Euler numbers congruences and Dirichlet L-functions

Nian-liang, Wang; Jun-zhuang, LI; Duan-sen, Liu

doi:10.1016/j.jnt.2009.01.004

Cited by 5 publications

(11 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…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%

“…Lemma 1 was derived by the first author in [7], which is a slight generalization of Yamamoto's result (cf. [4, (5.1), (5.2)]) for the characters primitive modulo p and q is a power of p. …”

Section: As Immediate Consequences For Any Positive Integer M We Havementioning

confidence: 94%

“…In this regard, 6) which is equal to 2h(−4p) by the class number formula (cf. [5,7]). Since the (1/4)th sum can be expressed as a multiple of L(1, χ), (1.6) can be interpreted as an expression of the Euler number in terms of the multiple of L (1, χ).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Nian-liang

Li²,

Liu³

2017

Kyushu J. Math.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…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%

Section: Resultsmentioning

confidence: 99%

“…Lemma 1 was derived by the first author in [7], which is a slight generalization of Yamamoto's result (cf. [4, (5.1), (5.2)]) for the characters primitive modulo p and q is a power of p. …”

Section: As Immediate Consequences For Any Positive Integer M We Havementioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Nian-liang

Li²,

Liu³

2017

Kyushu J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…where b m (n) are defined by n m as has been done in [38] and [37]. Thus we leave Theorem 1 at that and expect on another occasion to work out the explicit coefficients and prove the equivalence of these theorems.…”

Section: Theorem 1 the Notation Being The Same As Above We Havementioning

confidence: 94%

Weighted short-interval character sums

Kanemitsu

Nian-liang

2011

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Untitled

2011

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

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.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1522 SHIGERU KANEMITSU, HAILONG LI, AND NIANLIANG WANG equation. We have come upon this standpoint by viewing Szmidt, Urbanowicz and Zagier's formula as a form of a Lambert series.At the time when we wrote the first version of the present paper, we thought we would need the functional equation for the Hurwitz-Lerch L-function introduced by Morita [31] (for this, cf. the proof of Theorem 3), but it has turned out that we need that of the Dirichlet L-function and the complete character sum (for applications to short-interval sums). Thus we are led to give formulas for the special values of the relevant -function (cf. (2.5)) at positive integer arguments, which is essential in the case of imprimitive characters, whence those for the Dirichlet L-functions with imprimitive characters. This part is also based on Riemann, Hecke and Bochner's correspondence.Secondly, we will state some concrete formulas for the short-interval character sums with imprimitive characters for convenience of reference. This is because, to treat such sums, most of the authors refer to [7] or [18] but not [35], [38] or more recently [34]; cf. [37]for examples of such redundancies. Most of the literature contains formulas for primitive characters but one sometimes needs formulas for imprimitive characters as in the case of the analytic-number-theoretic application of discrete mean squares (cf. e.g. [22]) in which one needs a formula valid for all characters in order to use the orthogonality. In [38] and [34] the authors' intension was to apply the results to the quadratic fields and therefore all the characters they need are primitive Kronecker characters. Since Szmidt, Urbanowicz and Zagier's formula is valid for all characters, we are led to trying to generalize Yamamoto's method to this case. We note that Yamamoto's method allows us to treat shortinterval character sums with Clausen function weight (log sin integrals, cf. [30]) as well, which enables us to obtain class number relations for real quadrat...

show abstract

Euler numbers congruences and Dirichlet L-functions

Cited by 5 publications

References 12 publications

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

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

Weighted short-interval character sums

Untitled

Contact Info

Product

Resources

About