scholarly journals On the Product of L(1, χ)

1953 ◽  
Vol 5 ◽  
pp. 105-111 ◽  
Author(s):  
Tikao Tatuzawa
Keyword(s):  
Positive Integer ◽  
Positive Constant ◽  
Euler Function ◽  
Principal Character ◽  
Absolute Positive Constant ◽  
Domain Of Existence ◽  
Full Domain

Let k(≧3) be a positive integer and φ(k) be the Euler function. We denote by % one of the φ(k) characters formed with modulus k, and by % the principal character. Let Lis, %) be the L-series corresponding to % Throughout the paper we use c and c(ε) to denote respectively an absolute positive constant and a positive constant depending on parameter ε( >0) alone, not necessarily the same at their various occurrences. We use the symbol y = O(Z) for positive X when there exists a c satisfying Y ≦ in the full domain of existence of X and Y.

On Functional Properties of Incomplete Gaussian Sums

1991 ◽  
Vol 43 (1) ◽  
pp. 182-212 ◽  
Author(s):  
K. I. Oskolkov
Keyword(s):  
Functional Properties ◽  
Positive Constant ◽  
Absolute Constant ◽  
Special Function ◽  
Absolute Positive Constant ◽  
Image Position ◽  
Gaussian Sums

AbstractThe following special function of two real variables x2 and x1 is considered: and its connections with the incomplete Gaussian sums where ω are intervals of length |ω| ≤1. In particular, it is proved that for each fixed x2 and uniformly in X2 the function H(x2, x1) is of weakly bounded 2-variation in the variable x1 over the period [0, 1]. In terms of the sums W this means that for collections Ω = {ωk}, consisting of nonoverlapping intervals ωk ∪ [0,1) the following estimate is valid: where card denotes the number of elements, and c is an absolute positive constant. The exact value of the best absolute constant к in the estimate (which is due to G. H. Hardy and J. E. Littlewood) is discussed.

scholarly journals The mean values of Dirichlet L-functions at integer points and class numbers of cyclotomic fields

1994 ◽  
Vol 134 ◽  
pp. 151-172 ◽  
Author(s):  
Masanori Katsurada ◽  
Kohji Matsumoto
Keyword(s):  
Positive Integer ◽  
Dirichlet Character ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Mean Values ◽  
Integer Points ◽  
Principal Character ◽  
Dirichlet Characters ◽  
The Mean

Let q be a positive integer, and L(s, χ) the Dirichlet L-function corresponding to a Dirichlet character χ mod q. We putwhere χ runs over all Dirichlet characters mod q except for the principal character χ0.

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

scholarly journals The Solutions of Generalized Euler Function Equation φ_2(n-φ_2 (n))=2ω(n)

2021 ◽  
pp. 15-22
Author(s):  
Xu Yifan ◽  
Shen Zhongyan
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Euler Function ◽  
Prime Factors ◽  
Integer Solutions

By using the properties of Euler function, an upper bound of solutions of Euler function equation  is given, where  is a positive integer. By using the classification discussion and the upper bound we obtained, all positive integer solutions of the generalized Euler function equation  are given, where is the number of distinct prime factors of n.

Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers

2017 ◽  
Vol 27 (3) ◽  
pp. 358-386 ◽  
Author(s):  
CLAYTON COLLIER-CARTAINO ◽  
NATHAN GRABER ◽  
TAO JIANG
Keyword(s):  
Positive Integer ◽  
Positive Constant ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Turán Number ◽  
Formula Group ◽  
Image Position ◽  
Linear Hypergraphs

Anr-uniform hypergraph is called anr-graph. A hypergraph islinearif every two edges intersect in at most one vertex. Given a linearr-graphHand a positive integern, thelinear Turán numberexL(n,H) is the maximum number of edges in a linearr-graphGthat does not containHas a subgraph. For each ℓ ≥ 3, letCrℓdenote ther-uniform linear cycle of length ℓ, which is anr-graph with edgese1, . . .,eℓsuch that, for alli∈ [ℓ−1], |ei∩ei+1|=1, |eℓ∩e1|=1 andei∩ej= ∅ for all other pairs {i,j},i≠j. For allr≥ 3 and ℓ ≥ 3, we show that there exists a positive constantc=cr,ℓ, depending onlyrand ℓ, such that exL(n,Crℓ) ≤cn1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constantsa=am,randb=bm,r, depending only onmandr, such that\begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

2003 ◽  
Vol 12 (5-6) ◽  
pp. 477-494 ◽  
Author(s):  
Noga Alon ◽  
Michael Krivelevich ◽  
Benny Sudakov
Keyword(s):  
Positive Constant ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Ramsey Number ◽  
Absolute Positive Constant ◽  
Turán Number ◽  
Minimum Number ◽  
Ramsey Type ◽  
General Graphs ◽  
Refined Version

For a graph H and an integer n, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is r-degenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, . This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite r-degenerate graph H, This is motivated by a conjecture of Erdős that asserts that, for every such H, For two graphs G and H, the Ramsey number is the minimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either a red copy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, . Here we prove this conjecture for bipartite graphs G, and prove that for general graphs G with m edges, for some absolute positive constant c.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rödl, Kostochka, Gowers and Sudakov.

scholarly journals On the coprimality of some arithmetic functions

2010 ◽  
Vol 87 (101) ◽  
pp. 121-128
Author(s):  
Koninck De ◽  
Imre Kátai
Keyword(s):  
Positive Integer ◽  
Asymptotic Estimate ◽  
Counting Function ◽  
Arithmetic Functions ◽  
Euler Function ◽  
Number Of Divisors

Let ? stand for the Euler function. Given a positive integer n, let ?(n) stand for the sum of the positive divisors of n and let ?(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd (?(n), ?(n)) = gcd(?(n), ?(n)) = 1}. Moreover, setting l(n) : = gcd(?(n), ?(n+1)), we provide an asymptotic estimate for the size of #{n ? x: l(n) = 1}.

An Equation Involving the Smarandache Function

2012 ◽  
Vol 204-208 ◽  
pp. 4785-4788
Author(s):  
Bin Chen
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Elementary Number Theory ◽  
Euler Function ◽  
Elementary Number ◽  
Integer Solutions

For any Positive Integer N, LetΦ(n)andS(n)Denote the Euler Function and the Smarandache Function of the Integer N.In this Paper, we Use the Elementary Number Theory Methods to Get the Solutions of the Equation Φ(n)=S(nk) if the K=9, and Give its All Positive Integer Solutions.

scholarly journals Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level

2017 ◽  
Vol 2019 (18) ◽  
pp. 5703-5743 ◽  
Author(s):  
Peter Humphries
Keyword(s):  
Positive Integer ◽  
Prime Divisors ◽  
Spectral Multiplicity ◽  
Laplacian Eigenvalues ◽  
Positive Proportion ◽  
Principal Character ◽  
Squarefree Integer ◽  
Spectrum Of The Laplacian

Abstract We show that if a positive integer $q$ has $s(q)$ odd prime divisors $p$ for which $p^2$ divides $q$, then a positive proportion of the Laplacian eigenvalues of Maaß newforms of weight $0$, level $q$, and principal character occur with multiplicity at least $2^{s(q)}$. Consequently, the new part of the cuspidal spectrum of the Laplacian on $\Gamma_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Strömberg who proved this for $q = 9$ by different methods.

Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

2014 ◽  
Vol 58 (1) ◽  
pp. 125-147 ◽  
Author(s):  
N. A. Chernyavskaya ◽  
L. A. Shuster
Keyword(s):  
Liouville Equation ◽  
Positive Constant ◽  
Absolute Constant ◽  
Absolutely Continuous ◽  
Weighted Estimates ◽  
Negative Function ◽  
Almost Everywhere ◽  
Absolute Positive Constant ◽  
Sturm Liouville ◽  
Image Position

AbstractConsider the equationwhereƒ∈Lp(ℝ),p∈ (1, ∞) andBy a solution of (*), we mean any functionyabsolutely continuous together with (ry′) and satisfying (*) almost everywhere on ℝ. In addition, we assume that (*) is correctly solvable in the spaceLp(ℝ), i.e.(1) for any function, there exists a unique solutiony∈Lp(ℝ) of (*);(2) there exists an absolute constantc1(p) > 0 such that the solutiony∈Lp(ℝ) of (*) satisfies the inequalityWe study the following problem on the strengthening estimate (**). Let a non-negative functionbe given. We have to find minimal additional restrictions forθunder which the following inequality holds:Here,yis a solution of (*) from the classLp(ℝ), andc2(p) is an absolute positive constant.

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