The Discrete Mean Square of the Dirichlet L-Function at Nontrivial Zeros of Another Dirichlet L-Function

Garunkštis, Ramūnas; Kalpokas, Justas

doi:10.1142/s1793042113500085

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…Then by (4), we see that R ≪ T (T + t) −λ/2 + T 1/2 (T + t) δ−λ/2 . From this and formulas (3), (13), replacing λ/2 by λ, we obtain Proposition 6. Proposition 7.…”

Section: Proofsmentioning

confidence: 71%

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Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros

Garunkštis

Laurinčikas

2017

Period Math Hung

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“…Then by (4), we see that R ≪ T (T + t) −λ/2 + T 1/2 (T + t) δ−λ/2 . From this and formulas (3), (13), replacing λ/2 by λ, we obtain Proposition 6. Proposition 7.…”

Section: Proofsmentioning

confidence: 71%

“…The discrete mean value of the Dirichlet L-function at nontrivial zeros of another Dirichlet L-function were investigated by Garunkštis and Kalpokas [13]. See also Fujii [7,10], Conrey, Ghosh and Gonek [3,4], Steuding [22], and Garunkštis, Kalpokas, and Steuding [12].…”

Section: Discussionmentioning

confidence: 99%

Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros

Garunkštis

Laurinčikas

2017

Period Math Hung

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“…They remarked in [16] that similar results hold for Dirichlet L-functions with inequivalent characters under the Generalized Riemann Hypothesis. R. Garunkštis and J. Kalpokas [24] gave a lower bound for the proportion uniformly in the size of the conductors of the characters. Our result shows that under GRH, the values of ζ(s) at the zeros of another primitive L-function can be almost as large as the extreme large values of ζ(s) on the critical line without constraints.…”

Section: Large Values Of Dirichlet L-functions At Zeros Of a Class Of L-functions 1461mentioning

confidence: 99%

Large values of Dirichlet L-functions at zeros of a class of L-functions

2020

Can. J. Math.-J. Can. Math.

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In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is, $$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$ where $\chi $ is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of $GL(n)$ over ${\mathbb {Q}}$ .

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“…In Theorem 1.2 we fail to obtain positive proportion and we expect this to be a limitation of the method used. In [4] the authors look at the mean square of a single Dirichlet L-function at the zeros of another, and show that it is non-zero for at least cT of the zeros for some explicit c > 0. On the other hand, attempting to introduce a mollifier to overcome this limitation does not seem hopefuly either.…”

Section: Introductionmentioning

confidence: 99%

On the value distribution of two Dirichlet $\boldsymbol L$-functions

Laaksonen¹

2018

Funct. Approx. Comment. Math.

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We look at the values of two Dirichlet L-functions at the Riemann zeros (or a horizontal shift of them).Off the critical line we show that for a positive proportion of these points the pairs of values of the two L-functions are linearly independent over R, which, in particular, means that their arguments are different. On the critical line we show that, up to height T , the values are different for cT of the Riemann zeros for some positive c.the distribution of their zeros. In 1976 Fujii [3] showed that a positive proportion of zeros of L(s, ψ)L(s, χ) are distinct, where the characters are primitive and not necessarily of distinct moduli. A zero of the product is said to be distinct if it is a zero of only one of the two, or if it is a zero of both then it occurs with different multiplicities for each function. It is, in fact, believed that all zeros of Dirichlet L-functions to primitive characters are simple, and that two L-functions with distinct primitive characters do not share any non-trivial zeros at all. This comes from the Grand Simplicity Hypothesis (GSH), see [18]. The hypothesis is that the set {γ | L( 1 2 + iγ, χ) = 0 and χ is primitive} is linearly independent over Q. Since we are counting with multiplicities, it is implicit in the statement of the GSH that all zeros of Dirichlet L-functions are simple, and that γ = 0, i.e. L( 1 2 , χ) = 0. A similar result is expected for an even bigger class of functions. R. Murty and K. Murty [16] proved that two functions of the Selberg class S cannot share too many zeros (counted with multiplicity). They show that if F , G ∈ S then F = G provided thatwhere Z F (T ) denotes the set of zeros of F (s) in the region Re s ≥ 1/2 and |Im s| ≤ T , and ∆ is the symmetric difference. In 1986 Conrey et al. [1] proved that the Dedekind zeta function of a quadratic number field has infinitely many simple zeros. They were, however, unable to obtain the result for a positive proportion of the zeros.Apart from looking at the zeros, there has also been investigation into the a-values of ζ and L(s, χ), that is, the distribution of s such that ζ(s) = a (or L(s, χ) = a) for some fixed a ∈ C. In [6] Garunkštis and Steuding prove a discrete average for ζ ′ over the a-values of ζ, which implies that there are infinitely many simple a-points in the critical strip. On the critical line, however, we do not even know whether there are infinitely many a-points. For further results on

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The Discrete Mean Square of the Dirichlet L-Function at Nontrivial Zeros of Another Dirichlet L-Function

Cited by 5 publications

References 11 publications

Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros

Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros

Large values of Dirichlet L-functions at zeros of a class of L-functions

On the value distribution of two Dirichlet $\boldsymbol L$-functions

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