Zeros of Dirichlet polynomials

Roy, Arindam; Vatwani, Akshaa

doi:10.1090/tran/8261

Cited by 3 publications

(3 citation statements)

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“…Dans un article récent [19], Roy et Vatwani considèrent le cas de séries de Dirichlet où est une fonction arithmétique appartenant à une sous-famille (introduite dans [18]) de fonctions multiplicatives pour lesquelles les séries de Dirichlet précédentes sont absolument convergentes pour .…”

Section: éQuivalence Au Sens De Bohr Sommes Partielles De La Zeta De ...unclassified

Une note sur la densité des zéros des sommes partielles de la fonction zeta de Dedekind sur un corps quadratique

Dubon

2021

Can. Math. Bull.

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En s'appuyant sur la notion d'équivalence au sens de Bohr entre polynômes de Dirichlet et sur le fait que sur un corps quadratique la fonction zeta de Dedekind peut s'écrire comme produit de la fonction zeta de Riemann et d'une fonction L, nous montrons que, pour certaines valeurs du discriminant du corps quadratique, les sommes partielles de la fonction zeta de Dedekind ont leurs zéros dans des bandes verticales du plan complexe appelés bandes critiques et que les parties réelles de leurs zéros y sont denses.

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Section: éQuivalence Au Sens De Bohr Sommes Partielles De La Zeta De ...unclassified

Une note sur la densité des zéros des sommes partielles de la fonction zeta de Dedekind sur un corps quadratique

Dubon

2021

Can. Math. Bull.

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“…Nevertheless it is well known the regularity of the non-trivial zeros of ζ(s) in the sense that, all those found so far, are located on the line s = 1/2. Among the papers dealing with the issues raised on the distribution of the zeros of the partials sums of the Riemann zeta function, we suggest [7,8,10,15,[18][19][20]24]; on the implication of the truth of the Riemann Hypothesis when those zeros are close the line s = 1 [23,Theorem III], see [22,23] and [11,12]. On Dirichlet series, properties and abscissae of convergence, read [1,Chapter 8] and [5,6].…”

Section: Introductionmentioning

confidence: 99%

On the lower bounds of the partial sums of a Dirichlet series

Mora

Benítez

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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In this paper it is shown that for the ordinary Dirichlet series, $$\sum _{j=0}^{\infty }\frac{\alpha _{j}}{(j+1)^{s}}$$ ∑ j = 0 ∞ α j ( j + 1 ) s , $$\alpha _{0}=1$$ α 0 = 1 , of a class, say $${\mathcal {P}}$$ P , that contains in particular the series that define the Riemann zeta and the Dirichlet eta functions, there exists $$\lim _{n\rightarrow \infty }\rho _{n}/n$$ lim n → ∞ ρ n / n , where the $$\rho _{n}$$ ρ n ’s are the Henry lower bounds of the partial sums of the given Dirichlet series, $$P_{n}(s)=\sum _{j=0}^{n-1}\frac{\alpha _{j}}{(j+1)^{s}}$$ P n ( s ) = ∑ j = 0 n - 1 α j ( j + 1 ) s , $$n>2$$ n > 2 . Likewise it is given an estimate of the above limit. For the series of $${\mathcal {P}}$$ P having positive coefficients it is shown the existence of the $$\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n$$ lim n → ∞ a P n ( s ) / n , where the $$a_{P_{n}(s)}$$ a P n ( s ) ’s are the lowest bounds of the real parts of the zeros of the partial sums. Furthermore it has been proved that $$\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n=\lim _{n\rightarrow \infty }\rho _{n}/n$$ lim n → ∞ a P n ( s ) / n = lim n → ∞ ρ n / n .

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“…In the same way that is relevant the abscissa of convergence for a Dirichlet series [1,p. 165] (very interesting works on this subject and generalizations can be seen, for instance, in [4,5]), it is also relevant the essential bounds for a Dirichlet polynomial. In a recent article [14] both notions, i.e., the abscissa of convergence of an ordinary Dirichlet series, whose coefficients α n are defined by…”

Section: Introductionmentioning

confidence: 99%

Essential bounds of Dirichlet polynomials

Mora

Benítez

2021

RACSAM

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Zeros of Dirichlet polynomials

Abstract: We consider a certain class of multiplicative functions f : N → C f: \mathbb N \rightarrow \mathbb C and study the distribution of zeros of Dirichlet polynomials F N ( s ) = ∑ n ≤ N f ( n ) n − s F_N(s)… Show more

Cited by 3 publications

References 25 publications

Une note sur la densité des zéros des sommes partielles de la fonction zeta de Dedekind sur un corps quadratique

Une note sur la densité des zéros des sommes partielles de la fonction zeta de Dedekind sur un corps quadratique

On the lower bounds of the partial sums of a Dirichlet series

Essential bounds of Dirichlet polynomials

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