On the distribution of the partial sum of Euler's totient function in residue classes

Youness Lamzouri; M. Tip Phaovibul; Alexandru Zaharescu

Displaying similar documents to “On the distribution of the partial sum of Euler's totient function in residue classes”

Uniformly convex spiral functions and uniformly spirallike functions associated with Pascal distribution series

Gangadharan Murugusundaramoorthy, Basem Aref Frasin, Tariq Al-Hawary (2022)

Mathematica Bohemica

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The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes ${𝒮𝒫}_{p} (α, β)$ and ${𝒰𝒞𝒱}_{p} (α, β)$ of uniformly spirallike functions. Further, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

Products of factorials modulo p

Florian Luca, Pantelimon Stănică (2003)

Colloquium Mathematicae

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We show that if p ≠ 5 is a prime, then the numbers $1 / p (\binom{p}{m ₁, . . ., m_{t}}) | t \geq 1, m_{i} \geq 0 f o r i = 1, . . ., t a n d \sum_{i = 1}^{t} m_{i} = p$ cover all the nonzero residue classes modulo p.

Global approximations for the γ-order Lognormal distribution

Thomas L. Toulias (2013)

Discussiones Mathematicae Probability and Statistics

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A generalized form of the usual Lognormal distribution, denoted with $_{γ}$ , is introduced through the γ-order Normal distribution $_{γ}$ , with its p.d.f. defined into (0,+∞). The study of the c.d.f. of $_{γ}$ is focused on a heuristic method that provides global approximations with two anchor points, at zero and at infinity. Also evaluations are provided while certain bounds are obtained.

The number of binomial coefficients in residue classes modulo p and $p^{2}$ .

William A. Webb (1990)

Colloquium Mathematicae

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On residue formulas for characteristic numbers

Francisco Gómez Ruiz (2009)

Banach Center Publications

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We show that coefficients of residue formulas for characteristic numbers associated to a smooth toral action on a manifold can be taken in a quotient field $Q (X ₁, . . ., X_{r}) .$ This yields canonical identities over the integers and, reducing modulo two, residue formulas for Stiefel Whitney numbers.

Injectors of fitting classes of $\mathfrak{G}_{1}$ -groups

Federico Menegazzo, Martin L. Newell (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Fitting classes and injectors are discussed in the class of $\mathfrak{G}_{1}$ -groups. A necessary and sufficient condition for the existence of injectors is given; it is also shown that, when this condition holds, the injectors form a unique conjugacy class.

On the range of Carmichael's universal-exponent function

Florian Luca, Carl Pomerance (2014)

Acta Arithmetica

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Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds $x / {(l o g x)}^{. 36}$ for all large x, while for φ it is equal to $x / {(l o g x)}^{1 + o (1)}$ , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.

On approximation by Lüroth series

Karma Dajani, Cor Kraaikamp (1996)

Journal de théorie des nombres de Bordeaux

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Let $x \in] 0, 1]$ and $p_{n} / q_{n}, n \geq 1$ be its sequence of Lüroth Series convergents. Define the approximation coefficients $θ_{n} = θ_{n} (x)$ by $q_{n} x - p_{n}, n \geq 1$ . In [BBDK] the limiting distribution of the sequence ${(θ_{n})}_{n \geq 1}$ was obtained for a.e. $x$ using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each $n, θ_{n}$ is already distributed according to the limiting distribution. Using the natural extension we will study the distribution...

On asymmetric distributions of copula related random variables which includes the skew-normal ones

Ayyub Sheikhi, Fereshteh Arad, Radko Mesiar (2022)

Kybernetika

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Assuming that $C_{X, Y}$ is the copula function of $X$ and $Y$ with marginal distribution functions $F_{X} (x)$ and $F_{Y} (y)$ , in this work we study the selection distribution $Z \overset{d}{=} (X | Y \in T)$ . We present some special cases of our proposed distribution, among them, skew-normal distribution as well as normal distribution. Some properties such as moments and moment generating function are investigated. Also, some numerical analysis is presented for illustration.

On the computation of moments of the partial non-central $χ$ -square distribution function

Gil, Amparo, Segura, Javier, Temme, Nico C.

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Properties satisfied by the moments of the partial non-central $χ$ -square distribution function, also known as Nuttall Q-functions, and methods for computing these moments are discussed in this paper. The Nuttall Q-function is involved in the study of a variety of problems in different fields, as for example digital communications.

Multidimensional Gauss reduction theory for conjugacy classes of $SL (n, ℤ)$

Oleg Karpenkov (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we describe the set of conjugacy classes in the group $SL (n, ℤ)$ . We expand geometric Gauss Reduction Theory that solves the problem for $SL (2, ℤ)$ to the multidimensional case, where $ς$ -reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in $GL (n, ℤ)$ in terms of multidimensional Klein-Voronoi continued fractions.

On the sum of the first n values of the Euler function

R. Balasubramanian, Florian Luca, Dimbinaina Ralaivaosaona (2014)

Acta Arithmetica

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Let ϕ(n) be the Euler function of n. We put $E (n) = \sum_{m \leq n} ϕ (m) - (3 / π ²) n ²$ and give an asymptotic formula for the second moment of E(n).

The right tail exponent of the Tracy–Widom $β$ distribution

Laure Dumaz, Bálint Virág (2013)

Annales de l'I.H.P. Probabilités et statistiques

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The Tracy–Widom $β$ distribution is the large dimensional limit of the top eigenvalue of $β$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a \to \infty$ the tail of the Tracy–Widom distribution satisfies $P ({𝑇𝑊}_{β} g t; a) = a^{- (3 / 4) β + o (1)} exp (- \frac{2}{3} β a^{3 / 2}) .$

On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ , = φ: ℝ → ℝ|φ is continuous, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the...

On the limit distribution of the well-distribution measure of random binary sequences

Christoph Aistleitner (2013)

Journal de Théorie des Nombres de Bordeaux

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We prove the existence of a limit distribution of the normalized well-distribution measure $W (E_{N}) / \sqrt{N}$ (as $N \to \infty$ ) for random binary sequences $E_{N}$ , by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.

On the distribution of prime numbers which are of the form $x^{2} + y^{2} + 1$

Yoichi Motohashi (1970)

Acta Arithmetica

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A note on multilinear Muckenhoupt classes for multiple weights

Songqing Chen, Huoxiong Wu, Qingying Xue (2014)

Studia Mathematica

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This paper is devoted to investigating the properties of multilinear $A_{P ⃗}$ conditions and $A_{(P ⃗, q)}$ conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of $A_{P ⃗}$ and $A_{(P ⃗, q)}$ classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear $A_{(P ⃗, q)}$ classes in terms of the linear $A_{p}$ classes are established. These results essentially improve...

Automata, algebraicity and distribution of sequences of powers

Jean-Paul Allouche, Jean-Marc Deshouillers, Teturo Kamae, Tadahiro Koyanagi (2001)

Annales de l’institut Fourier

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Let $K$ be a finite field of characteristic $p$ . Let $K ((x))$ be the field of formal Laurent series $f (x)$ in $x$ with coefficients in $K$ . That is, $f (x) = \sum_{n = n_{0}}^{\infty} f_{n} x^{n}$ with $n_{0} \in 𝐙$ and $f_{n} \in K (n = n_{0}, n_{0} + 1, \dots)$ . We discuss the distribution of ${({f^{m}})}_{m = 0, 1, 2, \dots}$ for $f \in K ((x))$ , where ${f} : = \sum_{n = 0}^{\infty} f_{n} x^{n} \in K [[x]]$ denotes the nonnegative part of $f \in K ((x))$ . This is a little different from the real number case where the fractional part that excludes constant term (digit of order 0) is considered. We give an alternative proof of a result by De Mathan obtaining the generic...