Excursions in Multiplicative Number Theory 2022

This textbook offers a unique exploration of analytic number theory that is focused on explicit and realistic numerical bounds. By giving precise proofs in simplified settings, the author strategically builds practical tools and insights for exploring the behavior of arithmetical functions. An active learning style is encouraged across nearly three hundred exercises, making this an indispensable resource for both students and instructors. Designed to allow readers several different pathways to progress from basic notions to active areas of research, the book begins with a study of arithmetic functions and notions of arithmetical interest. From here, several guided “walks” invite readers to continue, offering explorations along three broad themes: the convolution method, the Levin–Faĭnleĭb theorem, and the Mellin transform. Having followed any one of the walks, readers will arrive at “higher ground”, where they will find opportunities for extensions and applications, such as the Selberg formula, Exponential sums with arithmetical coefficients, and the Large Sieve Inequality. Methodology is emphasized throughout, with frequent opportunities to explore numerically using computer algebra packages Pari/GP and Sage. Excursions in Multiplicative Number Theory is ideal for graduate students and upper-level undergraduate students who are familiar with the fundamentals of analytic number theory. It will also appeal to researchers in mathematics and engineering interested in experimental techniques in this active area. Preface Using this Book for a Course Contents Part I Approach: Multiplicativity 1 Arithmetic Convolution 1.1 Multiplicative Functions 1.2 Some Members of the Zoo 1.3 The Divisor Function 1.4 Computing Values of Multiplicative Functions 1.5 Convolution of Multiplicative Functions 1.6 Unitary Convolution 1.7 Taxonomy References 2 A Calculus on Arithmetical Functions 2.1 The Ring of Formal Dirichlet Series 2.2 Further Examples References 3 Analytical Dirichlet Series 3.1 Abscissa of Absolute Convergence 3.2 An Introduction to the Riemann Zeta-Function 3.3 Dirichlet Series and Convolution Product 3.4 Dirichlet Series and Multiplicativity 3.5 Dirichlet Series, Multiplicativity and Unitarian Convolution 3.6 A Popular Variation 3.7 Some Remarks Without Proof References 4 Growth of Arithmetical Functions 4.1 The Order of the Divisor Function 4.2 Extensions and Exercises References 5 An ``Algebraical'' Multiplicative Function 5.1 The Integers Modulo q 5.2 The Multiplicative Group Modulo f 5.3 The Subgroup of Squares 5.4 The Legendre Symbol 5.5 The L-Function of the Legendre Symbol References 6 Möbius Inversions 6.1 Pointwise Inversion 6.2 Functional Möbius Inversion 6.3 An Identity Factory 6.4 Another Moebius Inversion Formula References 7 Handling a Smooth Factor 7.1 Summing Smooth Functions 7.2 Some Notes on Integrals and Derivatives References Part II The Convolution Walk 8 The Convolution Method 8.1 Proof of Theorem A 8.2 An Exercise on Summation by Parts 8.3 A Warning and some Limitations References 9 Euler Products and Euler Sums 9.1 Exchanging Sum and Product Signs 9.2 The Witt Expansion 9.3 Explicit Witt Expansion of Polynomials 9.4 Euler Product Expansion in Terms of Zeta-Values 9.5 Refining Exercise [ProdExample]9-1 9.6 Euler Sums, with no log-Factor 9.7 Euler Sums, with log-Factor References 10 Some Practice 10.1 Four Examples 10.2 Expansion as Dirichlet Series 10.3 Expansion as Convolution Product 10.4 A General Lemma 10.5 A Final Example 10.6 Handling Square-free Integers References 11 Dirichlet's Hyperbola Principle 11.1 First Error Term for the Divisor Function 11.2 The Dirichlet Hyperbola Principle 11.3 A Better Remainder in the Divisor Problem References 12 The Mertens Estimates 12.1 The von Mangoldt Λ-Function 12.2 From the Logarithm to the von Mangoldt Function 12.2.1 An Upper Bound à la Chebyshevaut]Chebyshev, Pafnouty Lwowitsch 12.2.2 The Three Theorems of Mertens 12.3 A Bertrand Postulate Type Result References Part III The Levin–Faĭnleĭb Walk 13 The Levin–Faĭnleĭb Theorem and Analogues 13.1 A First Upper Bound 13.2 An Asymptotic Formula 13.3 Proof of Theorem B 13.4 A Stronger Theorem References 14 Variations on a Theme of Chebyshev References 15 Primes in Arithmetical Progressions 15.1 The Multiplicative Group of Z/qZ 15.2 Dirichlet Characters Modulo 3 and 4 15.3 The Theorems of Mertens Modulo 3 and 4 References 16 Computing a Famous Constant 16.1 The Hexagonal Versus the Square Lattice References 17 Euler Products with Primes in Progressions 17.1 The Witt Transform 17.2 Euler Products with Characters 17.3 On the Batemanaut]Bateman, Paul Trevier–Horn Conjecture References 18 The Chinese Remainder Theorem and Multiplicativity 18.1 The Chinese Remainder Theorem 18.2 On the Number of Divisors of n2+1 References 19 The Riemann Zeta Function 19.1 Upper Bounds in the Critical Strip 19.2 Computing Zeta References Part IV The Mellin Walk 20 The Mellin Transform 20.1 Some Examples 20.2 The complex Stirling Formula and the Cahen-Millen Formula 20.3 Mellin Transforms/Fourier Transforms 20.4 Truncated Transform 20.5 Smoothed Formulae 20.6 Smoothed Representations References 21 Proof of Theorem mathscrC 22 Roughing up: Removing a Smoothening 22.1 Exploring a Mean Value with Sage References 23 Proving the Prime Number Theorem 23.1 The Riemann zeta function near the line Re s = 1 23.2 The Prime Number Theorem for the Möbius Function 23.3 Proof of Theorem [PNT]23.1 References 24 The Selberg Formula 24.1 The Iseki–Tatuzawa Formula 24.2 A Different Proof 24.3 A Glimpse at the Asymptotic Sieve and Pk-Numbers References Part V Higher Ground: Applications/Extensions 25 Rankin's Trick and Brun's Sieve 25.1 Rankin's Trick in its Simplest Form 25.2 Rankin's Trick and Brun's Sieve References 26 Three Arithmetical Exponential Sums 26.1 Rational Approximations of the Golden Ratio and more 26.2 An Exponential Sum over Integers Free of Small Primes 26.3 An Exponential Sum over the Primes 26.4 An Exponential Sum with the Möbius Function References 27 Convolution Method and Non-Positive Functions References 28 The Large Sieve Inequality 28.1 An Approximate Parseval Equation 28.2 The Large Sieve Inequality 28.2.1 A Fourier Transform 28.2.2 Proof of a weak form of Theorem [LS]28.1 28.3 Introducing Farey Points References 29 Montgomery's Sieve 29.1 The Statement 29.2 Applications References Hints and Solutions for Selected Exercises Appendix Chart of Common Arithmetical Functions Appendix Getting Hold of the Bibliographical Items References Appendix Name Index Author Index Appendix Index Index This textbook offers a unique exploration of analytic number theory that is focused on explicit and realistic numerical bounds. By giving precise proofs in simplified settings, the author strategically builds practical tools and insights for exploring the behavior of arithmetical functions. An active learning style is encouraged across nearly three hundred exercises, making this an indispensable resource for both students and instructors. Designed to allow readers several different pathways to progress from basic notions to active areas of research, the book begins with a study of arithmetic functions and notions of arithmetical interest. From here, several guided “walks” invite readers to continue, offering explorations along three broad themes: the convolution method, the Levin–Faĭnleĭb theorem, and the Mellin transform. Having followed any one of the walks, readers will arrive at “higher ground”, where they will find opportunities for extensions and applications, such as the Selberg formula, Brun’s sieve, and the Large Sieve Inequality. Methodology is emphasized throughout, with frequent opportunities to explore numerically using computer algebra packages Pari/GP and Sage. Excursions in Multiplicative Number Theory is ideal for graduate students and upper-level undergraduate students who are familiar with the fundamentals of analytic number theory. It will also appeal to researchers in mathematics and engineering interested in experimental techniques in this active area. Allows readers several different pathways to progress from basic notions to active areas of research, the book begins with a study of arithmetic functions and notions of arithmetical interest. From here, several guided "walks" invite readers to continue, offering explorations along three broad themes: the convolution method, the Levin-Faĭnleĭb theorem, and the Mellin transform. Having followed any one of the walks, readers will arrive at "higher ground" where they will find opportunities for extensions and applications, such as the Selberg formula, Exponential sums with arithmetical coefficients, and the Large Sieve Inequality. Methodology is emphasized throughout, with frequent opportunities to explore numerically using computer algebra packages Pari/GP and Sage. Excursions in Multiplicative Number Theory is ideal for graduate students and upper-level undergraduate students who are familiar with the fundamentals of analytic number theory. It will also appeal to researchers in mathematics and engineering interested in experimental techniques in this active area