Relations between the random variable w_x and the Dirichlet divisor problem

2020/07/03 by Dmitry S. Pyatin, Pyatin, Dmitry S.
Computer Science · Economics, Econometrics and Finance · Mathematics · #Analytic Number Theory Research #Credit Risk and Financial Regulations #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2007.01920

openalex publication_date 2020/07/03 · openalex created_date 2021/10/11 · openalex updated_date 2026/07/28

Abstract

We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all n ∈ ℕ⁺: R(n) ≤ O(ψ(n)n^(1)/(4)) where R(n) = | ∑_x=1ⁿ\lfloor(n)/(x)\rfloor - nlogn - (2γ-1)n | and ψ(n) - any positive function that increases unboundedly as n → ∞ . The result is achieved under the hypothesis: \(n)/(x) \ ∼ w_x where w_x is uniformly distributed over [0,1) random variable with a values set \0, \frac 1 x, …, (x-1)/(x) \ and the value accepting probability p = (1)/(x) . The paper concludes with a numerical argument in support of the hypothesis being true. It is shown that the expectation: μ₁ [∑_x=1ⁿ((n)/(x) - (x-1)/(2x)) ]= (2n+1)H_{\lfloor√(n)\rfloor} - \lfloor√(n)\rfloor² - \lfloor√(n)\rfloor + C has deviation from D(n) is less than R(n) in absolute value for all n < 10⁵.

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