Displaying similar documents to “On the k -polygonal numbers and the mean value of Dedekind sums”

Some new sums related to D. H. Lehmer problem

Han Zhang, Wenpeng Zhang (2015)

Czechoslovak Mathematical Journal

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About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N ( k ; p ) denote the number of all 1 a i p - 1 such that a 1 a 2 a k 1 mod p and 2 a i + a ¯ i + 1 , i = 1 , 2 , , k . The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N ( k ; p ) , and give an interesting asymptotic formula...

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form W F , d ( a ) = x F ψ ( x d - a x ) , where F is a finite field, ψ: F → ℂ is the canonical additive character, g c d ( d , | F × | ) = 1 , and a F × . If we fix F and d, and examine the values of W F , d ( a ) as a runs through F × , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo | F × | ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...

Rademacher functions in Cesàro type spaces

Sergei V. Astashkin, Lech Maligranda (2010)

Studia Mathematica

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The Rademacher sums are investigated in the Cesàro spaces C e s p (1 ≤ p ≤ ∞) and in the weighted Korenblyum-Kreĭn-Levin spaces K p , w on [0,1]. They span l₂ space in C e s p for any 1 ≤ p < ∞ and in K p , w if and only if the weight w is larger than t l o g p / 2 ( 2 / t ) on (0,1). Moreover, the span of the Rademachers is not complemented in C e s p for any 1 ≤ p < ∞ or in K 1 , w for any quasi-concave weight w. In the case when p > 1 and when w is such that the span of the Rademacher functions is isomorphic to l₂, this span is...

Waring's number for large subgroups of ℤ*ₚ*

Todd Cochrane, Derrick Hart, Christopher Pinner, Craig Spencer (2014)

Acta Arithmetica

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Let p be a prime, ℤₚ be the finite field in p elements, k be a positive integer, and A be the multiplicative subgroup of nonzero kth powers in ℤₚ. The goal of this paper is to determine, for a given positive integer s, a value tₛ such that if |A| ≫ tₛ then every element of ℤₚ is a sum of s kth powers. We obtain t = p 22 / 39 + ϵ , t = p 15 / 29 + ϵ and for s ≥ 6, t = p ( 9 s + 45 ) / ( 29 s + 33 ) + ϵ . For s ≥ 24 further improvements are made, such as t 32 = p 5 / 16 + ϵ and t 128 = p 1 / 4 .

A-Browder-type theorems for direct sums of operators

Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)

Mathematica Bohemica

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We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties ( SBaw ) , ( SBab ) , ( SBw ) and ( SBb ) are not preserved under direct sums of operators. However, we prove that if S and T are bounded linear operators acting on Banach spaces and having the property ( SBab ) , then S T has the property ( SBab ) if and only if σ SBF + - ( S T ) = σ SBF + - ( S ) σ SBF + - ( T ) , where σ SBF + - ( T ) is the upper semi-B-Weyl spectrum of T . We obtain analogous preservation results for the properties ( SBaw ) ,...

A pure smoothness condition for Radó’s theorem for α -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

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Let Ω n be a bounded, simply connected -convex domain. Let α + n and let f be a function on Ω which is separately C 2 α j - 1 -smooth with respect to z j (by which we mean jointly C 2 α j - 1 -smooth with respect to Re z j , Im z j ). If f is α -analytic on Ω f - 1 ( 0 ) , then f is α -analytic on Ω . The result is well-known for the case α i = 1 , 1 i n , even when f a priori is only known to be continuous.

Persistence of iterated partial sums

Amir Dembo, Jian Ding, Fuchang Gao (2013)

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

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Let S n ( 2 ) denote the iterated partial sums. That is, S n ( 2 ) = S 1 + S 2 + + S n , where S i = X 1 + X 2 + + X i . Assuming X 1 , X 2 , ... , X n are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities p n ( 2 ) : = max 1 i n S i ( 2 ) l t ; 0 c 𝔼 | S n + 1 | ( n + 1 ) 𝔼 | X 1 | , with c 6 30 (and c = 2 whenever X 1 is symmetric). The converse inequality holds whenever the non-zero min ( - X 1 , 0 ) is bounded or when it has only finite third moment and in addition X 1 is squared integrable. Furthermore, p n ( 2 ) n - 1 / 4 for any non-degenerate squared integrable, i.i.d., zero-mean X i . In contrast, we show that for any 0 l t ; γ l t ; 1 / 4 there exist integrable,...

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions

Albert Baernstein II, Robert C. Culverhouse (2002)

Studia Mathematica

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Let X = i = 1 k a i U i , Y = i = 1 k b i U i , where the U i are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and a i , b i are real constants. We prove that if b ² i is majorized by a ² i in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp L ² - L p Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura (2013)

Acta Arithmetica

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Let p = 1 + 2 e + 1 q be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order 2 e + 1 (resp. order d φ dividing q), and let ψₙ be an even character of conductor p n + 1 and order pⁿ. We put χ = δφψₙ, whose value is contained in K = ( ζ ( p - 1 ) p ) . It is well known that the Bernoulli number B 1 , χ is not zero, which is shown in an analytic way. In the extreme cases d φ = 1 and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: T r n / 1 ( ξ B 1 , χ ) 0 for any...

The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces

Tamás Erdélyi (2003)

Studia Mathematica

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Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ( λ j ) j = 1 is a sequence of distinct positive numbers. Then s p a n 1 , x λ , x λ , . . . is dense in C[0,1] if and only if j = 1 ( λ j ) / ( λ j ² + 1 ) = . Moreover, if j = 1 ( λ j ) / ( λ j ² + 1 ) < , then every function from the C[0,1] closure of s p a n 1 , x λ , x λ , . . . can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

On the H-property and rotundity of Cesàro direct sums of Banach spaces

Saard Youyen, Suthep Suantai (2008)

Banach Center Publications

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In this paper, we define the direct sum ( i = 1 n X i ) c e s p of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ( i = 1 n X i ) c e s p has the H-property if and only if each X i has the H-property, and ( i = 1 n X i ) c e s p has the Schur property if and only if each X i has the Schur property. Moreover, we also show that ( i = 1 n X i ) c e s p is rotund if and only if each X i is rotund.