Course Notes for Analytic Number Theory
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Euler Product Asymptotics on Dirichlet L-Functions 3
EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS IKUYA KANEKO Abstract. We derive the asymptotic behaviour of partial Euler products for Dirichlet L-functions L(s,χ) in the critical strip upon assuming only the Generalised Riemann Hypothesis (GRH). Capturing the behaviour of the partial Euler products on the critical line is called the Deep Riemann Hypothesis (DRH) on the ground of its importance. Our result manifests the positional relation of the GRH and DRH. If χ is a non-principal character, we observe the √2 phenomenon, which asserts that the extra factor √2 emerges in the Euler product asymptotics on the central point s = 1/2. We also establish the connection between the behaviour of the partial Euler products and the size of the remainder term in the prime number theorem for arithmetic progressions. Contents 1. Introduction 1 2. Prelude to the asymptotics 5 3. Partial Euler products for Dirichlet L-functions 6 4. Logarithmical approach to Euler products 7 5. Observing the size of E(x; q,a) 8 6. Appendix 9 Acknowledgement 11 References 12 1. Introduction This paper was motivated by the beautiful and profound work of Ramanujan on asymptotics for the partial Euler product of the classical Riemann zeta function ζ(s). We handle the family of Dirichlet L-functions ∞ L(s,χ)= χ(n)n−s = (1 χ(p)p−s)−1 with s> 1, − ℜ 1 p X Y arXiv:1902.04203v1 [math.NT] 12 Feb 2019 and prove the asymptotic behaviour of partial Euler products (1.1) (1 χ(p)p−s)−1 − 6 pYx in the critical strip 0 < s < 1, assuming the Generalised Riemann Hypothesis (GRH) for this family. -
Methods of Measuring Population Change
Methods of Measuring Population Change CDC 103 – Lecture 10 – April 22, 2012 Measuring Population Change • If past trends in population growth can be expressed in a mathematical model, there would be a solid justification for making projections on the basis of that model. • Demographers developed an array of models to measure population growth; four of these models are usually utilized. 1 Measuring Population Change Arithmetic (Linear), Geometric, Exponential, and Logistic. Arithmetic Change • A population growing arithmetically would increase by a constant number of people in each period. • If a population of 5000 grows by 100 annually, its size over successive years will be: 5100, 5200, 5300, . • Hence, the growth rate can be calculated by the following formula: • (100/5000 = 0.02 or 2 per cent). 2 Arithmetic Change • Arithmetic growth is the same as the ‘simple interest’, whereby interest is paid only on the initial sum deposited, the principal, rather than on accumulating savings. • Five percent simple interest on $100 merely returns a constant $5 interest every year. • Hence, arithmetic change produces a linear trend in population growth – following a straight line rather than a curve. Arithmetic Change 3 Arithmetic Change • The arithmetic growth rate is expressed by the following equation: Geometric Change • Geometric population growth is the same as the growth of a bank balance receiving compound interest. • According to this, the interest is calculated each year with reference to the principal plus previous interest payments, thereby yielding a far greater return over time than simple interest. • The geometric growth rate in demography is calculated using the ‘compound interest formula’. -
On Analytic Number Theory
Lectures on Analytic Number Theory By H. Rademacher Tata Institute of Fundamental Research, Bombay 1954-55 Lectures on Analytic Number Theory By H. Rademacher Notes by K. Balagangadharan and V. Venugopal Rao Tata Institute of Fundamental Research, Bombay 1954-1955 Contents I Formal Power Series 1 1 Lecture 2 2 Lecture 11 3 Lecture 17 4 Lecture 23 5 Lecture 30 6 Lecture 39 7 Lecture 46 8 Lecture 55 II Analysis 59 9 Lecture 60 10 Lecture 67 11 Lecture 74 12 Lecture 82 13 Lecture 89 iii CONTENTS iv 14 Lecture 95 15 Lecture 100 III Analytic theory of partitions 108 16 Lecture 109 17 Lecture 118 18 Lecture 124 19 Lecture 129 20 Lecture 136 21 Lecture 143 22 Lecture 150 23 Lecture 155 24 Lecture 160 25 Lecture 165 26 Lecture 169 27 Lecture 174 28 Lecture 179 29 Lecture 183 30 Lecture 188 31 Lecture 194 32 Lecture 200 CONTENTS v IV Representation by squares 207 33 Lecture 208 34 Lecture 214 35 Lecture 219 36 Lecture 225 37 Lecture 232 38 Lecture 237 39 Lecture 242 40 Lecture 246 41 Lecture 251 42 Lecture 256 43 Lecture 261 44 Lecture 264 45 Lecture 266 46 Lecture 272 Part I Formal Power Series 1 Lecture 1 Introduction In additive number theory we make reference to facts about addition in 1 contradistinction to multiplicative number theory, the foundations of which were laid by Euclid at about 300 B.C. Whereas one of the principal concerns of the latter theory is the deconposition of numbers into prime factors, addi- tive number theory deals with the decomposition of numbers into summands. -
Ch. 15 Power Series, Taylor Series
Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex. -
Analytic Number Theory
A course in Analytic Number Theory Taught by Barry Mazur Spring 2012 Last updated: January 19, 2014 1 Contents 1. January 244 2. January 268 3. January 31 11 4. February 2 15 5. February 7 18 6. February 9 22 7. February 14 26 8. February 16 30 9. February 21 33 10. February 23 38 11. March 1 42 12. March 6 45 13. March 8 49 14. March 20 52 15. March 22 56 16. March 27 60 17. March 29 63 18. April 3 65 19. April 5 68 20. April 10 71 21. April 12 74 22. April 17 77 Math 229 Barry Mazur Lecture 1 1. January 24 1.1. Arithmetic functions and first example. An arithmetic function is a func- tion N ! C, which we will denote n 7! c(n). For example, let rk;d(n) be the number of ways n can be expressed as a sum of d kth powers. Waring's problem asks what this looks like asymptotically. For example, we know r3;2(1729) = 2 (Ramanujan). Our aim is to derive statistics for (interesting) arithmetic functions via a study of the analytic properties of generating functions that package them. Here is a generating function: 1 X n Fc(q) = c(n)q n=1 (You can think of this as a Fourier series, where q = e2πiz.) You can retrieve c(n) as a Cauchy residue: 1 I F (q) dq c(n) = c · 2πi qn q This is the Hardy-Littlewood method. We can understand Waring's problem by defining 1 X nk fk(q) = q n=1 (the generating function for the sequence that tells you if n is a perfect kth power). -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
7. Properties of Uniformly Convergent Sequences
46 1. THE THEORY OF CONVERGENCE 7. Properties of uniformly convergent sequences Here a relation between continuity, differentiability, and Riemann integrability of the sum of a functional series or the limit of a functional sequence and uniform convergence is studied. 7.1. Uniform convergence and continuity. Theorem 7.1. (Continuity of the sum of a series) N The sum of the series un(x) of terms continuous on D R is continuous if the series converges uniformly on D. ⊂ P Let Sn(x)= u1(x)+u2(x)+ +un(x) be a sequence of partial sum. It converges to some function S···(x) because every uniformly convergent series converges pointwise. Continuity of S at a point x means (by definition) lim S(y)= S(x) y→x Fix a number ε> 0. Then one can find a number δ such that S(x) S(y) <ε whenever 0 < x y <δ | − | | − | In other words, the values S(y) can get arbitrary close to S(x) and stay arbitrary close to it for all points y = x that are sufficiently close to x. Let us show that this condition follows6 from the hypotheses. Owing to the uniform convergence of the series, given ε > 0, one can find an integer m such that ε S(x) Sn(x) sup S Sn , x D, n m | − |≤ D | − |≤ 3 ∀ ∈ ∀ ≥ Note that m is independent of x. By continuity of Sn (as a finite sum of continuous functions), for n m, one can also find a number δ > 0 such that ≥ ε Sn(x) Sn(y) < whenever 0 < x y <δ | − | 3 | − | So, given ε> 0, the integer m is found. -
Of the Riemann Hypothesis
A Simple Counterexample to Havil's \Reformulation" of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 10025 [email protected] The Riemann Hypothesis (RH) is \the greatest mystery in mathematics" [3]. It is a conjecture about the Riemann zeta function. The zeta function allows us to pass from knowledge of the integers to knowledge of the primes. In his book Gamma: Exploring Euler's Constant [4, p. 207], Julian Havil claims that the following is \a tantalizingly simple reformulation of the Rie- mann Hypothesis." Havil's Conjecture. If 1 1 X (−1)n X (−1)n cos(b ln n) = 0 and sin(b ln n) = 0 na na n=1 n=1 for some pair of real numbers a and b, then a = 1=2. In this note, we first state the RH and explain its connection with Havil's Conjecture. Then we show that the pair of real numbers a = 1 and b = 2π=ln 2 is a counterexample to Havil's Conjecture, but not to the RH. Finally, we prove that Havil's Conjecture becomes a true reformulation of the RH if his conclusion \then a = 1=2" is weakened to \then a = 1=2 or a = 1." The Riemann Hypothesis In 1859 Riemann published a short paper On the number of primes less than a given quantity [6], his only one on number theory. Writing s for a complex variable, he assumes initially that its real 1 part <(s) is greater than 1, and he begins with Euler's product-sum formula 1 Y 1 X 1 = (<(s) > 1): 1 ns p 1 − n=1 ps Here the product is over all primes p. -
1.1 Constructing the Real Numbers
18.095 Lecture Series in Mathematics IAP 2015 Lecture #1 01/05/2015 What is number theory? The study of numbers of course! But what is a number? • N = f0; 1; 2; 3;:::g can be defined in several ways: { Finite ordinals (0 := fg, n + 1 := n [ fng). { Finite cardinals (isomorphism classes of finite sets). { Strings over a unary alphabet (\", \1", \11", \111", . ). N is a commutative semiring: addition and multiplication satisfy the usual commu- tative/associative/distributive properties with identities 0 and 1 (and 0 annihilates). Totally ordered (as ordinals/cardinals), making it a (positive) ordered semiring. • Z = {±n : n 2 Ng.A commutative ring (commutative semiring, additive inverses). Contains Z>0 = N − f0g closed under +; × with Z = −Z>0 t f0g t Z>0. This makes Z an ordered ring (in fact, an ordered domain). • Q = fa=b : a; 2 Z; b 6= 0g= ∼, where a=b ∼ c=d if ad = bc. A field (commutative ring, multiplicative inverses, 0 6= 1) containing Z = fn=1g. Contains Q>0 = fa=b : a; b 2 Z>0g closed under +; × with Q = −Q>0 t f0g t Q>0. This makes Q an ordered field. • R is the completion of Q, making it a complete ordered field. Each of the algebraic structures N; Z; Q; R is canonical in the following sense: every non-trivial algebraic structure of the same type (ordered semiring, ordered ring, ordered field, complete ordered field) contains a copy of N; Z; Q; R inside it. 1.1 Constructing the real numbers What do we mean by the completion of Q? There are two possibilities: 1. -
Arxiv:Math/0201082V1
2000]11A25, 13J05 THE RING OF ARITHMETICAL FUNCTIONS WITH UNITARY CONVOLUTION: DIVISORIAL AND TOPOLOGICAL PROPERTIES. JAN SNELLMAN Abstract. We study (A, +, ⊕), the ring of arithmetical functions with unitary convolution, giving an isomorphism between (A, +, ⊕) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell- Everett[4] between the ring (A, +, ·) of arithmetical functions with Dirichlet convolution and the power series ring C[[x1,x2,x3,... ]] on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units. 1. Introduction The ring of arithmetical functions with Dirichlet convolution, which we’ll denote by (A, +, ·), is the set of all functions N+ → C, where N+ denotes the positive integers. It is given the structure of a commutative C-algebra by component-wise addition and multiplication by scalars, and by the Dirichlet convolution f · g(k)= f(r)g(k/r). (1) Xr|k Then, the multiplicative unit is the function e1 with e1(1) = 1 and e1(k) = 0 for k> 1, and the additive unit is the zero function 0. Cashwell-Everett [4] showed that (A, +, ·) is a UFD using the isomorphism (A, +, ·) ≃ C[[x1, x2, x3,... ]], (2) where each xi corresponds to the function which is 1 on the i’th prime number, and 0 otherwise. Schwab and Silberberg [9] topologised (A, +, ·) by means of the norm 1 arXiv:math/0201082v1 [math.AC] 10 Jan 2002 |f| = (3) min { k f(k) 6=0 } They noted that this norm is an ultra-metric, and that ((A, +, ·), |·|) is a valued ring, i.e. -
Math Club: Recurrent Sequences Nov 15, 2020
MATH CLUB: RECURRENT SEQUENCES NOV 15, 2020 In many problems, a sequence is defined using a recurrence relation, i.e. the next term is defined using the previous terms. By far the most famous of these is the Fibonacci sequence: (1) F0 = 0;F1 = F2 = 1;Fn+1 = Fn + Fn−1 The first several terms of this sequence are below: 0; 1; 1; 2; 3; 5; 8; 13; 21; 34;::: For such sequences there is a method of finding a general formula for nth term, outlined in problem 6 below. 1. Into how many regions do n lines divide the plane? It is assumed that no two lines are parallel, and no three lines intersect at a point. Hint: denote this number by Rn and try to get a recurrent formula for Rn: what is the relation betwen Rn and Rn−1? 2. The following problem is due to Leonardo of Pisa, later known as Fibonacci; it was introduced in his 1202 book Liber Abaci. Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year? 3. Daniel is coming up the staircase of 20 steps. He can either go one step at a time, or skip a step, moving two steps at a time. -
Nine Chapters of Analytic Number Theory in Isabelle/HOL
Nine Chapters of Analytic Number Theory in Isabelle/HOL Manuel Eberl Technische Universität München 12 September 2019 + + 58 Manuel Eberl Rodrigo Raya 15 library unformalised 173 18 using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods Much of the formalised material is not particularly analytic. Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Some of these results have already been formalised by other people (Avigad, Harrison, Carneiro, . ) – but not in the context of a large library. What is Analytic Number Theory? Studying the multiplicative and additive structure of the integers using analytic methods In this work: only multiplicative number theory (primes, divisors, etc.) Much of the formalised material is not particularly analytic.