@article{Lag02,
  title={An {E}lementary {P}roblem {E}quivalent to the {R}iemann {H}ypothesis},
  author={Lagarias, Jeffrey Clark},
  journal={The American Mathematical Monthly},
  volume={109},
  number={6},
  pages={534--543},
  year={2002},
  publisher={Taylor \& Francis},
  doi = {10.1080/00029890.2002.11919883},
}

@article{Her18,
  title={Robin's {I}nequality for {N}ew {F}amilies of {I}ntegers},
  author={Hertlein, Alexander},
  journal={Integers},
  volume={18},
  year={2018}
}

@book{Dick19,
  title={History of the {T}heory of {N}umbers},
  author={Dickson, Leonard Eugene},
  number={256},
  year={1919},
  publisher={Carnegie Institution of Washington},
  address={Washington, D.C, United States}
}

@article{Och12,
  title={Odd perfect numbers are greater than $10^{1500}$},
  author={Ochem, Pascal and Rao, Micha{\"e}l},
  journal={Mathematics of Computation},
  volume={81},
  number={279},
  pages={1869--1877},
  year={2012},
  doi={10.1090/S0025-5718-2012-02563-4}
}

@article{Nil15,
  title={Odd perfect numbers, {D}iophantine equations, and upper bounds},
  author={Nielsen, Pace},
  journal={Mathematics of Computation},
  volume={84},
  number={295},
  pages={2549--2567},
  year={2015},
  doi={10.1090/S0025-5718-2015-02941-X}
}

@article{hagis1982some,
  title={Some results concerning quasiperfect numbers},
  author={Hagis Jr, Peter and Cohen, Graeme L},
  journal={Journal of the Australian Mathematical Society},
  volume={33},
  number={2},
  pages={275--286},
  year={1982},
  publisher={Cambridge University Press},
  doi={10.1017/S1446788700018401}
}

@article{dimitrov2024inequalities,
  title={Inequalities involving arithmetic functions},
  author={Dimitrov, Stoyan},
  journal={Lithuanian Mathematical Journal},
  volume={64},
  number={4},
  pages={421--452},
  year={2024},
  publisher={Springer},
  doi={10.1007/s10986-024-09655-x}
}

@article{Cohen78,
  title={On odd perfect numbers},
  author={Cohen, Graeme L.},
  journal={The Fibonacci Quarterly},
  volume={16},
  number={6},
  pages={523--527},
  year={1978},
  doi={10.1080/00150517.1978.12430302}
}

@article{Zelinsky25,
  title={The Sum of the Reciprocals of the Prime Divisors of an Odd Perfect or Odd Primitive Non-Deficient Number},
  author={Zelinsky, Joshua},
  journal={Integers},
  volume={25},
  pages={\#A59},
  year={2025},
  doi={10.5281/zenodo.15756036}
}

@book{Apostol76,
  title={Introduction to Analytic Number Theory},
  author={Apostol, Tom M.},
  year={1976},
  publisher={Springer-Verlag},
  address={New York, United States},
  series={Undergraduate Texts in Mathematics},
  doi={10.1007/978-1-4757-5579-4}
}

@article{RS62,
  title={Approximate formulas for some functions of prime numbers},
  author={Rosser, J. Barkley and Schoenfeld, Lowell},
  journal={Illinois Journal of Mathematics},
  volume={6},
  number={1},
  pages={64--94},
  year={1962},
  doi={10.1215/ijm/1255631807}
}

@article{Norton60,
  title={Remarks on the number of factors of an odd perfect number},
  author={Norton, Karl K.},
  journal={Acta Arithmetica},
  volume={6},
  number={4},
  pages={365--374},
  year={1960},
  doi={10.4064/aa-6-4-365-374}
}

@article{nielsen2007,
  author  = {Nielsen, Pace P.},
  title   = {Odd perfect numbers have at least nine distinct prime factors},
  journal = {Mathematics of Computation},
  volume  = {76},
  number  = {260},
  pages   = {2109--2126},
  year    = {2007},
  doi     = {10.1090/S0025-5718-07-01990-4}
}

@article{ochemrao2014,
  author  = {Ochem, Pascal and Rao, Micha{\"e}l},
  title   = {On the number of prime factors of an odd perfect number},
  journal = {Mathematics of Computation},
  volume  = {83},
  number  = {289},
  pages   = {2435--2439},
  year    = {2014},
  doi     = {10.1090/S0025-5718-2013-02776-7}
}

@article{laatsch1986,
  author  = {Laatsch, Richard},
  title   = {Measuring the abundancy of integers},
  journal = {Mathematics Magazine},
  volume  = {59},
  number  = {2},
  pages   = {84--92},
  year    = {1986},
  doi     = {10.1080/0025570X.1986.11977227}
}

@article{weiner2000,
  author  = {Weiner, Paul A.},
  title   = {The abundancy ratio, a measure of perfection},
  journal = {Mathematics Magazine},
  volume  = {73},
  number  = {4},
  pages   = {307--310},
  year    = {2000},
  doi     = {10.1080/0025570X.2000.11996860}
}

@article{ryan2003,
  author  = {Ryan, Richard F.},
  title   = {A simpler dense proof regarding the abundancy index},
  journal = {Mathematics Magazine},
  volume  = {76},
  number  = {4},
  pages   = {299--301},
  year    = {2003},
  doi     = {10.1080/0025570X.2003.11953199}
}

@article{holdenerstanton2007,
  author  = {Stanton, William G. and Holdener, Judy A.},
  title   = {Abundancy ``outlaws'' of the form $\frac{\sigma(N)+t}{N}$},
  journal = {Journal of Integer Sequences},
  volume  = {10},
  number  = {9},
  pages   = {Article 07.9.6},
  year    = {2007},
  url     = {https://cs.uwaterloo.ca/journals/JIS/VOL10/Holdener/holdener44.html}
}

@article{holdener2006,
  author  = {Holdener, Judy A.},
  title   = {Conditions equivalent to the existence of odd perfect numbers},
  journal = {Mathematics Magazine},
  volume  = {79},
  number  = {5},
  pages   = {389--391},
  year    = {2006},
  doi     = {10.1080/0025570X.2006.11953437}
}

@article{dris2012,
  author  = {Dris, Jose Arnaldo B.},
  title   = {The abundancy index of divisors of odd perfect numbers},
  journal = {Journal of Integer Sequences},
  volume  = {15},
  number  = {4},
  pages   = {Article 12.4.4},
  year    = {2012},
  url     = {https://cs.uwaterloo.ca/journals/JIS/VOL15/Dris/dris8.html}
}

@article{chenchen2014,
  author  = {Chen, Feng-Juan and Chen, Yong-Gao},
  title   = {On the index of an odd perfect number},
  journal = {Colloquium Mathematicum},
  volume  = {136},
  number  = {1},
  pages   = {41--49},
  year    = {2014},
  doi     = {10.4064/cm136-1-4}
}

@book{hardywright2008,
  author    = {Hardy, G. H. and Wright, E. M.},
  title     = {An Introduction to the Theory of Numbers},
  edition   = {6},
  publisher = {Oxford University Press},
  address   = {Oxford},
  year      = {2008},
  isbn      = {978-0-19-921986-5}
}

@article{sympy2017,
  author  = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and {\v{C}}ert{\'i}k, Ond{\v{r}}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, A. T. and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou{\v{c}}ka, {\v{S}}t{\v{e}}p{\'a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony},
  title   = {{SymPy}: symbolic computing in {P}ython},
  journal = {PeerJ Computer Science},
  volume  = {3},
  pages   = {e103},
  year    = {2017},
  doi     = {10.7717/peerj-cs.103}
}

@article{numpy2020,
  author  = {Harris, Charles R. and Millman, K. Jarrod and van der Walt, St{\'e}fan J. and Gommers, Ralf and Virtanen, Pauli and Cournapeau, David and Wieser, Eric and Taylor, Julian and Berg, Sebastian and Smith, Nathaniel J. and Kern, Robert and Picus, Matti and Hoyer, Stephan and van Kerkwijk, Marten H. and Brett, Matthew and Haldane, Allan and del R{\'i}o, Jaime Fern{\'a}ndez and Wiebe, Mark and Peterson, Pearu and G{\'e}rard-Marchant, Pierre and Sheppard, Kevin and Reddy, Tyler and Weckesser, Warren and Abbasi, Hameer and Gohlke, Christoph and Oliphant, Travis E.},
  title   = {Array programming with {NumPy}},
  journal = {Nature},
  volume  = {585},
  number  = {7825},
  pages   = {357--362},
  year    = {2020},
  doi     = {10.1038/s41586-020-2649-2}
}

@misc{mpmath2023,
  author       = {Johansson, Fredrik and others},
  title        = {mpmath: a {P}ython library for arbitrary-precision floating-point arithmetic (version 1.3.0)},
  year         = {2023},
  howpublished = {\url{https://mpmath.org/}}
}

@article{Kuhnel50,
  title={Versch{\"a}rfung der notwendigen {B}edingungen f{\"u}r die {E}xistenz von ungeraden vollkommenen {Z}ahlen},
  author={K{\"u}hnel, Ullrich},
  journal={Mathematische Zeitschrift},
  volume={52},
  pages={202--211},
  year={1950}
}

@article{SandorAtanassov24,
  title={Some new arithmetic functions},
  author={S{\'a}ndor, J{\'o}zsef and Atanassov, Krassimir T.},
  journal={Notes on Number Theory and Discrete Mathematics},
  volume={30},
  number={4},
  pages={851--856},
  year={2024}
}

@article{Mandal25,
  title={A note on newly introduced arithmetic functions $\varphi^{+}$ and $\sigma^{+}$},
  author={Mandal, Sagar},
  journal={Notes on Number Theory and Discrete Mathematics},
  volume={31},
  number={2},
  pages={404--409},
  year={2025},
  doi={10.7546/nntdm.2025.31.2.404-409}
}