全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量
162

相关文章

更多...

On Algebraic Methods for the Riemann Hypothesis

DOI: 10.4236/apm.2025.151001, PP. 1-35

Keywords: Gamma Function, Zeta Function, Bernoulli Numbers, Robin Inequality, Twin Primes, Primes

Full-Text   Cite this paper   Add to My Lib

Abstract:

Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.

References

[1]  Riemann, B. (1859) Über die Anzahl der Primzahlen unter einer gegebenen Grösse. [Monatsberichte der Berliner Akademie, November 1859]. 136-144.
[2]  Robin. G. (1984) Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. Journal de Mathématiques Pures et Appliquées, 63, 187-217.
[3]  Euler, L. (1755) Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, Vol. 1.
[4]  Gauss, C.F. (1863) Theoria residuorum biquadraticorum, Commentatio secunda. Königlichen Gesellschaft der Wis-senschaften zu Göttingen, 95-148.
[5]  Solé, P. and Planat, M. (2012) The Robin Inequality for 7-Free Integers. Integers, 12, 301-308.
https://doi.org/10.1515/integ.2011.103
[6]  Rosser, J.B. and Schoenfeld, L. (1962) Approximate Formulas for Some Functions of Prime Numbers. Illinois Journal of Mathematics, 6, 64-94.
https://doi.org/10.1215/ijm/1255631807
[7]  Heath-Brown, D.R. (1984) The Divisor Function at Consecutive Integers. Mathematika, 31, 141-149.
https://doi.org/10.1112/s0025579300010743
[8]  Pinner, C.G. (1997) Repeated Values of the Divisor Function. The Quarterly Journal of Mathematics, 48, 499-502.
https://doi.org/10.1093/qmath/48.4.499
[9]  Choie, Y.J., Lichiardopol, N., Moree, P. and Sol, P. (2007) On Robin’s Criterion for the Riemann Hypothesis. Tome, 19, 357-372.
[10]  Chakrabarty, S. (2019) The Repeated Divisor Function and Possible Correlation with Highly Composite Numbers. arXiv: 1704.00007.
[11]  Nicolas, J.L. (1983) Petites valeurs de la fonction d’Euler et hypothèse de Riemann, Séminaire de Théorie des nombres. Progress in Mathematics, 38, 207-218.
[12]  Anthony, M.M. (2016) The Towering Zeta Function. Advances in Pure Mathematics, 6, 351-392.
https://doi.org/10.4236/apm.2016.65026
[13]  Anthony, M.M. (2024) On Perron’s Formula and the Prime Numbers. Advances in Pure Mathematics, 14, 487-494.
https://doi.org/10.4236/apm.2024.146027
[14]  Gradshteyn, I.S. and Ryzhik, I.M. (2007) Table of Integrals, Series, and Products. 7th Edition, Academic Press.
[15]  Dusart, P. (1999) The kth Prime Is Greater than . Mathematics of Computation, 68, 411-415.
https://doi.org/10.1090/s0025-5718-99-01037-6
[16]  Hardy, G.H. (1999) Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. AMS Chelsea Publishing.
[17]  Euler, L. (1760) Observation de summis divisorum, Novi Commentarii. Academiae scientiarum Imperialis Petropoli tanae, 5, 59-74.

Full-Text

  
  

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133