On a conjecture of the Euler numbers

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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“…cf. [3][4][5][6][7][8][9][10]. In view of the allowance summand l our results supersede most of the previous results.…”

Section: Introductionsupporting

confidence: 68%

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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“…G. Liu [9][10][11], and G. Liu and W. Zhang [12], Sun [14] and W. Zhang and Z. Xu [16]. A classical result on Euler numbers congruence modulo powers of two is due to Stern [13].…”

Section: Introductionmentioning

confidence: 99%

Euler numbers congruences and Dirichlet L-functions

Nian-liang¹,

Jun-zhuang²,

Duan-sen³

2009

Journal of Number Theory

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“…Specifically, in [5], T. Kim found many valuable results involving Euler numbers and polynomials connected with zeta functions. Other papers in regard to the Bernoulli polynomials and Euler polynomials can be found in [15][16][17][18][19]; we will not go into detail here.…”

Section: Introductionmentioning

confidence: 99%