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Variant of a theorem of Erdős on the sum-of-proper-divisors function

2013
*
Mathematics of Computation
*

In 1973,

doi:10.1090/s0025-5718-2013-02775-5
fatcat:tmdbzzjatvdmhg5jtmasxsiyoe
*Erdős*proved that*a*positive proportion*of*numbers are not*of**the*form σ(n) − n,*the**sum**of**the**proper**divisors**of*n. ... We prove*the*analogous result where σ is replaced with*the**sum*-*of*-unitary-*divisors**function*σ * (which*sums**divisors*d*of*n such that (d, n/d) = 1), thus solving*a*problem*of*te Riele from 1976. ... Acknowledgments*The*authors thank Paul Pollack for helping them find some*of**the*references related to Romanov's*theorem*. ...##
###
Shifts of the prime divisor function of Alladi and Erdős
[article]

2017
*
arXiv
*
pre-print

We introduce

arXiv:1710.10875v1
fatcat:zfohxybqhzf7vegk7w3eulljni
*a*variation*on**the*prime*divisor**function*B(n)*of*Alladi and*Erdős*,*a*close relative*of**the**sum**of**proper**divisors**function*s(n). ... We prove that no unbounded sequences occur, analogous to*the*Catalan-Dickson conjecture, and give evidence towards*the*analogue*of**the**Erdős*-Granville-Pomerance-Spiro conjecture*on**the*pre-image*of*s(n ... They can be viewed as*variants**of**the**sum**of**proper**divisors**function*s(n) = d||n d, discussed since Pythagoras. ...##
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Reflections on Paul Erdős on His Birth Centenary, Part II

2015
*
Notices of the American Mathematical Society
*

Paul

doi:10.1090/noti1223
fatcat:oo4u652qrrfizkj22kko6kmwqa
*Erdős*and*the*Probabilistic Method Probabilistic Beginnings*The*probabilistic method is*one**of**the*most significant contributions*of*Paul*Erdős*. ...*The*method is*a*powerful technique with numerous applications in combinatorics, graph theory, additive number theory and geometry.*Theorem*1 ([25], [1]). ... Acknowledgments*The*author takes pleasure in expressing here warm thanks to K. Alladi and R. de la Bretèche for their help during*the*preparation*of*this paper. ...##
###
Sums of proper divisors follow the Erdős–Kac law
[article]

2021
*
arXiv
*
pre-print

Let s(n)=∑_d| n, d<n d denote

arXiv:2106.10756v1
fatcat:whcn5727tvhl7crmp4f2y5gfui
*the**sum**of**the**proper**divisors**of*n. ...*The*same method applies with s(n) replaced by any*of*several other unconventional arithmetic*functions*, such as β(n):=∑_p| n p, n-ϕ(n), and n+τ(n) (τ being*the**divisor**function*). ... Hence,*the**Erdős*-Kac*theorem*for ω ′ (s(n)) is*a*consequence*of**the*corresponding*theorem*for ω(s(n)). ...##
###
Erdős and the Integers

1999
*
Journal of Number Theory
*

Many proofs,

doi:10.1006/jnth.1999.2395
fatcat:zrv3hokrdjfepbbel5vqu2ofom
*variants*, and extensions*of**the**Erdo*s Wintner*theorem*have been given. De la Cal (1992) extended it to*functions*with values in*a*Banach space. ... For an account*of**the*method and further developments see Adhikari (1998) .*Erdo*s and Shapiro (1955) consider*the*related*function**divisors*. _(n) denotes*the**sum**of**divisors**of*n.*A*_ B. ...##
###
An elemental Erdős-Kac theorem for algebraic number fields
[article]

2016
*
arXiv
*
pre-print

Fix

arXiv:1603.05352v1
fatcat:dy3wscucpnealnf74xse4jppiu
*a*number field K. For each nonzero α∈Z_K, let ν(α) denote*the*number*of*distinct, nonassociate irreducible*divisors**of*α. ... Here D, as well as*the*constants*of*proportionality, depend only*on**the*class group*of*K. ... Clark for helpful discussions about arithmetic in Prin(Z K ), and he thanks Carl Pomerance for suggesting*the*consideration*of**the*behavior*of**the*Prin(Z K )*divisor**function*. ...##
###
Some of Erdős' Unconventional Problems in Number Theory, Thirty-four Years Later
[chapter]

2013
*
Bolyai Society Mathematical Studies
*

We shall make use

doi:10.1007/978-3-642-39286-3_23
fatcat:f2fiuzuq3bccpmxrocpm3jjjqa
*of*this extra information later*on*. 2. See [29] for*a*short proof. ...*The*author takes pleasure in expressing here warm thanks to R. Balasubramanian, N. Bingham, R. de la Bretèche, C. Dartyge, I.Z. Ruzsa and T. Stoll for their help during*the*preparation*of*this paper. ...*Erdős*attacks*the*problem from another viewpoint: primitive abundant numbers, i.e. abundant numbers having no abundant*proper**divisor*. ...##
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DIVISOR-SUM FIBERS

2018
*
Mathematika
*

Let $s(\cdot )$ denote

doi:10.1112/s0025579317000535
fatcat:vcnu5y5rqjbkte2irwp4rvhofe
*the**sum*-*of*-*proper*-*divisors**function*, that is, $s(n)=\*sum*_{d\mid n,~d<n}d$ . ... Finally, we make some remarks*on*solutions $n$ to congruences*of**the*form $\unicode[STIX]{x1D70E}(n)\equiv*a*~(\text{mod}~n)$ , proposing*a*modification*of**a*conjecture appearing in recent work*of**the*first ... Acknowledgments*The*first author is supported by*the*National Science Foundation under Grant No. DMS-1402268. ...##
###
Book Review: Intégration et théorie des nombres

1988
*
Bulletin of the American Mathematical Society
*

In 1949,

doi:10.1090/s0273-0979-1988-15648-0
fatcat:pwqeevyne5dxvfhitgeyxu3nti
*Erdös*and Selberg found an elementary proof*of**the*prime number*theorem*,*one*that did not apply*functions**of**a*complex variable. ...*The*work*of**Erdös*and Kac was clarified and extended by Kubilius, who in 1954/55 constructed*a*finite probability space*on*which to model*the*behavior*of*additive arithmetic*functions*by*sums**of*independent ...##
###
On the range of Carmichael's universal-exponent function

2014
*
Acta Arithmetica
*

We also improve

doi:10.4064/aa162-3-6
fatcat:damb2ettqzfbhomhlvayvsvej4
*on*an earlier result*of**the*first 1 Mathematics Subject Classification: 11N37 ... It is closely related to Euler's ϕ-*function*, but we show here that*the*image*of*λ is much denser than*the*image*of*ϕ. ... If v = λ(n) and n is minimal with this property, then for each prime p n, we have λ(n/p)*a**proper**divisor**of*v. Let V λ = λ(N),*the*set*of*all values*of*λ. ...##
###
Powerful amicable numbers

2011
*
Colloquium Mathematicum
*

Let s(n) := d|n, d<n d denote

doi:10.4064/cm122-1-10
fatcat:eammgi5htbgyni42exxfojq7l4
*the**sum**of**the**proper**divisors**of**the*natural number n. ... As shown by*Erdős*and Szekeres in 1935,*the*number*of*-full n ≤ x is asymptotically c x 1/ , as x → ∞. Here c is*a*positive constant depending*on*. ...*The*author thanks Carl Pomerance for helpful comments*on**the*content and presentation*of*this paper. This work was conducted during*a*visit*of**the*author to Dartmouth College. ...##
###
On sets with small additive doubling in product sets

2015
*
Journal of Number Theory
*

In particular, it follows that under this condition

doi:10.1016/j.jnt.2015.04.029
fatcat:sdc3b7txlray7f5zeoom5pxbcm
*the*additive energy*of*B n · B n is asymptotically o(|B n | 6 ), which, in turn, gives*the*classical*Erdős*Multiplication Table*Theorem*as*a*special ...*On**the*other hand, we present examples for which N < π(M ) + M 2/3 . ...*One**of**the*notable examples is*the*special treatment*of*perfect numbers by Euclid in Elements. Recall that*a*number is called perfect if it is equal to*the**sum**of*its*proper**divisors*. ...##
###
Remarks on Fibers of the Sum-of-Divisors Function
[chapter]

2015
*
Analytic Number Theory
*

Let σ denote

doi:10.1007/978-3-319-22240-0_18
fatcat:3dasqsstsneapc6rst6iqe4sjm
*the*usual*sum*-*of*-*divisors**function*. We show that every positive real number can be approximated arbitrarily closely by*a*fraction m/n with σ (m) = σ (n). ... This answers in*the*affirmative*a*question*of**Erdős*. We also show that for almost all*of**the*elements v*of*σ (N),*the*members*of**the*fiber σ −1 (v) all share*the*same largest prime factor. ... Acknowledgements*The*author thanks Kevin Ford for guiding him through*the*arguments*of*[6] . He also thanks*the*referee for useful feedback that led to improvements in*the*exposition. ...##
###
A Spectral Lower Bound for the Divisorial Gonality of Metric Graphs

2015
*
International mathematics research notices
*

Let Γ be

doi:10.1093/imrn/rnv213
fatcat:7ip2ejlia5etrf63yog5vyd6l4
*a*compact metric graph, and denote by ∆*the*Laplace operator*on*Γ with*the*first non-trivial eigenvalue λ1. ... We prove*the*following Yang-Li-Yau type inequality*on*divisorial gonality γ div*of*Γ. ...*A*. likes to thank David Cohen-Steiner for his interest in*the*subject and for fruitful discussion and collaboration*on*related questions. Part*of*this research was conducted while O.*A*. and J. ...##
###
A spectral lower bound for the divisorial gonality of metric graphs
[article]

2014
*
arXiv
*
pre-print

Let Γ be

arXiv:1407.5614v2
fatcat:t76hx7l7jvg6dkmiojgloguvga
*a*compact metric graph, and denote by Δ*the*Laplace operator*on*Γ with*the*first non-trivial eigenvalue λ_1. ... We prove*the*following Yang-Li-Yau type inequality*on*divisorial gonality γ_div*of*Γ. ...*A*. likes to thank David Cohen-Steiner for his interest in*the*subject and for fruitful discussion and collaboration*on*related questions. Part*of*this research was conducted while O.*A*. and J. ...
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