4: Number Theoretic Functions
- Page ID
- 60313
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Multiplicative Functions
- This page discusses number theoretic functions, specifically multiplicative and completely multiplicative functions for positive integers. It defines multiplicative functions that satisfy \(f(ab) = f(a)f(b)\) for coprime \(a\) and \(b\), while completely multiplicative functions do not. Examples include the σ function for summing k-th powers of divisors, the Mobius function for square-freeness, and Euler's phi function, which counts integers relatively prime to \(n\).
- 4.2: Additive Functions
- This page covers additive functions, defining their properties where \(f(ab) = f(a) + f(b)\) for coprime \(a\) and \(b\). It differentiates between additive and completely additive functions. Additionally, it introduces two key functions: \(\omega(n)\), which counts distinct prime divisors, and \(\Omega(n)\), which counts total prime powers dividing \(n\). Both functions illustrate unique additive properties, with \(\omega\) being additive and \(\Omega\) completely additive.
- 4.3: Mobius inversion
- This page discusses Lemmas 4.12 and 4.13, and Theorem 4.14 concerning number theory and Möbius inversion. Lemma 4.12 defines the function \(\epsilon(n)\) associated with the Möbius function, revealing it equals zero for \(n > 1\). Lemma 4.13 demonstrates the equivalence between sets \(S_n\) and \(T_n\). Theorem 4.
- 4.4: Euler’s Phi or Totient Function
- This page covers key concepts about Euler's phi function, including Lemma 4.15 which shows that any natural number can be expressed as the sum of phi values at its divisors. Theorem 4.16 provides an explicit formula for the phi function based on prime power factorization and confirms its multiplicative nature. Corollary 4.17 reinforces this property, highlighting its importance in number theory applications.
- 4.5: Dirichlet and Lambert Series
- This page covers the Dirichlet convolution of arithmetic functions and its applications in number theory, defining crucial functions such as \(\epsilon(n)\) and discussing convolution identities. It introduces Dirichlet and Lambert series, focusing on the Riemann zeta function as a key example.
- 4.6: Exercise
- This page provides exercises in number theory, focusing on multiplicative functions and their properties. It covers tasks related to classifying functions, computing arithmetic functions (like divisors and Euler's totient), and exploring amicable and perfect numbers. Additionally, it delves into inclusion-exclusion principles, Dirichlet convolution, and the Möbius function, exploring relationships among number-theoretic functions and their multiplicative nature.