SavedAdd your journal

Journals / Turkish Journal of Mathematics / 2020 / Cilt: 44 - Sayı: 3

Higher-order character Dedekind sum

Mümün CAN
Pages
998–1015
Publication Date
2022-11-04
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Moreover, we offer Petersson–Knopp type identities for this sum.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we are interested in higher-order character Dedekind sum $sum_{v=0}^{ck;-1};X_1;(nu);{mathfrak B}_{p,;x_2};left(afrac{v;+;z}c+xright);+;{mathfrak B}_q;left(bfrac{v;+;z}{ck}+yright);,;a,;b,;c;in;mathbb{N};and;x,;y,;z;in;mathbb{R}$ where χ1 and χ2 are primitive characters of modulus k, Bp (x) and Bp,χ2(x) are Bernoulli and generalized Bernoullifunctions, respectively.
📊
Key Findings & Data:We employ the Fourier series technique to demonstrate reciprocity formulas for this sum.
💡
Conclusion & Impact:Moreover, we offer Petersson–Knopp type identities for this sum.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

In this paper, we are interested in higher-order character Dedekind sum $sum_{v=0}^{ck;-1};X_1;(nu);{mathfrak B}_{p,;x_2};left(afrac{v;+;z}c+xright);+;{mathfrak B}_q;left(bfrac{v;+;z}{ck}+yright);,;a,;b,;c;in;mathbb{N};and;x,;y,;z;in;mathbb{R}$ where χ1 and χ2 are primitive characters of modulus k, Bp (x) and Bp,χ2(x) are Bernoulli and generalized Bernoullifunctions, respectively. We employ the Fourier series technique to demonstrate reciprocity formulas for this sum. Derivedformulas are analogues of Mikolás’ reciprocity formula. Moreover, we offer Petersson–Knopp type identities for this sum.

🔖

Cite This Article

CAN, M. (2022). Higher-order character Dedekind sum. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 3, 998–1015. https://ajindex.com/articles/607514

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (19)

  1. Alkan E, Xiong M, Zaharescu A. A bias phenomenon on the behavior of Dedekind sums. Mathematical Research Letters 2008; (15): 1039-1052
  2. Apostol TM. Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Mathematical Journal 1950; (17): 147-157
  3. Apostol TM. Introduction to Analytic Number Theory. New York, NY, USA: Springer-Verlag, 1976
  4. Apostol TM, Vu TH. Identities for sums of Dedekind type. Journal of Number Theory 1982; (14): 391-396
  5. Bayad A, Raouj A. Reciprocity formulae for multiple Dedekind–Rademacher sums. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics 2011; (349): 131-136
  6. Beck M, Chavez A. Bernoulli–Dedekind sums. Acta Arithmetica 2011; (149): 65-82
  7. Berndt BC. Character transformation formulae similar to those for the Dedekind Eta–function. In: ‘Analytic Number Theory’, Proceedings of Symposia in Pure Mathematics XXIV. Providence, RI, USA: American Mathematical Society, 1973, pp. 9-30
  8. Berndt BC. Generalized Dedekind eta-functions and generalized Dedekind sums. Transactions of the American Mathematical Society 1973; (178): 495-508
  9. Berndt BC. Character analogues of Poisson and Euler–MacLaurin summation formulas with applications. Journal of Number Theory 1975; (7): 413-445
  10. Berndt BC. On Eisenstein series with characters and the values of Dirichlet L-functions. Acta Arithmetica 1975; (28): 299-320
  11. Berndt BC. Generalized Eisenstein series and modified Dedekind sums. Journal für die reine und angewandte Mathematik 1975; (272): 182-193
  12. Berndt BC. Reciprocity theorems for Dedekind sums and generalizations. Advances in Mathematics 1977; (23): 285-316
  13. Boztaş MÇ, Can M. Transformation formulas of a character analogue of log θ2 (z). The Ramanujan Journal 2019; (48): 323-349
  14. Can M, Cenkci M, Kurt V, Simsek Y. Twisted Dedekind type sums associated with Barnes’ type multiple FrobeniusEuler l-functions. Advanced Studies in Contemporary Mathematics 2009; (18): 135-160
  15. Can M, Kurt V. Character analogues of certain Hardy–Berndt sums. International Journal of Number Theory 2014; (10): 737-762
  16. Carlitz L. Some theorems on generalized Dedekind sums. Pacific Journal of Mathematics 1953; (3): 513-522
  17. Carlitz L. Generalized Dedekind sums. Mathematische Zeitschrift 1964; (85): 83-90
  18. Cenkci M. On p-adic character Dedekind sums. Palestine Journal of Mathematics 2015; (4): 502-507
  19. Cenkci M, Can M, Kurt V. Degenerate and charact