Banks 2007 prime numbers beatty sequences

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theorem_5_1: Banks and Shparlinski's asymptotic formula for the von Mangoldt sum over the values q floor(alpha n + beta) + a, n up to N, uniform for coprime 0 <= a < q up to a small power of N when alpha is irrational of finite type; with Corollaries 5.2 and 5.3 it gives the main terms (q/phi(q)) N and, for (a, q) = (0, 1) or (1, 2), qN, the latter case being the primes of Long's conjecture.

theorem_5_4: Banks and Shparlinski's asymptotic formula for the von Mangoldt sum over the Beatty values floor(alpha n + beta), n up to N, lying in a residue class a mod q, uniform for q up to a small power of N when alpha is irrational of finite type; with Corollaries 5.5 and 5.6 it gives the expected count of primes in a Beatty sequence, the one-prime statement behind Problem 972.

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William D. Banks, Igor E. Shparlinski, Prime numbers with Beatty sequences. arXiv preprint (2007). arXiv:0708.1015.

The copy read for this card is arXiv:0708.1015v1 (7 August 2007), the only arXiv version listed on 2026-09-18; the paper appeared as Colloq. Math. 115 (2009), no. 2, 147-157, doi:10.4064/cm115-2-1 (Crossref record; not compared), so locators here are the preprint's. Read status: claims checked for Theorem 5.1 (p. 7) and for Theorem 5.4 with Corollaries 5.5 and 5.6 (p. 11), each read clause by clause on the page images; the proofs (Sections 3-5) were not checked.

Motivated by Long's conjecture that there are infinitely many primes p = 2floor(alphan) + 1 for irrational 1 < alpha < 2, the paper proves asymptotic formulas that are uniform in a growing modulus. Theorem 5.1 shows that for alpha positive irrational of finite type there is kappa > 0 with sum_{n <= N} Lambda(qfloor(alphan + beta) + a) = alpha^{-1} sum_{m <= floor(alphaN + beta)} Lambda(qm + a) + O(N^{1 - kappa}) uniformly for 0 <= a < q <= N^kappa with gcd(a, q) = 1, and Theorem 5.4 is the analog for the sum of Lambda(floor(alphan + beta)) over the n with floor(alphan + beta) congruent to a mod q. Corollaries 5.2 and 5.5 turn these, for q up to a power of log N, into the main terms (q/phi(q)) N for the weighted count of primes qfloor(alphan + beta) + a and N/phi(q) for the weighted count of Beatty values in the class a mod q, each with error O(N exp(-C sqrt(log N))); Corollaries 5.3 and 5.6 sharpen the error to O(N exp(-c (log N)^{3/5} (log log N)^{-1/5})) when (a, q) = (0, 1) or (1, 2), with main terms qN and N. The method combines discrepancy bounds for fractional parts of irrational multiples with exponential-sum estimates for the von Mangoldt function over arithmetic progressions (Theorems 4.1 and 4.2). For problem 972, the paper establishes results with one prime condition at a time. It does not address the simultaneous primality constraint of that problem.

Source: https://arxiv.org/abs/0708.1015. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:0708.1015), every other right reserved.

Bears on. #972 (Theorem 5.4 and Corollaries 5.5-5.6, p. 11: the count of primes among the Beatty values floor(alpha*n + beta), in a residue class and in total, for alpha irrational of finite type, the one-prime statement; the problem's two-prime question is not addressed; theorem_5_4)

Results to transcribe.

• Theorem 5.1: For alpha positive irrational of finite type there is kappa > 0 such that sum_{n <= N} Lambda(qfloor(alphan + beta) + a) = alpha^{-1} sum_{m <= floor(alphaN+beta)} Lambda(qm + a) + O(N^{1-kappa}), uniformly for 0 <= a < q <= N^kappa with gcd(a,q) = 1.
• Theorem 5.4: The analogous formula for sum over n <= N with floor(alphan + beta) = a mod q of Lambda(floor(alphan + beta)), again with error O(N^{1-kappa}).
• Corollaries 5.2, 5.5 (recorded on the Theorem 5.1 and 5.4 pages): For q up to (log N)^B the sums have main terms (q/phi(q)) N (Corollary 5.2) and N/phi(q) (Corollary 5.5) with error O(N exp(-C sqrt(log N))); Corollaries 5.3 and 5.6 give qN and N with error O(N exp(-c (log N)^{3/5} (log log N)^{-1/5})) for (a, q) = (0, 1) or (1, 2).
• Theorems 4.1, 4.2: Exponential-sum estimates for the von Mangoldt function twisted by e(kgammam) over arithmetic progressions, uniform for q up to M^{eps/4}, for gamma irrational of finite type.

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