Displaying similar documents to “Dirichlet series induced by the Riemann zeta-function”

The size of the Lerch zeta-function at places symmetric with respect to the line ( s ) = 1 / 2

Ramūnas Garunkštis, Andrius Grigutis (2019)

Czechoslovak Mathematical Journal

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Let ζ ( s ) be the Riemann zeta-function. If t 6 . 8 and σ > 1 / 2 , then it is known that the inequality | ζ ( 1 - s ) | > | ζ ( s ) | is valid except at the zeros of ζ ( s ) . Here we investigate the Lerch zeta-function L ( λ , α , s ) which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters λ = α it is still possible to obtain a certain version of the inequality | L ( λ , λ , 1 - s ¯ ) | > | L ( λ , λ , s ) | .

Mean values related to the Dedekind zeta-function

Hengcai Tang, Youjun Wang (2024)

Czechoslovak Mathematical Journal

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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.

On discrete mean values of Dirichlet L -functions

Ertan Elma (2021)

Czechoslovak Mathematical Journal

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Let χ be a nonprincipal Dirichlet character modulo a prime number p 3 and let 𝔞 χ : = 1 2 ( 1 - χ ( - 1 ) ) . Define the mean value p ( - s , χ ) : = 2 p - 1 ψ ( mod p ) ψ ( - 1 ) = - 1 L ( 1 , ψ ) L ( - s , χ ψ ¯ ) ( σ : = s > 0 ) . We give an identity for p ( - s , χ ) which, in particular, shows that p ( - s , χ ) = L ( 1 - s , χ ) + 𝔞 χ 2 p s L ( 1 , χ ) ζ ( - s ) + o ( 1 ) ( p ) for fixed 0 < σ < 1 2 and | t : = s | = o ( p ( 1 - 2 σ ) / ( 3 + 2 σ ) ) .

The mean square of the divisor function

Chaohua Jia, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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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 ) ) . ...

An arithmetic Riemann-Roch theorem for pointed stable curves

Gérard Freixas Montplet (2009)

Annales scientifiques de l'École Normale Supérieure

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Let ( 𝒪 , Σ , F ) be an arithmetic ring of Krull dimension at most 1, 𝒮 = Spec 𝒪 and ( π : 𝒳 𝒮 ; σ 1 , ... , σ n ) an n -pointed stable curve of genus g . Write 𝒰 = 𝒳 j σ j ( 𝒮 ) . The invertible sheaf ω 𝒳 / 𝒮 ( σ 1 + + σ n ) inherits a hermitian structure · hyp from the dual of the hyperbolic metric on the Riemann surface 𝒰 . In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ω 𝒳 / 𝒮 ( σ 1 + ... + σ n ) hyp . The theorem is applied to modular curves X ( Γ ) , Γ = Γ 0 ( p ) or Γ 1 ( p ) , p 11 prime, with sections given by the cusps. We show Z ' ( Y ( Γ ) , 1 ) e a π b Γ 2 ( 1 / 2 ) c L ( 0 , Γ ) , with p 11 m o d 12 when Γ = Γ 0 ( p ) . Here Z ( Y ( Γ ) , s ) is the Selberg...

Representation growth of linear groups

Michael Larsen, Alexander Lubotzky (2008)

Journal of the European Mathematical Society

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Let Γ be a group and r n ( Γ ) the number of its n -dimensional irreducible complex representations. We define and study the associated representation zeta function 𝒵 Γ ( s ) = n = 1 r n ( Γ ) n - s . When Γ is an arithmetic group satisfying the congruence subgroup property then 𝒵 Γ ( s ) has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite...

A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken (2017)

Czechoslovak Mathematical Journal

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If ( M , ) is a manifold with a symmetric linear connection, then T * M can be endowed with the natural Riemann extension g ¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g ¯ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure 𝒫 on ( T * M , g ¯ ) and prove that 𝒫 is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if g ¯ reduces...

A remark on separate holomorphy

Marek Jarnicki, Peter Pflug (2006)

Studia Mathematica

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Let X be a Riemann domain over k × . If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set P k such that every slice X a of X with a∉ P is a region of holomorphy with respect to the family f | X a : f .

On Popov's explicit formula and the Davenport expansion

Quan Yang, Jay Mehta, Shigeru Kanemitsu (2023)

Czechoslovak Mathematical Journal

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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...