The convergence of Euler products over <i>p</i>-adic number fields

Delbourgo, Daniel

doi:10.1017/s0013091507000636

Cited by 9 publications

(19 citation statements)

References 7 publications

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“…The expansion is particularly simple for c = 2 , and this parameter can be used for p ≠ 2 and Dirichlet characters with odd conductor. For this case we obtain similar results as in [1,2], and [4]. In Sect.…”

Section: Introductionsupporting

confidence: 83%

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Dirichlet series expansions of p-adic L-functions

Knospe

Washington

2021

Abh. Math. Semin. Univ. Hambg.

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We study p-adic L-functions $$L_p(s,\chi )$$ L p ( s , χ ) for Dirichlet characters $$\chi $$ χ . We show that $$L_p(s,\chi )$$ L p ( s , χ ) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of $$\chi $$ χ . The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for $$c=2$$ c = 2 , where we obtain a Dirichlet series expansion that is similar to the complex case.

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Section: Introductionsupporting

confidence: 83%

“…In Sect. 3, we give an explicit formula for the values of E 1,c and derive the Dirichlet series expansion from (2). The expansion is particularly simple for c = 2 , and this parameter can be used for p ≠ 2 and Dirichlet characters with odd conductor.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet series expansions of p-adic L-functions

Knospe

Washington

2021

Abh. Math. Semin. Univ. Hambg.

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“…To our great surprise, we discovered that the formula for these approximations continued to hold true even without restricting p, in all the cases that we tried. This indicates that the first named author was over-zealous in the conditions imposed in [3], and so these formulae are likewise valid in the case where p | φ(f) with p 2f.…”

Section: Locating the Zeros Ofmentioning

confidence: 92%

“…The key ingredient is to utilise the p-adic approximations developed by the first author in [2,3], which seem well suited to resolving problems of type (I) and (II) (see previous page), relatively quickly. In total, the computations in this paper took approximately five months to run on PARI/GP.…”

Section: On λ-Invariants Attached To Cyclic Cubic Number Fieldsmentioning

confidence: 99%

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On -invariants attached to cyclic cubic number fields

Delbourgo¹,

Chao²

2015

LMS J. Comput. Math.

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We describe an algorithm for finding the coefficients of F (X) modulo powers of p, where p = 2 is a prime number and F (X) is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic λ-invariants attached to those cubic extensions K/Q with cyclic Galois group A3 (up to field discriminant <10 7 ), and also tabulate the class number of K(e 2πi/p ) for p = 5 and p = 7. If the λ-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the p-adic L-function and deduce Λ-monogeneity for the class group tower over the cyclotomic Zp-extension of K.

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Sum expressions for Kubota–Leopoldt -adic -functions

Zhao

2022

Proceedings of the Edinburgh Mathematical Society

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When $p$ is an odd prime, Delbourgo observed that any Kubota–Leopoldt $p$ -adic $L$ -function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on $p$ , and without the Euler factor when the character is non-trivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero–Greenberg formula for the derivative $L_p'(0,\chi )$ . We will also discuss the convergence of sum expressions in terms of elementary $p$ -adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.

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The convergence of Euler products over p-adic number fields

Cited by 9 publications

References 7 publications

Dirichlet series expansions of p-adic L-functions

Dirichlet series expansions of p-adic L-functions

On -invariants attached to cyclic cubic number fields

Sum expressions for Kubota–Leopoldt -adic -functions

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