DOCSLIB.ORG

Mean Square of the Hurwitz Zeta-Function and Other Remarks R Balasubramanian, K Ramachandra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mean Square of the Hurwitz Zeta-Function and Other Remarks R Balasubramanian, K Ramachandra Mean square of the Hurwitz zeta-function and other remarks R Balasubramanian, K Ramachandra To cite this version: R Balasubramanian, K Ramachandra. Mean square of the Hurwitz zeta-function and other remarks. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2004, 27, pp.8 - 27. hal-01109904 HAL Id: hal-01109904 https://hal.archives-ouvertes.fr/hal-01109904 Submitted on 27 Jan 2015 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. 8 Hardy-Ramanujan Journal Vol.27 (2004) 8-27 MEAN SQUARE OF THE HURWITZ ZETA-FUNCTION AND OTHER REMARKS BY R.BALASUBRAMANIAN AND K.RAMACHANDRA (Accepted on 26-04-2005) ABSTRACT. The Hurwitz zeta-function associated with the parameter a(0 < a · 1) is a generalisation of the Riemann zeta-function namely the case a = 1: It is de¯ned by X1 ³(s; a) = (n + a)¡s; (s = σ + it; σ > 1) (A:1) n=0 and its analytic continuation. In fact à ! X1 Z n+1 1¡s ¡s du a ³(s; a) = (n + a) ¡ s + (A:2) n=0 n (u + a) s ¡ 1 gives the analytic continuation to (σ > 0). (This remark is due to E.LANDAU (see [EL]). A repetition of this process several times shows that a1¡s ³(s; a) ¡ (A:3) s ¡ 1 can be continued as an entire function to the whole plane. In Re(s) ¸ ¡1; t ¸ 2; ³(s; a) ¡ a¡s = O(t3) and by the functional equation (see x2) it is 0 1 à ! 1 ¡Re(s) jsj 2 O @ A 2¼ in Re(s) ·¡1; t ¸ 2: From these facts we can deduce an `Approximate functional equation' (see x3), which is a generalisation of the approximate functional equation for ³(s). (The two are not very much di®erent). Combining this with an important theorem due to van-der- CORPUT, (see [ECT], Theorem 5.9), we prove 1 Z T +T 3 ¡ 1 1 ¡ 1 ¡it 2 3 T 3 j³( + it; a) ¡ a 2 j dt ¿ (log T ) (A:4) T 2 Mean Square of the ... 9 uniformly in a(0 < a · 1): From this we deduce similar results for quasi- L functions (see [RB, KR]1) and more general functions: Let a1; a2;::: be any periodic sequence of complex numbers for which the sum over a period is zero. Let b1; b2;::: be any sequence of complex Pn " numbers for which j=2 jbj ¡ bj¡1j + jbnj · n for every " > 0 and every n ¸ n0("). Then we prove 1 Z T +T 3 X1 ¡ 1 anbn 2 " T 3 j j dt · T (A:5) 1 +it T n=1 (n + a) 2 for every " > 0 and every T ¸ T0(") Here as usual 0 < a · 1 and T0(") is independent of a. x1. INTRODUCTORY REMARKS. These concern problems on ³(s; a) which need serious attention. We begin with M.N.HUXLEY'S recent important achievement [MNH] 1 32 ³( + it) = O (t®+"); ® = ; t ¸ 2: (1.1) 2 " 205 89 The earlier important record was also due to him and it was ® = 570 : We wish him every 1 64 success in his venture regarding improvement of (A.5) with 3 replaced by 205 . The result Z T 1 1 2 T T T 1 j³( + it)j dt = log + (2γ ¡ 1) + O(T 3 ) (1.2) 2¼ 0 2 2¼ 2¼ 2¼ P1 1 R n+1 du is due to R.BALASUBRAMANIAN [RB]. (Here as usual γ = n=1( n ¡ n u ) is the EULER constant). It remains to consider Z T X1 a b j n n j2dt 1 +it 0 n=1 (n + a) 2 1 and obtain an analogue of (1.2). Some work in the direction of extending (1.2) to ³( 2 + ¡ 1 ¡it it; a) ¡ a 2 is in progress and it is due to a student of M.JUTILA. This extension gives (A.4) as a corollary and (A.4) implies 1 ¡ 1 ¡it 1 3 ³( + it; a) ¡ a 2 = O(t 6 (log t) ): (1.3) 2 The result 2 Z T +T 3 ¡ 2 1 ¡ 2 ¡it 4 " T 3 j³( + it; a) ¡ a 2 j dt ¿" T (1.4) T 2 uniformly in a seems to be hopeless at present although N.ZAVOROTNYI [NZ] has proved Z T 1 4 2 +" j³( + it)j dt = TP (log T ) + O(T 3 ): (1.5) 0 2 10 RB, KR where P (x) is a certain polynomial of degree 4 in x. (For further work by Y.MOTOHASHI 2 and A.IVIC see [AI]). Even (1.4) with 1 in place of 3 also seems to be hopeless. Mean value results such as (A.4) imply (by convexity principles) X ¡σ¡it 2 1 +" à max ! j³(σ + it; a) ¡ a j ¿ T 3 I t"I 1 σ ¸ 2 1 where I runs over all (abutting) unit intervals into which [T;T +T 3 ] can be divided. (Similar remarks apply to all the mean value upper bounds over short intervals considered in this paper). It gives, as a corollary, the theorem 1 ¡ 1 ¡it 1 +" j³( + it; a) ¡ a 2 j < t 6 2 for all t ¸ t0("): (This can certainly be obtained by the WEYL-HARDY-LITTLEWOOD method (see [ECT] Theorem 5.3)). The most important result in the theory of the Riemann zeta-function is, undoubtedly, the result (1¡σ)3=2 2 1 j³(σ + it)j · C t C (log t) 3 ; ( · σ · 1; t ¸ 3) (1.6) 1 2 2 where C1 > 0 and C2 > 0 are absolute constants. This result is due to the Soviet Mathe- matician I.M.VINOGRADOV. (See the chapter on VINOGRADOFF's (both the names are the same) method in [ECT]). He proved (1.6) with (log t)2=3 replaced by a higher power of log t: De¯nitely he knew (1.6) and published 2 ³(1 + it) = O((log t) 3 ) (1.7) also. Both (1.6) and (1.7) are nearly 50 years old. It is a great challenge to improve even (1.7). It is well-known that quasi Riemann hypothesis (³(s) 6= 0 in σ ¸ 1 ¡ ± for some constant ± > 0) implies easily ³(1 + it) = O(log log t); t ¸ 100: (1.8) It is not hard to prove (1.6) for ³(σ + it; a) ¡ a¡σ¡it, (VINOGRADOFF's work gives (1.6) for this function as well). For the latest economical constants C1 = 76:2 and C2 = 4:45 in (1.6) see KEVIN FORD [KF]. See also [KR, AS]1. x2 FUNCTIONAL EQUATION. (We borrow the functional equation for ³(s; a) from x2.17 of [ECT]). We have X1 ³(s; a) = (n + a)¡s; (σ > 1); n=0 Mean Square of the ... 11 X1 Z n+1 1¡s ¡s du a = ((n + a) ¡ s ) + ; (σ > 0); (2.1) n=0 n (u + a) s ¡ 1 a1¡s and repetition of this process several times shows that ³(s; a) ¡ s¡1 is entire. Also in σ ¸ ¡1; t ¸ 2 we have ³(s; a) ¡ a¡s = O(t3). Again (see[ECT]) we have the functional equation ( 2¡(1 ¡ s) s¼ X1 cos(2¼ma) ³(s; a) = 1¡s sin( ) 1¡s (2¼) 2 m=1 m ) s¼ X1 sin(2¼ma) + cos( ) 1¡s ; (σ < 0): (2.2) 2 m=1 m s¼ Note that jtan ( 2 )j lies between two positive constants for all σ provided t ¸ 2; and so the conversion factors 2¡(1 ¡ s) s¼ 2¡(1 ¡ s) s¼ à (s) = sin ( ) and à (s) = cos( ) (2.3) 1 (2¼)1¡s 2 2 (2¼)1¡s 2 are nearly the same (as that for the Riemann zeta-function) so far as the magnitudes are concerned. It is useful to record here that for t ¸ 2; we have à ! 1 ¡σ jsj + 1 2 jà (s)j + jà (s)j ¿ (2.4) 1 2 2¼ where the implied constant is absolute. x3. APPROXIMATE FUNCTIONAL EQUATION 1 4 10 Let 0 < " · 100 ;T ¸ T0(");L = log T;¿ = L ; h = " L; 1 · X · T; 1 T 1 · Y · T; 1 · H · T 2 and XY ¸ (1 + "): (3.1) 2¼ 1 P1 ¡s Let w = u + iv; s = 2 + it; T · t · T + H; ³(s; a) = n=0(n + a) ; (0 < a · 1); σ > 1: 1 (We put s = 2 + it for trivial simpli¯cation. The results are true with trivial modi¯cations with s = σ + it for any σ uniformly in any ¯xed interval a0 · σ · b0): Then we have à ! X1 µn + a¶h (n + a)¡sExp ¡ n=1 x 1 Z w dw = (³(s + w; a) ¡ a¡s¡w)¡( + 1)Xw + O(T ¡1) (3.2) 2¼i u=2;jvj·¿ h w 5L w ¡jIm(w)j Here we move the line of integration to u = ¡ " . (We use j¡( h + 1)j ¿ exp(¡ h ) in h ¡s jRe(w)j · 2 ): The pole at w = 0 contributes ³(s; a) ¡ a and the horizontal portions of 12 RB, KR the contour contribute O(T ¡1): We then use the functional equation and the fact that for jIm(z)j ¸ 2, we have à ! 1 ¡Re(z) jzj + 1 2 jà (z)j + jà (z)j ¿ (3.3) 1 2 2¼ a result mentioned already.
Load more

Recommended publications
  • Almost Periodic Sequences and Functions with Given Values
    ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 1–16 ALMOST PERIODIC SEQUENCES AND FUNCTIONS WITH GIVEN VALUES Michal Veselý Abstract. We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set X in a metric space, it is proved the existence of an almost periodic sequence {ψk}k∈Z such that {ψk; k ∈ Z} = X and ψk = ψk+lq(k), l ∈ Z for all k and some q(k) ∈ N which depends on k. 1. Introduction The aim of this paper is to construct almost periodic sequences and functions which attain values in a metric space. More precisely, our aim is to find almost periodic sequences and functions whose ranges contain or consist of arbitrarily given subsets of the metric space satisfying certain conditions. We are motivated by the paper [3] where a similar problem is investigated for real-valued sequences. In that paper, using an explicit construction, it is shown that, for any bounded countable set of real numbers, there exists an almost periodic sequence whose range is this set and which attains each value in this set periodically. We will extend this result to sequences attaining values in any metric space. Concerning almost periodic sequences with indices k ∈ N (or asymptotically almost periodic sequences), we refer to [4] where it is proved that, for any precompact sequence {xk}k∈N in a metric space X , there exists a permutation P of the set of positive integers such that the sequence {xP (k)}k∈N is almost periodic.
    [Show full text]
  • Discrete Signals and Their Frequency Analysis
    Discrete signals and their frequency analysis. Valentina Hubeika, Jan Cernock´yˇ DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz • recapitulation – fundamentals on discrete signals. • periodic and harmonic sequences • discrete signal processing • convolution • Fourier transform with discrete time • Discrete Fourier Transform 1 Sampled signal ⇒ discrete signal During sampling we consider only values of the signal at sampling period multiplies T : x(nT ), nT = ... − 2T, −T, 0,T, 2T, 3T,... For a discrete signal, we forget about real time and simply count the samples. Discrete time becomes : x[n], n = ... − 2, −1, 0, 1, 2, 3,... Thus we often call discrete signals just sequences. 2 Important discrete signals Unit step and unit impulse: 1 for n ≥ 0 1 for n =0 σ[n]= δ[n]= 0 elsewhere 0 elsewhere 3 Periodic discrete signals their behaviour repeats after N samples, the smallest possible N is denoted as N1 and is called fundamental period. Harmonic discrete signals (harmonic sequences) x[n]= C1 cos(ω1n + φ1) (1) • C1 is a positive constant – magnitude. • ω1 is a spositive constant – normalized angular frequency. As n is just a number, the unit of ω1 is [rad]. Note, that in the previous lecture we denoted with the same simbol an angular frequency of continuous signals. Although in the last lecture we used ′ symbol ω1 for discrete time, we will not do it any longer. You will recognize continuous time frequency if there is real time associated with it (for instance cos(ω1t)). If you see discrete time n (for instance cos(ω1n)) you should know we are talking about normalized angular frequency.
    [Show full text]
  • Arxiv:1904.09040V2 [Math.NT]
    PERIODICITIES FOR TAYLOR COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS PAVEL GUERZHOY, MICHAEL H. MERTENS, AND LARRY ROLEN Abstract. Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around τ0 = i of the classical Jacobi theta function θ3. Here, we generalize the phenomenon observed by Romik to a broader class of modular forms of half-integral weight and, in particular, prove the conjecture. 1. Introduction Fourier coefficients of modular forms are well-known to encode many interesting quantities, such as the number of points on elliptic curves over finite fields, partition numbers, divisor sums, and many more. Thanks to these connections, the arithmetic of modular form Fourier coefficients has long enjoyed a broad study, and remains a very active field today. However, Fourier expansions are just one sort of canonical expansion of modular forms. Petersson also defined [17] the so-called hyperbolic and elliptic expansions, which instead of being associated to a cusp of the modular curve, are associated to a pair of real quadratic numbers or a point in the upper half-plane, respectively. A beautiful exposition on these different expansions and some of their more recent connections can be found in [8]. In particular, there Imamoglu and O’Sullivan point out that Poincar´eseries with respect to hyperbolic expansions include the important examples of Katok [11] and Zagier [26], which are the functions which Kohnen later used [15] to construct the holomorphic kernel for the Shimura/Shintani lift.
    [Show full text]
  • Construction of Almost Periodic Sequences with Given Properties
    Electronic Journal of Differential Equations, Vol. 2008(2008), No. 126, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) CONSTRUCTION OF ALMOST PERIODIC SEQUENCES WITH GIVEN PROPERTIES MICHAL VESELY´ Abstract. We define almost periodic sequences with values in a pseudometric space X and we modify the Bochner definition of almost periodicity so that it remains equivalent with the Bohr definition. Then, we present one (easily modifiable) method for constructing almost periodic sequences in X . Using this method, we find almost periodic homogeneous linear difference systems that do not have any non-trivial almost periodic solution. We treat this problem in a general setting where we suppose that entries of matrices in linear systems belong to a ring with a unit. 1. Introduction First of all we mention the article [9] by Fan which considers almost periodic sequences of elements of a metric space and the article [22] by Tornehave about almost periodic functions of the real variable with values in a metric space. In these papers, it is shown that many theorems that are valid for complex valued sequences and functions are no longer true. For continuous functions, it was observed that the important property is the local connection by arcs of the space of values and also its completeness. However, we will not use their results or other theorems and we will define the notion of the almost periodicity of sequences in pseudometric spaces without any conditions, i.e., the definition is similar to the classical definition of Bohr, the modulus being replaced by the distance.
    [Show full text]
  • Florentin Smarandache
    FLORENTIN SMARANDACHE SEQUENCES OF NUMBERS INVOLVED IN UNSOLVED PROBLEMS 1 141 1 1 1 8 1 1 8 40 8 1 1 8 1 1 1 1 12 1 1 12 108 12 1 1 12 108 540 108 12 1 1 12 108 12 1 1 12 1 1 1 1 16 1 1 16 208 16 1 1 16 208 1872 208 16 1 1 16 208 1872 9360 1872 208 16 1 1 16 208 1872 208 16 1 1 16 208 16 1 1 16 1 1 2006 Introduction Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. ( on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes/squares/cubes/factorials, almost primes, mobile periodicals, functions, tables, prime/square/factorial bases, generalized factorials, generalized palindromes, etc. ) have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, University of Craiova Library, and Arhivele Statului (Filiala Vâlcea, Romania). It is based on the old article “Properties of Numbers” (1975), updated many times. Special thanks to C. Dumitrescu & V. Seleacu from the University of Craiova (see their edited book "Some Notions and Questions in Number Theory", Erhus Univ. Press, Glendale, 1994), M. Perez, J. Castillo, M. Bencze, L. Tutescu, E, Burton who helped in collecting and editing this material. The Author 1 Sequences of Numbers Involved in Unsolved Problems Here it is a long list of sequences, functions, unsolved problems, conjectures, theorems, relationships, operations, etc.
    [Show full text]
  • Complexity of Periodic Sequences
    Complexity of Periodic Sequences Wieb Bosma1 Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, the Netherlands, email: [email protected] Abstract. Periodic sequences form the easiest sub-class of k-automatic sequences. Two characterizations of k-automatic sequences lead to two different complexity measures: the sizes of the minimal automaton with output generating the sequence on input either the k-representations of numbers or their reverses. In this note we analyze this exactly. 1 Introduction 1 By definition, an infinite k-automatic sequence a = (an)n=0 = a0a1a2a3 ··· is the output of a deterministic finite automaton with output (DFAO) upon feeding the index n as input for an. In [2] two obvious complexity measure for such sequences are compared. The first, denoted kakk, is simply the size (that is, the R number of states) of the smallest DFAO that produces a; the second, kakk (the reversed size) is the size of the smallest DFAO that produces a when the input for n is the reverse of the k-ary representation of n. In general the two measures may differ, even exponentially, in size. In this note we attempt to analyze the exact values of both complexity measures for periodic sequences a. It turns out R 2 that in this case kakk = O(n) and kakk is O(n ); we will be more precise in the statement of the main theorems. R The main tool for the analysis is the basic result that kakk is essentially equal to the size of the k-kernel Kk(a); this kernel may be defined to be the smallest set of infinite sequences containing a as well as every pj(b) for any b 2 Kk(a) 1 and any j with 0 ≤ j < k, where pj(b) = (bj+n)n=0 = bjbj+nbj+2nbj+3n ··· .
    [Show full text]
  • Finite Automata and Arithmetic
    Finite automata and arithmetic J.-P. Allouche ∗ 1 Introduction The notion of sequence generated by a finite automaton, (or more precisely a finite automaton with output function, i. e. a \uniform tag system") has been introduced and studied by Cobham in 1972 (see [19]; see also [24]). In 1980, Christol, Kamae, Mend`esFrance and Rauzy, ([18]), proved that a sequence with values in a finite field is automatic if and only if the related formal power series is algebraic over the rational functions with coefficients in this field: this was the starting point of numerous results linking automata theory, combinatorics and number theory. Our aim is to survey some results in this area, especially transcendence results, and to provide the reader with examples of automatic sequences. We will also give a bibliography where more detailed studies can be found. See in particular the survey of Dekking, Mend`esFrance and van der Poorten, [22], or the author's, [2], where many relations between finite automata and number theory (and between finite automata and other mathematical fields) are described. For applications of finite automata to physics see [6]. In the first part of this paper we will recall the basic definitions and give the theorem of Christol, Kamae, Mend`esFrance and Rauzy. We will also give five typical examples of sequences generated by finite automata. In the second part we will discuss transcendence results related to au- tomata theory, giving in particular some results concerning the Carlitz zeta function. We will indicate in the third part of this paper the possible generaliza- tions of these automatic sequences.
    [Show full text]
  • On the Periodic Hurwitz Zeta-Function. a Javtokas, a Laurinčikas
    On the periodic Hurwitz zeta-function. A Javtokas, A Laurinčikas To cite this version: A Javtokas, A Laurinčikas. On the periodic Hurwitz zeta-function.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2006, 29, pp.18 - 36. hal-01111465 HAL Id: hal-01111465 https://hal.archives-ouvertes.fr/hal-01111465 Submitted on 30 Jan 2015 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. Hardy-Ramanujan Journal Vol.29 (2006) 18-36 On the periodic Hurwitz zeta-function A. Javtokas, A. Laurinˇcikas Abstract.In the paper an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved. 1. Introduction Let N, N0, R and C denote the sets of all positive integers, non-negative integers, real and complex numbers, respectively, and let a = a ; m Z f m 2 g be a periodic sequence of complex numbers with period k 1. Denote by s = σ + it a complex variable. The periodic zeta-function ζ(s≥; a), for σ > 1, is defined by 1 a ζ(s; a) = m ; ms m=1 X and by analytic continuation elsewhere. Define 1 k−1 a = a : k m m=0 X A.
    [Show full text]
  • 2. Infinite Series
    2. INFINITE SERIES 2.1. A PRE-REQUISITE:SEQUENCES We concluded the last section by asking what we would get if we considered the “Taylor polynomial of degree for the function ex centered at 0”, x2 x3 1 x 2! 3! As we said at the time, we have a lot of groundwork to consider first, such as the funda- mental question of what it even means to add an infinite list of numbers together. As we will see in the next section, this is a delicate question. In order to put our explorations on solid ground, we begin by studying sequences. A sequence is just an ordered list of objects. Our sequences are (almost) always lists of real numbers, so another definition for us would be that a sequence is a real-valued function whose domain is the positive integers. The sequence whose nth term is an is denoted an , or if there might be confusion otherwise, an n 1, which indicates that the sequence starts when n 1 and continues forever. Sequences are specified in several different ways. Perhaps the simplest way is to spec- ify the first few terms, for example an 2, 4, 6, 8, 10, 12, 14,... , is a perfectly clear definition of the sequence of positive even integers. This method is slightly less clear when an 2, 3, 5, 7, 11, 13, 17,... , although with a bit of imagination, one can deduce that an is the nth prime number (for technical reasons, 1 is not considered to be a prime number). Of course, this method com- pletely breaks down when the sequence has no discernible pattern, such as an 0, 4, 3, 2, 11, 29, 54, 59, 35, 41, 46,..
    [Show full text]
  • ON PARTITIONS and PERIODIC SEQUENCES Introduction
    Ø Ñ ÅØÑØÐ ÈÙ ÐØÓÒ× DOI: 10.2478/tmmp-2019-0002 Tatra Mt. Math. Publ. 73 (2019), 9–18 ON PARTITIONS AND PERIODIC SEQUENCES Milan Paˇsteka´ Department of Mathematics, Faculty of Education, University of Trnava, Trnava, SLOVAKIA ABSTRACT. In the first part we associate a periodic sequence with a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved. Introduction This paper is inspired by the papers [CIV], [CV], [K] where the uniformly distributed sequences of partitions of the unit interval are studied. For technical reasons we will study the systems of finite sequences of the unit interval — an equivalent of partitions. Let VN = vN (1),...,vN (BN ) , N =1, 2, 3,..., limN→∞ BN = ∞ be a sys- tem of finite sequences of elements of interval [0, 1]. We say that this system is uniformly distributed if and only if B 1 1 N lim f vN (n) = f(x)dx. N→∞ BN n=1 0 By the standard way we can prove 1º ÌÓÖÑ The system VN ,N =1, 2, 3,... is uniformly distributed if and only if 1 lim n ≤ BN ; vN (n) <x = x N→∞ BN for every x ∈ [0, 1]. c 2019 Mathematical Institute, Slovak Academy of Sciences. 2010 M a t h e m a t i c s Subject Classification: 11A25. K e y w o r d s: statistical independence, uniform distribution, polyadic continuity. Supported by Vega Grant 2/0109/18. Licensed under the Creative Commons Attribution-NC-ND4.0 International Public License.
    [Show full text]
  • Properties of Periodic Fibonacci-Like Sequences
    Properties of Periodic Fibonacci-like Sequences Alexander V. Evako Email address: [email protected] Abstract Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence . It is shown that this sequence is periodic with the period ( ) if | | . Keywords: Generalized Fibonacci-like sequence, Periodicity, Discrete Dynamical System, PDE, Graph, Network, Digital Space 1. Preliminaries Generalized Fibonacci sequences are widely used in many areas of science including applied physics, chemistry and biology. Over the past few decades, there has been a growth of interest in studying properties periodic Fibonacci sequences modulo m, Pell sequence and Lucas sequence. Many interesting results have been received in this area by researchers (see, e.g., [1, 4, 8, 10, 14]). Fibonacci like sequences are basic elements in computational solutions of partial differential equations using finite difference approximations of the PDE [ 6, 11], and in partial differential equations on networks and graphs [2, 7, 9]. The goals for this study is to investigate properties of the Fibonacci-like sequences, which are structural elements in discrete dynamic processes that can be encoded into graphs. The property we are interested in this study is the periodicity of a the Fibonacci-like sequence. In Section 2, we study a periodic continuous function ( ), where ( ) ( ) ( ). We show that if ( ) is periodic with period then | | . We apply the obtained results to the sequence, which is defined by recurrence relation with , where A, B C, a and b are real numbers.
    [Show full text]
  • On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes
    On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Benigno Rafael Parra Avila November 2009 © Benigno Rafael Parra Avila. All Rights Reserved. 2 This dissertation titled On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes by BENIGNO RAFAEL PARRA AVILA has been approved for the Department of Mathematics and the College of Arts and Sciences by Sergio R. Lopez-Permouth´ Professor of Mathematics Benjamin M. Ogles Dean, College of Arts and Sciences 3 Abstract PARRA AVILA, BENIGNO RAFAEL, Ph.D., November 2009, Mathematics On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes (74 pp.) Director of Dissertation: Sergio R. Lopez-Permouth´ Let R be a commutative ring with identity. A power series f 2 R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Zm for some positive integer m. Let F be a field; we prove the equivalence between two notions of rationality in F[[x1;:::; xn]], and hence in F((x1;:::; xn)), and thus extend Kronecker’s criterion for rationality from F[[x]] to the multivariable setting. We introduce the notion of sequential code, a natural generalization of cyclic and even constacyclic codes, over a (not necessarily finite) field and explore fundamental properties of sequential codes as well as connections with periodicity of sequences and with the related notion of linear recurrence sequences.
    [Show full text]