[Paper Review] Prime numbers with Beatty sequences

William D. Banks, Igor E. Shparlinski|ArXiv.org|Aug 7, 2007

Analytic Number Theory Research18 references3 citations

TL;DR

This paper establishes asymptotic formulas for the number of primes in generalized Beatty sequences of the form $ p = q\lfloor \alpha n + \beta \rfloor + a $, where $ \alpha $ is a positive irrational of finite type and $ q $ grows as a power of $ N $. Using exponential sum estimates and discrepancy bounds, it proves that the count is asymptotically proportional to $ \frac{q}{\varphi(q)} \pi(N) $, extending earlier results to larger moduli and confirming a refined version of Long's conjecture on primes of the form $ 2\lfloor \alpha n \rfloor + 1 $. The key contribution is a nontrivial error term even when $ q $ grows polynomially with $ N $, under Diophantine conditions on $ \alpha $.

ABSTRACT

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[αn]+1$, where $10$ depends only on $α$. We also prove a similar result for primes $p=[αn+β]$ such that $p\equiv a\pmod q$.

Motivation & Objective

• To establish uniform asymptotic estimates for the number of primes in sequences of the form $ p = q\lfloor \alpha n + \beta \rfloor + a $, where $ \alpha $ is irrational of finite type and $ q $ grows with $ N $.
• To generalize Ribenboim’s argument for primes in Beatty sequences to allow $ q $ to grow as a power of $ N $, under Diophantine conditions on $ \alpha $.
• To analyze the distribution of such primes in arithmetic progressions modulo $ q $, particularly for $ (a,q) = (0,1) $ and $ (1,2) $, corresponding to the Long conjecture.
• To derive nontrivial error terms in the prime counting function even when $ q $ grows polynomially with $ N $, using exponential sum methods and discrepancy bounds.
• To extend results to the case where the Beatty sequence itself is required to lie in a fixed arithmetic progression modulo $ q $, proving a similar asymptotic formula.

Proposed method

• Use exponential sum estimates involving the von Mangoldt function $ \Lambda(n) $ to analyze the distribution of primes in Beatty sequences.
• Apply discrepancy bounds for fractional parts $ \{ \gamma m \} $ with irrational $ \gamma $, particularly leveraging the finite type condition on $ \alpha $ to control error terms.
• Employ the identity $ \psi_{\Delta}(x) = \sum_{k \in \mathbb{Z}} g_k e^{2\pi i k x} $ with smooth trigonometric polynomials to approximate the characteristic function of intervals.
• Use partial summation and the Siegel–Walfisz theorem to estimate sums over arithmetic progressions, especially for small $ q $.
• Apply the Korobov–Vinogradov zero-free region bound to achieve stronger error terms in special cases like $ (a,q) = (0,1) $ or $ (1,2) $.
• Use a transformation via $ t = \lceil \alpha^{-1} \rceil $ to reduce the case $ \alpha < 1 $ to the case $ \alpha > 1 $, ensuring consistency across all $ \alpha > 0 $.

Experimental results

Research questions

• RQ1 Can the number of primes of the form $ p = q\lfloor \alpha n + \beta \rfloor + a $ be asymptotically estimated when $ q $ grows with $ N $, for irrational $ \alpha $ of finite type?
• RQ2 Does the asymptotic density of such primes in arithmetic progressions modulo $ q $ remain proportional to $ \frac{q}{\varphi(q)} \pi(N) $ even when $ q $ grows as a power of $ N $?
• RQ3 What is the best possible error term in the asymptotic formula for the number of such primes, and how does it depend on the Diophantine properties of $ \alpha $?
• RQ4 Can the same methods be adapted to count primes $ p = \lfloor \alpha n + \beta \rfloor $ such that $ p \equiv a \pmod{q} $, and what is the resulting asymptotic?
• RQ5 How do the error terms in the prime counting function behave when $ q $ is allowed to grow, and can they be improved using deep zero-free regions of the zeta function?

Key findings

• For $ \alpha > 0 $ irrational of finite type, there exists $ \kappa > 0 $ such that the number of primes $ p = q\lfloor \alpha n + \beta \rfloor + a $ with $ n \leq N $ and $ 0 \leq a < q \leq N^\kappa $, $ \gcd(a,q) = 1 $, satisfies $ \mathcal{N}_{\alpha,\beta;q,a}(N) = (1 + o(1)) \frac{q}{\varphi(q)} \pi(N) $, with the $ o(1) $ term depending only on $ \alpha $ and $ \beta $.
• When $ q \leq (\log N)^B $ for fixed $ B > 0 $, the error term in the sum $ \sum_{n \leq N} \Lambda(q\lfloor \alpha n + \beta \rfloor + a) $ is $ O\left( N \exp(-C\sqrt{\log N}) \right) $, with $ C > 0 $ depending on $ \alpha, \beta, B $.
• For the special cases $ (a,q) = (0,1) $ and $ (1,2) $, the error term improves to $ O\left( N \exp(-c(\log N)^{3/5}(\log\log N)^{-1/5}) \right) $, matching the best-known error in the prime number theorem.
• A similar asymptotic formula holds for the number of $ n \leq N $ such that $ \lfloor \alpha n + \beta \rfloor \equiv a \pmod{q} $ and is prime: $ \sum_{\substack{n \leq N \\ \lfloor \alpha n + \beta \rfloor \equiv a \pmod{q}}} \Lambda(\lfloor \alpha n + \beta \rfloor) = \alpha^{-1} \sum_{\substack{m \leq \lfloor \alpha N + \beta \rfloor \\ m \equiv a \pmod{q}}} \Lambda(m) + O(N^{1-\kappa}) $, with $ \kappa > 0 $ depending only on $ \alpha $.
• The error term in the above formula is $ O(N \exp(-C\sqrt{\log N})) $ when $ q \leq (\log N)^B $, and $ O(N \exp(-c(\log N)^{3/5}(\log\log N)^{-1/5})) $ for $ (a,q) = (0,1) $ or $ (1,2) $.
• The results are uniform in $ a $ and $ q $, and the implied constants depend only on $ \alpha $ and $ \beta $, not on $ q $, provided $ q \leq N^\kappa $.