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

Carl Pomerance, Hee-Sung Yang
2013 Mathematics of Computation  
In 1973, 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.  ... 
doi:10.1090/s0025-5718-2013-02775-5 fatcat:tmdbzzjatvdmhg5jtmasxsiyoe

Shifts of the prime divisor function of Alladi and Erdős [article]

Snehal Shekatkar, Tian An Wong
2017 arXiv   pre-print
We introduce 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.  ... 
arXiv:1710.10875v1 fatcat:zfohxybqhzf7vegk7w3eulljni

Reflections on Paul Erdős on His Birth Centenary, Part II

Krishnaswami Alladi, Steven Krantz, Noga Alon, D. A. Goldston, András Sárközy, József Szabados, Gérald Tenenbaum, Stephan Ramon Garcia, Amy L. Shoemaker
2015 Notices of the American Mathematical Society  
Paul 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.  ... 
doi:10.1090/noti1223 fatcat:oo4u652qrrfizkj22kko6kmwqa

Sums of proper divisors follow the Erdős–Kac law [article]

Paul Pollack, Lee Troupe
2021 arXiv   pre-print
Let s(n)=∑_d| n, d<n d denote 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)).  ... 
arXiv:2106.10756v1 fatcat:whcn5727tvhl7crmp4f2y5gfui

Erdős and the Integers

Imre Z. Ruzsa
1999 Journal of Number Theory  
Many proofs, 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.  ... 
doi:10.1006/jnth.1999.2395 fatcat:zrv3hokrdjfepbbel5vqu2ofom

An elemental Erdős-Kac theorem for algebraic number fields [article]

Paul Pollack
2016 arXiv   pre-print
Fix 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.  ... 
arXiv:1603.05352v1 fatcat:dy3wscucpnealnf74xse4jppiu

Some of Erdős' Unconventional Problems in Number Theory, Thirty-four Years Later [chapter]

Gérald Tenenbaum
2013 Bolyai Society Mathematical Studies  
We shall make use 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.  ... 
doi:10.1007/978-3-642-39286-3_23 fatcat:f2fiuzuq3bccpmxrocpm3jjjqa

DIVISOR-SUM FIBERS

Paul Pollack, Carl Pomerance, Lola Thompson
2018 Mathematika  
Let $s(\cdot )$ denote 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.  ... 
doi:10.1112/s0025579317000535 fatcat:vcnu5y5rqjbkte2irwp4rvhofe

Book Review: Intégration et théorie des nombres

P. D. T. A. Elliott
1988 Bulletin of the American Mathematical Society  
In 1949, 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  ... 
doi:10.1090/s0273-0979-1988-15648-0 fatcat:pwqeevyne5dxvfhitgeyxu3nti

On the range of Carmichael's universal-exponent function

Florian Luca, Carl Pomerance
2014 Acta Arithmetica  
We also improve 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.  ... 
doi:10.4064/aa162-3-6 fatcat:damb2ettqzfbhomhlvayvsvej4

Powerful amicable numbers

Paul Pollack
2011 Colloquium Mathematicum  
Let s(n) := d|n, d<n d denote 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.  ... 
doi:10.4064/cm122-1-10 fatcat:eammgi5htbgyni42exxfojq7l4

On sets with small additive doubling in product sets

Dmitrii Zhelezov
2015 Journal of Number Theory  
In particular, it follows that under this condition 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.  ... 
doi:10.1016/j.jnt.2015.04.029 fatcat:sdc3b7txlray7f5zeoom5pxbcm

Remarks on Fibers of the Sum-of-Divisors Function [chapter]

Paul Pollack
2015 Analytic Number Theory  
Let σ denote 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.  ... 
doi:10.1007/978-3-319-22240-0_18 fatcat:3dasqsstsneapc6rst6iqe4sjm

A Spectral Lower Bound for the Divisorial Gonality of Metric Graphs

Omid Amini, Janne Kool
2015 International mathematics research notices  
Let Γ be 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.  ... 
doi:10.1093/imrn/rnv213 fatcat:7ip2ejlia5etrf63yog5vyd6l4

A spectral lower bound for the divisorial gonality of metric graphs [article]

Omid Amini, Janne Kool
2014 arXiv   pre-print
Let Γ be 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.  ... 
arXiv:1407.5614v2 fatcat:t76hx7l7jvg6dkmiojgloguvga
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