The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Displaying similar documents to “On the Riesz means of n/ϕ(n) - III”

On the spinor zeta functions problem: higher power moments of the Riesz mean

Haiyan Wang (2013)

Acta Arithmetica

Similarity:

Let F be a Siegel cusp form of integral weight k on the Siegel modular group Sp₂(ℤ) of genus 2. The coefficients of the spinor zeta function Z F ( s ) are denoted by cₙ. Let D ρ ( x ; Z F ) be the Riesz mean of cₙ. Kohnen and Wang obtained the truncated Voronoï-type formula for D ρ ( x ; Z F ) under the Ramanujan-Petersson conjecture. In this paper, we study the higher power moments of D ρ ( x ; Z F ) , and then derive an asymptotic formula for the hth (h=3,4,5) power moments of D ( x ; Z F ) by using Ivić’s large value arguments and other techniques. ...

On Popov's explicit formula and the Davenport expansion

Quan Yang, Jay Mehta, Shigeru Kanemitsu (2023)

Czechoslovak Mathematical Journal

Similarity:

We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function a n with the periodic Bernoulli polynomial weight B ¯ ϰ ( n x ) and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order 0 or 1 gives the well-known explicit formula for respectively...

Mean values related to the Dedekind zeta-function

Hengcai Tang, Youjun Wang (2024)

Czechoslovak Mathematical Journal

Similarity:

Let K / be a nonnormal cubic extension which is given by an irreducible polynomial g ( x ) = x 3 + a x 2 + b x + c . Denote by ζ K ( s ) the Dedekind zeta-function of the field K and a K ( n ) the number of integral ideals in K with norm n . In this note, by the higher integral mean values and subconvexity bound of automorphic L -functions, the second and third moment of a K ( n ) is considered, i.e., n x a K 2 ( n ) = x P 1 ( log x ) + O ( x 5 / 7 + ϵ ) , n x a K 3 ( n ) = x P 4 ( log x ) + O ( X 321 / 356 + ϵ ) , where P 1 ( t ) , P 4 ( t ) are polynomials of degree 1, 4, respectively, ϵ > 0 is an arbitrarily small number.

The mean square of the divisor function

Chaohua Jia, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

Similarity:

Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that n x d ² ( n ) = x P ( l o g x ) + E ( x ) , where P(y) is a cubic polynomial in y and E ( x ) = O ( x 3 / 5 + ε ) , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), E ( x ) = O ( x 1 / 2 + ε ) . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce E ( x ) = O ( x 1 / 2 ( l o g x ) l o g l o g x ) . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove E ( x ) = O ( x 1 / 2 ( l o g x ) ) . ...

L¹ representation of Riesz spaces

Bahri Turan (2006)

Studia Mathematica

Similarity:

Let E be a Riesz space. By defining the spaces L ¹ E and L E of E, we prove that the center Z ( L ¹ E ) of L ¹ E is L E and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality L ¹ E = Z ( E ) ' . Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in L ¹ E which are different from the representations appearing in the literature.

Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa (2014)

Journal of the European Mathematical Society

Similarity:

For 1 n < d integers and ρ > 2 , we prove that an n -dimensional Ahlfors-David regular measure μ in d is uniformly n -rectifiable if and only if the ρ -variation for the Riesz transform with respect to μ is a bounded operator in L 2 ( μ ) . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L 2 ( μ ) boundedness of the Riesz transform to the uniform rectifiability of μ .

An interpolatory estimate for the UMD-valued directional Haar projection

Richard Lechner

Similarity:

We prove an interpolatory estimate linking the directional Haar projection P ( ε ) to the Riesz transform in the context of Bochner-Lebesgue spaces L p ( ; X ) , 1 < p < ∞, provided X is a UMD-space. If ε i = 1 , the result is the inequality | | P ( ε ) u | | L p ( ; X ) C | | u | | L p ( ; X ) 1 / | | R i u | | L p ( ; X ) 1 - 1 / , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of L p ( ; X ) . In order to obtain the interpolatory result (1) we analyze stripe operators S λ , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

Similarity:

Let ( X , d , μ ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure μ . Let L be a non-negative self-adjoint operator of order m on L 2 ( X ) . Assume that the semigroup e - t L generated by L satisfies the Davies-Gaffney estimate of order m and L satisfies the Plancherel type estimate. Let H L p ( X ) be the Hardy space associated with L . We show the boundedness of Stein’s square function 𝒢 δ ( L ) arising from Bochner-Riesz means associated to L from Hardy spaces H L p ( X ) to L p ( X ) , and also study...

Sharp inequalities for Riesz transforms

Adam Osękowski (2014)

Studia Mathematica

Similarity:

We establish the following sharp local estimate for the family R j j = 1 d of Riesz transforms on d . For any Borel subset A of d and any function f : d , A | R j f ( x ) | d x C p | | f | | L p ( d ) | A | 1 / q , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, C p = [ 2 q + 2 Γ ( q + 1 ) / π q + 1 k = 0 ( - 1 ) k / ( 2 k + 1 ) q + 1 ] 1 / q , 1 < p < 2, and C p = [ 4 Γ ( q + 1 ) / π q k = 0 1 / ( 2 k + 1 ) q ] 1 / q , 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

Similarity:

For 1 ≤ q ≤ α ≤ p ≤ ∞, ( L q , l p ) α is a complex Banach space which is continuously included in the Wiener amalgam space ( L q , l p ) and contains the Lebesgue space L α . We study the closure ( L q , l p ) c , 0 α in ( L q , l p ) α of the space of test functions (infinitely differentiable and with compact support in d ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space W ¹ ( ( L q , l p ) α ) (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

The Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Studia Mathematica

Similarity:

Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form C Λ ( G ) or L ¹ Λ ( G ) and which quotients of the form C ( G ) / C Λ ( G ) or L ¹ ( G ) / L ¹ Λ ( G ) have the Daugavet property. We show that C Λ ( G ) is a rich subspace of C(G) if and only if Γ Λ - 1 is a semi-Riesz set. If L ¹ Λ ( G ) is a rich subspace of L¹(G), then C Λ ( G ) is a rich subspace of C(G) as well. Concerning quotients, we prove that C ( G ) / C Λ ( G ) has the Daugavet property if Λ is a Rosenthal set, and that L ¹ Λ ( G ) is a poor subspace of L¹(G)...

The σ -property in C ( X )

Anthony W. Hager (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

The σ -property of a Riesz space (real vector lattice) B is: For each sequence { b n } of positive elements of B , there is a sequence { λ n } of positive reals, and b B , with λ n b n b for each n . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “ σ ” obtains for a Riesz space of continuous real-valued functions C ( X ) . A basic result is: For discrete X , C ( X ) has σ iff the cardinal | X | < 𝔟 , Rothberger’s bounding number. Consequences...