(a_0, a_1, \cdots, a_{k-1}\), and further \(\beta_k = 1/(k! \phi(|a_0|))\). A slightly different asymptotic formula but with the same first term is proved in the less interesting range \(x/a_* \leq Q \leq x^B\), \(B>1\) arbitrary. The author also proves a short interval version of his results. The case \(k=1\) of the above is the Barban-Davenport-Halberstam Theorem in the asymptotic form first proved by Montgomery and further developed by Hooley. The author's complicated proof of his result uses the method Montgomery introduced which depends on Lavrik's estimate for the exceptional set for prime pairs. The argument also uses Hooley's method in places together with a Bombieri-Vinogradov theorem for \(k\)-tuplets developed in an earlier paper of the author [Tsukuba J. Math. 17, No. 1, 43-57 (1993; Zbl 0797.11076)].
prime k-tuples in long arithmetic progressions
scientific article; zbMATH DE number 755501
scientific article