Möbius Inversion from the Point of View of Arithmetical Semigroup Flows
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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. -
Generalized Riemann Hypothesis Léo Agélas
Generalized Riemann Hypothesis Léo Agélas To cite this version: Léo Agélas. Generalized Riemann Hypothesis. 2019. hal-00747680v3 HAL Id: hal-00747680 https://hal.archives-ouvertes.fr/hal-00747680v3 Preprint submitted on 29 May 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Generalized Riemann Hypothesis L´eoAg´elas Department of Mathematics, IFP Energies nouvelles, 1-4, avenue de Bois-Pr´eau,F-92852 Rueil-Malmaison, France Abstract (Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most impor- tant unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems (Borwien et al.(2008), Sabbagh(2002)). In this paper, we give a proof of Gen- eralized Riemann Hypothesis which implies the proof of Riemann Hypothesis and Goldbach's weak conjecture (also known as the odd Goldbach conjecture) one of the oldest and best-known unsolved problems in number theory. 1. Introduction The Riemann hypothesis is one of the most important conjectures in math- ematics. -
Limiting Values and Functional and Difference Equations †
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan; [email protected] * Correspondence: [email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions. -
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. -
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. -
3.4 Multiplicative Functions
3.4 Multiplicative Functions Question how to identify a multiplicative function? If it is the convolution of two other arithmetic functions we can use Theorem 3.19 If f and g are multiplicative then f ∗ g is multiplicative. Proof Assume gcd ( m, n ) = 1. Then d|mn if, and only if, d = d1d2 with d1|m, d2|n (and thus gcd( d1, d 2) = 1 and the decomposition is unique). Therefore mn f ∗ g(mn ) = f(d) g d dX|mn m n = f(d1d2) g d1 d2 dX1|m Xd2|n m n = f(d1) g f(d2) g d1 d2 dX1|m Xd2|n since f and g are multiplicative = f ∗ g(m) f ∗ g(n) Example 3.20 The divisor functions d = 1 ∗ 1, and dk = 1 ∗ 1 ∗ ... ∗ 1 ν convolution k times, along with σ = 1 ∗j and σν = 1 ∗j are all multiplicative. Note 1 is completely multiplicative but d is not, so convolution does not preserve complete multiplicatively. Question , was it obvious from its definition that σ was multiplicative? I suggest not. Example 3.21 For prime powers pa+1 − 1 d(pa) = a+1 and σ(pa) = . p−1 Hence pa+1 − 1 d(n) = (a+1) and σ(n) = . a a p−1 pYkn pYkn 11 Recall how in Theorem 1.8 we showed that ζ(s) has a Euler Product, so for Re s > 1, 1 −1 ζ(s) = 1 − . (7) ps p Y When convergent, an infinite product is non-zero. Hence (7) gives us (as already seen earlier in the course) Corollary 3.22 ζ(s) =6 0 for Re s > 1. -
Aspects of Quantum Field Theory and Number Theory, Quantum Field Theory and Riemann Zeros, © July 02, 2015
juan guillermo dueñas luna ASPECTSOFQUANTUMFIELDTHEORYAND NUMBERTHEORY ASPECTSOFQUANTUMFIELDTHEORYANDNUMBER THEORY juan guillermo dueñas luna Quantum Field Theory and Riemann Zeros Theoretical Physics Department. Brazilian Center For Physics Research. July 02, 2015. Juan Guillermo Dueñas Luna: Aspects of Quantum Field Theory and Number Theory, Quantum Field Theory and Riemann Zeros, © July 02, 2015. supervisors: Nami Fux Svaiter. location: Rio de Janeiro. time frame: July 02, 2015. ABSTRACT The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1=2. Motivated by the Hilbert- Pólya conjecture which states that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory, a lot of activity has been devoted to establish a bridge between number theory and physics. Using the techniques of the spectral zeta function we show that prime numbers and the Rie- mann zeros have a different behaviour as the spectrum of a linear operator associated to a system with countable infinite number of degrees of freedom. To explore more connections between quantum field theory and number theory we studied three systems involving these two sequences of numbers. First, we discuss the renormalized zero-point energy for a massive scalar field such that the Riemann zeros appear in the spectrum of the vacuum modes. This scalar field is defined in a (d + 1)-dimensional flat space-time, assuming that one of the coordinates lies in an finite interval [0, a]. For even dimensional space-time we found a finite reg- ularized energy density, while for odd dimensional space-time we are forced to introduce mass counterterms to define a renormalized vacuum energy. -
The Mobius Function and Mobius Inversion
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Number Theory Mathematics via Primary Historical Sources (TRIUMPHS) Winter 2020 The Mobius Function and Mobius Inversion Carl Lienert Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_number Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, Number Theory Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Recommended Citation Lienert, Carl, "The Mobius Function and Mobius Inversion" (2020). Number Theory. 12. https://digitalcommons.ursinus.edu/triumphs_number/12 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Number Theory by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. The Möbius Function and Möbius Inversion Carl Lienert∗ January 16, 2021 August Ferdinand Möbius (1790–1868) is perhaps most well known for the one-sided Möbius strip and, in geometry and complex analysis, for the Möbius transformation. In number theory, Möbius’ name can be seen in the important technique of Möbius inversion, which utilizes the important Möbius function. In this PSP we’ll study the problem that led Möbius to consider and analyze the Möbius function. Then, we’ll see how other mathematicians, Dedekind, Laguerre, Mertens, and Bell, used the Möbius function to solve a different inversion problem.1 Finally, we’ll use Möbius inversion to solve a problem concerning Euler’s totient function. -
Prime Factorization, Zeta Function, Euler Product
Analytic Number Theory, undergrad, Courant Institute, Spring 2017 http://www.math.nyu.edu/faculty/goodman/teaching/NumberTheory2017/index.html Section 2, Euler products version 1.2 (latest revision February 8, 2017) 1 Introduction. This section serves two purposes. One is to cover the Euler product formula for the zeta function and prove the fact that X p−1 = 1 : (1) p The other is to develop skill and tricks that justify the calculations involved. The zeta function is the sum 1 X ζ(s) = n−s : (2) 1 We will show that the sum converges as long as s > 1. The Euler product formula is Y −1 ζ(s) = 1 − p−s : (3) p This formula expresses the fact that every positive integers has a unique rep- resentation as a product of primes. We will define the infinite product, prove that this one converges for s > 1, and prove that the infinite sum is equal to the infinite product if s > 1. The derivative of the zeta function is 1 X ζ0(s) = − log(n) n−s : (4) 1 This formula is derived by differentiating each term in (2), as you would do for a finite sum. We will prove that this calculation is valid for the zeta sum, for s > 1. We also can differentiate the product (3) using the Leibnitz rule as 1 though it were a finite product. In the following calculation, r is another prime: 8 9 X < d −1 X −1= ζ0(s) = 1 − p−s 1 − r−s ds p : r6=p ; 8 9 X <h −2i X −1= = − log(p)p−s 1 − p−s 1 − r−s p : r6=p ; ( ) X −1 = − log(p) p−s 1 − p−s ζ(s) p ( 1 !) X X ζ0(s) = − log(p) p−ks ζ(s) : (5) p 1 If we divide by ζ(s), the sum on the right is a sum over prime powers (numbers n that have a single p in their prime factorization). -
Lagrange Inversion Theorem for Dirichlet Series Arxiv:1911.11133V1
Lagrange Inversion Theorem for Dirichlet series Alexey Kuznetsov ∗ November 26, 2019 Abstract We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4]. Keywords: Dirichlet series, Lagrange Inversion Theorem, convolution polynomials 2010 Mathematics Subject Classification : Primary 30B50, Secondary 11M41 1 Introduction 0 The Lagrange Inversion Theorem states that if φ(z) is analytic near z = z0 with φ (z0) 6= 0 then the function η(w), defined as the solution to φ(η(w)) = w, is analytic near w0 = φ(z0) and is represented there by the power series X bn η(w) = z + (w − w )n; (1) 0 n! 0 n≥0 where the coefficients are given by n−1 n d z − z0 bn = lim n−1 : (2) z!z0 dz φ(z) − φ(z0) 0 This result can be stated in the following simpler form. We assume that φ(z0) = z0 = 0 and φ (0) = 1. We define A(z) = z=φ(z) and check that A is analytic near z = 0 and is represented there by the power series arXiv:1911.11133v1 [math.NT] 25 Nov 2019 X n A(z) = 1 + anz : (3) n≥1 Let us define B(z) implicitly via A(zB(z)) = B(z): (4) Then the function η(z) = zB(z) solves equation η(z)=A(η(z)) = z, which is equivalent to φ(η(z)) = z. Applying Lagrange Inversion Theorem in the form (2) we obtain X an(n + 1) B(z) = zn; (5) n + 1 n≥0 ∗Dept. -
Prime Numbers and the Riemann Hypothesis
Prime numbers and the Riemann hypothesis Tatenda Kubalalika October 2, 2019 ABSTRACT. Denote by ζ the Riemann zeta function. By considering the related prime zeta function, we demonstrate in this note that ζ(s) 6= 0 for <(s) > 1=2, which proves the Riemann hypothesis. Keywords and phrases: Prime zeta function, Riemann zeta function, Riemann hypothesis, proof. 2010 Mathematics Subject Classifications: 11M26, 11M06. Introduction. Prime numbers have fascinated mathematicians since the ancient Greeks, and Euclid provided the first proof of their infinitude. Central to this subject is some innocent-looking infinite series known as the Riemann zeta function. This is is a function of the complex variable s, defined in the half-plane <(s) > 1 by 1 X ζ(s) := n−s n=1 and in the whole complex plane by analytic continuation. Euler noticed that the above series can Q −s −1 be expressed as a product p(1 − p ) over the entire set of primes, which entails that ζ(s) 6= 0 for <(s) > 1. As shown by Riemann [2], ζ(s) extends to C as a meromorphic function with only a simple pole at s = 1, with residue 1, and satisfies the functional equation ξ(s) = ξ(1 − s), 1 −z=2 1 R 1 −x w−1 where ξ(z) = 2 z(z − 1)π Γ( 2 z)ζ(z) and Γ(w) = 0 e x dx. From the functional equation and the relationship between Γ and the sine function, it can be easily noticed that 8n 2 N one has ζ(−2n) = 0, hence the negative even integers are referred to as the trivial zeros of ζ in the literature. -
On the Fourier Transform of the Greatest Common Divisor
On the Fourier transform of the greatest common divisor Peter H. van der Kamp Department of Mathematics and Statistics La Trobe University Victoria 3086, Australia January 17, 2012 Abstract The discrete Fourier transform of the greatest common divisor m X ka id[b a](m) = gcd(k; m)αm ; k=1 with αm a primitive m-th root of unity, is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan's sum, id[b a] = id ∗ c•(a), and on the other hand it can be written as a generalised convolution product, id[b a] = id ∗a φ. We show that id[b a](m) counts the number of elements in the set of ordered pairs (i; j) such that i·j ≡ a mod m. Furthermore we generalise a dozen known identities for the totient function, to identities which involve the discrete Fourier transform of the greatest common divisor, including its partial sums, and its Lambert series. 1 Introduction In [15] discrete Fourier transforms of functions of the greatest common divisor arXiv:1201.3139v1 [math.NT] 16 Jan 2012 were studied, i.e. m X ka bh[a](m) = h(gcd(k; m))αm ; k=1 where αm is a primitive m-th root of unity. The main result in that paper is 1 the identity bh[a] = h ∗ c•(a), where ∗ denoted Dirichlet convolution, i.e. X m bh[a](m) = h( )cd(a); (1) d djm 1Similar results in the context of r-even function were obtained earlier, see [10] for details.