4.3 Dirichlet's divisor problem and related estimates

Dirichlet's divisor problem looks at how the number of divisors grows as numbers get bigger. It's all about finding patterns in the chaos of prime factors and their combinations.

The hyperbola method gives us a cool way to visualize and count divisors. By plotting divisor pairs on a graph, we can estimate their growth and spot interesting mathematical relationships.

Dirichlet's Divisor Problem and Hyperbola Method

Understanding Dirichlet's Divisor Problem and Function

• Dirichlet's divisor problem investigates the asymptotic behavior of the summatory divisor function
• Divisor function d(n) counts the number of positive divisors of a positive integer n
• Summatory divisor function D(x) sums d(n) for all positive integers n up to x
• Dirichlet's divisor problem aims to find the best possible estimate for the error term in the asymptotic formula for D(x)
• Asymptotic formula for D(x) expressed as $D(x) = x \log x + (2\gamma - 1)x + \Delta(x)$
• γ represents the Euler-Mascheroni constant approximately equal to 0.57721
• Δ(x) denotes the error term, which Dirichlet sought to minimize

Exploring the Hyperbola Method

• Hyperbola method provides a geometric approach to count divisors of integers
• Utilizes the fact that divisors of n come in pairs (a, b) where ab = n
• Plots divisor pairs as points on a coordinate plane, forming a hyperbola
• Dirichlet's hyperbola method refines this approach for more accurate estimates
• Counts integer points under the hyperbola xy = x in the first quadrant
• Divides the region under the hyperbola into three parts: square, rectangular, and hyperbolic
• Square part corresponds to pairs (a, b) where both a and b are less than or equal to √x
• Rectangular part includes pairs where one coordinate exceeds √x
• Hyperbolic part contains the remaining points close to the hyperbola

Analyzing the Error Term

• Error term Δ(x) in Dirichlet's divisor problem measures the deviation from the main term
• Dirichlet proved that Δ(x) = O(x^(1/2)) (Big O notation)
• Subsequent improvements reduced the exponent in the error term
• Current best known bound for the error term $\Delta(x) = O(x^{131/416+\epsilon})$ for any ε > 0
• Conjectured that the true order of magnitude of Δ(x) is O(x^(1/4+ε)) for any ε > 0
• Lau and Tsang showed that Δ(x) changes sign infinitely often
• Omega results establish lower bounds for the error term
• Hardy proved that Δ(x) = Ω(x^(1/4)) (Omega notation)
• Improved omega result $\Delta(x) = \Omega_{\pm}(x^{1/4}(\log x)^{1/4}(\log \log x)^{(3+\log 4)/(4 \log 2)-1/4-\epsilon})$ for any ε > 0

Circle Problem and Lattice Point Counting

Exploring the Circle Problem

• Circle problem investigates the number of lattice points inside or on a circle centered at the origin
• Lattice points defined as points with integer coordinates in the Cartesian plane
• Let r(n) denote the number of ways to express n as a sum of two squares
• Circle problem closely related to studying the behavior of r(n)
• Gauss circle problem specifically examines the error term in estimating lattice points
• Number of lattice points inside or on a circle of radius √x denoted by N(x)
• Main term in the asymptotic formula for N(x) given by πx
• Error term P(x) defined as N(x) - πx
• Gauss proved that |P(x)| = O(x^(1/2))

Techniques in Lattice Point Counting

• Lattice point counting employs various analytic and geometric methods
• Dirichlet's hyperbola method adapted for lattice point problems
• Exponential sums play a crucial role in estimating error terms
• Weyl's inequality used to bound certain exponential sums
• van der Corput's method provides tools for estimating oscillatory integrals
• Large sieve inequality applied to obtain average results over sequences
• Smoothing techniques used to reduce the problem to estimating smoother functions
• Voronoi summation formula connects lattice point counts to Bessel functions
• Hardy-Littlewood circle method applied to study the distribution of r(n)

Advancements in Error Term Estimates

• Exponent pairs technique introduced to improve error term bounds
• Exponent pair (κ, λ) relates to the error term estimate O(x^κ + x^λ)
• van der Corput's AB-process generates new exponent pairs from existing ones
• A-process: If (κ, λ) is an exponent pair, then (λ/(2λ+2), (κ+1)/(2λ+2)) is also an exponent pair
• B-process: If (κ, λ) is an exponent pair, then ((2κ+2)/(4κ+3), (2λ+1)/(4κ+3)) is also an exponent pair
• Huxley's method combines ideas from harmonic analysis and geometry
• Current best known bound for P(x) $O(x^{131/416+\epsilon})$ for any ε > 0
• Conjectured that |P(x)| = O(x^(1/4+ε)) for any ε > 0
• Hardy's omega result $P(x) = \Omega_{\pm}(x^{1/4})$
• Improved omega result by Soundararajan $P(x) = \Omega_{\pm}(x^{1/4}(\log x)^{1/4}(\log \log x)^{(3+\log 4)/(4 \log 2)-1/4-\epsilon})$ for any ε > 0