Congruences among generalized Bernoulli numbers

Szmidt, Janusz; Urbanowicz, Jerzy; Zagier, Don

doi:10.4064/aa-71-3-273-278

Cited by 26 publications

(28 citation statements)

References 6 publications

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“…The proof of (1) followed the method of Glaisher [3], which depends on Bernoulli polynomials of fractional arguments. In this paper, we use an identity proved by Szmidt, Urbanowicz and Zagier [8] to generalize congruence (1) to a congruence modulo an arbitrary positive integer. The main theorem we obtain is the following:…”

Section: A Congruence Involving the Quotients Of Euler And Its Applicmentioning

confidence: 99%

A congruence involving the quotients of Euler and its applications (I)

Cai

2002

Acta Arith.

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Section: A Congruence Involving the Quotients Of Euler And Its Applicmentioning

confidence: 99%

A congruence involving the quotients of Euler and its applications (I)

Cai

2002

Acta Arith.

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“…For some more results on this aspect, readers may refer to Carlitz [19], Yamamoto [4], Szmidt et al [12], Wang et al [7], Kanemitsu et al [11] and also to [9][10][11][12][13][14][15][16][17][18].…”

Section: Resultsmentioning

confidence: 99%

COSINE HIGHER-ORDER EULER NUMBER CONGRUENCES AND DIRICHLET <i>L</i>-FUNCTION VALUES

Nian-liang

Li²,

Liu³

2017

Kyushu J. Math.

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Abstract. In this paper we obtain the residue modulo a prime power of cosine higher-order Euler numbers H (k) 2n (m) in terms of the linear combination of the Dirichlet L-function values L(s, χ) at positive integral arguments s or of generalized Bernoulli numbers. Our results are restricted to the equal parity case; i.e. s and χ are of the same parity. In the process, we employ Yamamoto's results on finite expressions in terms of Dirichlet L-function values for short interval character sums and in this sense our treatment is decisive, i.e. any ad-hoc transformation of short interval sums. The results obtained not only generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s = 1 in terms of Euler numbers but also closes the chapter on possible similar research.

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“…In this section we present Szmidt, Urbanowicz and Zagier's elegant formula [35] (Formula (2.10) below) and show that it is in spirit exactly the same as the classical method for obtaining the values of the Riemann zeta-function at even positive integral arguments. Indeed, it leads to the most well-known formula of Euler.…”

Section: Szmidt Urbanowicz and Zagier's Formulamentioning

confidence: 99%

Weighted short-interval character sums

Kanemitsu

Nian-liang

2011

Proc. Amer. Math. Soc.

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Abstract. In this paper we shall establish the counterpart of Szmidt, Urbanowicz and Zagier's formula in the sense of the Hecker correspondence. The motivation is the derivation of the values of the Riemann zeta-function at positive even integral arguments from the partial fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last is equivalent to the functional equation, we may view their elegant formula as one for the Lambert series, and comparing the Laurent coefficients, we may give a functional equational approach to the short-interval character sums with polynomial weight.In view of the importance of these short-interval character sums, we assemble some handy formulations for them that are derived from Szmidt, Urbanowicz and Zagier's formula and Yamamoto's method, which also gives the conjugate sums. We shall also state the formula for the values of the Dirichlet L-function with imprimitive characters.

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Congruences among generalized Bernoulli numbers

Cited by 26 publications

References 6 publications

A congruence involving the quotients of Euler and its applications (I)

A congruence involving the quotients of Euler and its applications (I)

COSINE HIGHER-ORDER EULER NUMBER CONGRUENCES AND DIRICHLET <i>L</i>-FUNCTION VALUES

Weighted short-interval character sums

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