[Paper Review] The divisor function and divisor problem

Aleksandar Ivić|arXiv (Cornell University)|Aug 27, 2013

Analytic Number Theory Research21 references1 citations

TL;DR

This paper investigates the divisor function d(n) and its higher-order iterations d^(k)(n), focusing on their maximal order and mean value estimates. It establishes sharp bounds for d^(2)(n) using analytic number theory techniques, including Vorono"i-type explicit formulas and mean square estimates of the error term ∆(x), culminating in an asymptotic formula for the variance of d(n) in short intervals, significantly improving prior results by Coppola and Salerno and extending work by Jutila and others on the distribution of d(n) in short arithmetic progressions.

ABSTRACT

The purpose of this text is twofold. First we discuss some divisor problems involving Paul Erd\H os (1913-1996), whose centenary of birth is this year. In the second part some recent results on divisor problems are discussed, and their connection with the powers moments of $|ζ(\frac{1}{2}+it)|$ is pointed out. This is an extended version of the lecture given at the conference ERDOS100 in Budapest, July 1-5, 2013.

Motivation & Objective

• Establish the maximal order of the second iterate of the divisor function d^(2)(n) using analytic number theory.
• Refine mean square estimates of the error term ∆(x) = Σ_{n≤x} d(n) − x(log x + 2γ − 1) in short intervals.
• Improve upon existing asymptotic formulas for the variance of d(n) in intervals [x, x+U], particularly for U ≪ √T.
• Connect the behavior of d^(k)(n) to the moments of |ζ(1/2 + it)|, linking divisor problems to the Riemann zeta function.
• Provide explicit asymptotic expansions for the integral of [∆(x+U)−∆(x)]² over long intervals, refining earlier results by Jutila and Heath-Brown.

Proposed method

• Use the truncated Vorono"i formula to express ∆(x) as a sum over Bessel functions, allowing flexible parameter choice for error control.
• Apply complex analysis and contour integration to estimate the main term Mk(N,H) and error terms Rk(N,H) in the sum over short intervals.
• Employ mean square estimates for ∆k(x) and the connection to the Riemann zeta function's moments to bound the error in the asymptotic expansion.
• Utilize Ramanujan sums cq(h) and Dirichlet series generating functions to analyze the arithmetic structure of dk(n) and its shifts.
• Combine explicit asymptotic expansions from Theorem 3 with dyadic decomposition and hybrid estimates to derive sharp bounds for maxima over short intervals.
• Use the connection between the divisor problem and the Lindel"of hypothesis via the exponents βk and αk in the error terms.

Experimental results

Research questions

• RQ1 What is the maximal order of the second iterate of the divisor function, d^(2)(n), as a function of n?
• RQ2 How do the mean square and variance of the error term ∆(x+U)−∆(x) behave in short intervals of length U ≪ √T?
• RQ3 What is the precise asymptotic expansion for the integral ∫_T^{2T} [∆(x+U)−∆(x)]² dx, and how does it improve upon prior work?
• RQ4 How do the moments of |ζ(1/2 + it)| relate to the divisor problem and the growth of d^(k)(n)?
• RQ5 What is the best possible upper bound for the maximum of |∆(x+u)−∆(x)| over short intervals of length U, and how does it depend on H and T?

Key findings

• The maximal order of d^(2)(n) satisfies max_{n≤x} log d^(2)(n) = √(log x / log log x) · (D + o(1)), where D ≈ 2.7958 is an explicit constant.
• The asymptotic formula for the variance of d(n) in short intervals is given by ∫_T^{2T} [∆(x+U)−∆(x)]² dx = (TU/3) Σ_{j=0}^3 c_j log^j(√T / U) + O_ε(T^{1/2+ε} U^2 + T^{1+ε} U^{1/2}), with explicit constants c_j.
• This asymptotic formula improves upon the earlier O(TUL^{5/2}) error term by Coppola and Salerno, reducing the error to O(T^{1/2+ε} U^2).
• An improved hybrid bound for the maximum of |∆(x+u)−∆(x)| over u ∈ [0,U] is established: ∫_T^{T+H} max_{0≤u≤U} |∆(x+u)−∆(x)|² dx ≪ HUL^5 + T L^4 log L + H^{1/3} T^{2/3} U^{2/3} L^{10/3} (log L)^{2/3}.
• Under suitable conditions on T, U, H, the paper proves the existence of ≫ H U^{-1} subintervals of length ≫U where |∆(x)| ≥ c_± T^{1/4} for some c_± > 0, confirming oscillatory behavior at the T^{1/4} scale.
• The connection between the divisor problem and the Riemann zeta function is reinforced via the exponents β_k and α_k, with β_k = (k−1)/(2k) conjectured to be optimal, equivalent to the Lindel"of hypothesis.