This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.
References
[1]
Riemann, B. (1859) Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, 671-680.
[2]
Robin, G. (1984) Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. Journal de Mathématiques Pures et Appliquées, 63, 187-182.
[3]
Euler, L. (1775) Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, Volume 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. IllinoisJournalofMathematics, 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. TheQuarterlyJournalofMathematics, 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. Max-Planck-Institute.
[10]
Chakrabarty, S. (2019) The Repeated Divisor Function and Possible Correlation with Highly Composite Numbers.
[11]
Nicolas, J.-L. (1981-1982) Petites valeurs de la fonction d’Euler et hypothèse de Riemann, Séminaire de Théorie des nombres. D.P.P., 207-218, Progress in Mathematics, Vol. 38, Birkhäuser, 1983.
Anthony, M.M. (2024) On Perron’s Formula and the Prime Numbers. AdvancesinPureMathematics, 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 k(lnk + ln ln k − 1) for k ≥ 2. MathematicsofComputation, 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. E243 in the Enestro ̈m index.
[18]
Bronstein, M., Corless, R.M., Davenport, J.H. and Jeffrey, D.J. (2008) Algebraic Properties of the Lambert W Function from a Result of Rosenlicht and of Liouville. IntegralTransformsandSpecialFunctions, 19, 709-712. https://doi.org/10.1080/10652460802332342
[19]
Caley, T.S. (2007) A Review of the von Staudt Clausen Theorem. Dal Housie University.
[20]
Stanford, N. (2013) Dirichlet’s Theorem and Applications. University of Connecticut-Storrs.
[21]
Beardon, A.F. (2021) Winding Numbers, Unwinding Numbers, and the Lambert W Function. ComputationalMethodsandFunctionTheory, 22, 115-122. https://doi.org/10.1007/s40315-021-00398-1
