The Lindelöf class of L-functions II

Dixit, Anup B.; Murty, V. Kumar

doi:10.1142/s1793042119501215

Cited by 4 publications

(6 citation statements)

References 8 publications

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“…27-28]), it is worth asking whether the polynomial Euler product assumption can be dispensed with. One can also enquire whether Ramanujan-Hardy-Littlewood type identities are valid for larger classes of L-functions, for instance the Lindel öf class [19,7].…”

Section: Discussionmentioning

confidence: 99%

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Riesz type criteria for $L$-functions in the Selberg class

Gupta¹,

Vatwani²

2022

Preprint

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We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan-Hardy-Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type G n 0 0 n .

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Section: Discussionmentioning

confidence: 99%

“…We will use Lemma (α) on p. 56 of [25] with f (s) = F (s), s 0 = 2 + it and r = 4(2 + c F ). Then from lemma 3.4 (in particular ( 5) and ( 9)) of [11], it is clear that the hypothesis…”

Section: Ramanujan-hardy-littlewood-type Identity For Selberg Classmentioning

confidence: 91%

Riesz type criteria for $L$-functions in the Selberg class

Gupta¹,

Vatwani²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…We will use Lemma (α) on page 56 of [25] with f (s) = F(s), s 0 = 2 + it and r = 4(2 + c F ). Then, from Lemma 3.4 (in particular, ( 5) and ( 9)) of [11], it is clear that the hypothesis…”

Section: Lemma 41 Let F Satisfy Axioms (I)-(iv) Of the Class S If ρ =...mentioning

confidence: 92%

“…We will use Lemma on page 56 of [25] with , and . Then, from Lemma 3.4 (in particular, (5) and (9)) of [11], it is clear that the hypothesis holds with for some constant A depending on F . Lemma of [25, p. 56] then yields for , and so, in particular, for due to the choice of r .…”

Section: Ramanujan–hardy–littlewood-type Identity For Selberg Classmentioning

confidence: 99%

Riesz-type criteria for L-functions in the Selberg class

Gupta¹,

Vatwani²

2023

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

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We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$ .

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“…One can consider linear combination of elements in S to see this. A family of L-functions based on a growth condition was introduced by V. K. Murty in [17] (see [11] for more details). The Igusa zeta-function, and the zeta function of groups have Euler products but may not have a functional equation (see [19]).…”

Section: Introductionmentioning

confidence: 99%

LARGE VALUES OF L-FUNCTIONS ON THE 1-LINE

Dixit

Mahatab

2020

Bull. Aust. Math. Soc.

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We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

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The Lindelöf class of L-functions II

Cited by 4 publications

References 8 publications

Riesz type criteria for $L$-functions in the Selberg class

Riesz type criteria for $L$-functions in the Selberg class

Riesz-type criteria for L-functions in the Selberg class

LARGE VALUES OF L-FUNCTIONS ON THE 1-LINE

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