Unit fraction representations using shifted Golomb primes

Rishika Agrawal

REVIEW

Unit fraction representations using shifted Golomb primes

Rishika Agrawal

Number Theory · Submitted 4 Nov 2026 · 33 references

Pith's verdict: A real theorem for a genuinely sparse sequence — the supporting AP and non-smooth estimates are the durable part — but Proposition 6.1's stated hypotheses contradict each other and don't cover the way Section 6 uses it.

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arXiv ID2611.06798 (version 1)

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CitationRishika Agrawal. "Unit fraction representations using shifted Golomb primes." arXiv:2611.06798 (2026). https://pith.science/paper/E7D6AT42

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Editor's note

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Short version: this proves something new for a sequence people have mostly left alone since Erdős, and the machinery underneath it is the part I'd actually cite. There is one formal hole: Proposition 6.1's hypotheses are self-contradictory and don't cover the application in Section 6.

What's new. Golomb primes have relative density 1/log log N among the primes and reciprocal sum log log log N — thin enough that Bloom's shifted-prime theorem genuinely doesn't apply, and the paper says so plainly. Theorem 1.4 (equidistribution in progressions for moduli up to a power of log) and Theorem 1.5 (at most ~26% of shifted Golomb primes have a prime-power divisor above x^0.99) both look correct and reusable. The block decomposition with a shared modulus P of size about (log log N)^2 is a real idea, not a repackaging of Bloom, and the arithmetic hangs together: 0.003 − 0.000777 = 0.002223, summed over 333 log_2 N blocks gives 0.741·log 1000 > 5, and the rounding errors y_j/P are absorbed by b | P. I checked those and they work.

The soft spot. Proposition 6.1 requires N^{0.997} ≤ M ≤ N/10^4, which cannot both hold for any relevant N, and Section 6 applies it with M = N_{j+1} ≈ N_j. Reading Appendix A, I found no step that needs an upper bound on M/N: the minor-arc split |A\A_I| < M/log N, K = M N^{-2/log_2 N}, the qL/M ≤ η/2 estimate, and Lemma A.2's j_0 = ⌊log(N/M)/log 2⌋ all survive at M ≈ N, with j_0 = 0. So I'm fairly confident the upper bound is a typo for M ≤ N. A referee still has to force the fix, because as printed the main theorem does not follow.

Smaller things. The line "condition (iv) holds since R(A_j) ≫ 1/log_2 N_j" is sloppy: it needs 1/log_2 N ≥ (log N)^{-1/101}, which is true only past about exp(exp(1350)). The stress-test calls that a second real gap; I'd call it a missing "for N sufficiently large", since Section 6 already picks N in terms of a, b, h, y. Remark 4.3's unpublished-preprint citation should be cut or flagged as speculation. The AI-use note is unusually explicit and, as far as I can tell, accurate — the suggested contribution was a strategy, and the argument is written out in full.

Who it's for: analytic number theorists working on Egyptian fractions and sieve estimates. The two supporting theorems will likely be cited on their own even by people who ignore the main result.

Recommendation: send it out. The result deserves referee time, Appendix A needs a careful read, and the fix is one line.

Referee report

Significance

If the argument holds, this settles Question 1.1 for a set that is zero density in the primes and whose reciprocal sum is only log log log N, considerably thinner than shifted primes (Bloom) or Pomerance's example, and it is the natural next test case after Bloom's theorem. The by-products have independent value: Theorem 1.4 is the first equidistribution result for Golomb primes in progressions to modulus a power of log x, and Theorem 1.5 gives an explicit positive proportion of smooth shifted Golomb primes, which is the input such sparsity-driven problems need. I credit the paper for reproving the Croot–Bloom–Liu–Sawhney proposition in full in Appendix A rather than citing it, for proving the necessity of the congruence hypotheses at the start of §6, and for stating plainly in §1 that the method does not extend to arbitrary positive-density subsets. There is no code or formalization, so the argument is checked by hand; the numerical margins in Theorems 1.5 and 6 are tight but explicit.

Major objections (2)

[§6 and Proposition 6.1] Proposition 6.1 is stated with the hypotheses N^{0.997} ≤ M ≤ N/10^4, which cannot both hold for any N < 10^{1333}, so as printed the proposition is vacuous. Section 6 applies it with N = N_j and M = N_{j+1} ≈ N_j (the pruning paragraph and the subsequent verification of y_j/P ∈ [R(A_j)/log N_j, (log N_j/(log N_j+1))R(A_j)]), so M ≤ N/10^4 fails by four orders of magnitude and, as written, Theorem 1.3 does not follow from the stated proposition. I checked Appendix A line by line: |A| ≥ M R(A) ≥ N^{0.95} from condition (iv), the m_1 estimate, K = M N^{-2/log_2 N}, the inequality qL/M ≤ η/2, the step K < M < Q used to force Q ∤ x(t), and the decomposition (M,N] ⊆ ∪_j [X_j, 2X_j) in Lemma A.2 with j_0 = ⌊log(N/M)/log 2⌋ = 0 — none of these uses more than N^{0.997} ≤ M and M ≤ N. The upper bound is therefore inert, but a hypothesis error inside the load-bearing proposition has to be corrected in the text: restate the hypothesis as N^{0.997} ≤ M ≤ N (or M ≤ N/2) and say explicitly that Appendix A uses only the lower bound together with M ≤ N.
[§6, paragraph after Eq. (6.1)] The sentence 'condition (iv) holds since R(A_j) ≫ 1/log_2 N_j' does not by itself verify condition (iv). The constant in R(A_j) ≥ (1+o(1))·0.002223/log_2 N_j is absolute, so verifying R(A_j) ≥ (log N_j)^{-1/101} amounts to 0.002223·(log N_j)^{1/101} ≥ log_2 N_j, which fails at, for instance, N = 10^{100} (left side ≈ 0.02, right side ≈ 332) and first holds only near log N ≈ 10^{700}. The claim is true for all sufficiently large N, and §6 already allows 'N large in terms of a, b, h and y', so no new mathematics is required; but the exponent 1/101 should not be inferred from a ≫ estimate, and the paper should insert the threshold explicitly (and note that Proposition 6.1's 'N sufficiently large in terms of h' may be read as 'in terms of h and absolute constants'). I verified that no other step of §6 or of Appendix A imposes an upper bound on N, so this repair is confined to one line.

Minor objections (6)

[§2 outline, §6, Lemma 6.2] The definition N_{j+1} = N e^{-0.003j} is inconsistent with the relation log_2 N_{j+1} = log_2 N_j − 0.003 used in Lemma 6.2 and in §6. With the literal e^{-0.003j} and J = ⌊333 log_2 N⌋ the last block would have N_{J+1} < 1, and the sum Σ_j 0.002223/log_2 N_j would diverge near j ≈ 231 log_2 N. The intended definition is N_{j+1} = N 2^{-0.003j}, which is exactly what the displayed bound (0.741+o(1)) log 1000 in §6 uses; please fix all three occurrences.
[Lemma 3.5 (statement)] The statement sets z = x^{1/4}(log x)^{100}, whereas the proof of the lemma applies [10, Theorem 5.6] with z = x^{1/4}/(log x)^{100} and later quotes z^2 = x^{1/2}/(log x)^{200}; §5 likewise uses the version with division. The statement should read z = x^{1/4}/(log x)^{100}.
[Theorem 1.5 and Lemma 6.6] The constant is asserted to satisfy 0.08e^{2γ}/0.99^2 < 0.259; this is true but the margin is about 10^{-4} (0.08·3.1722/0.9801 = 0.2589). Since Lemma 6.6 uses 1 − 0.259 = 0.741 as an explicit factor, please display the numerical evaluation so the margin can be checked without recomputation.
[Lemma 6.2 (displayed error term)] The error term in the final display appears as O_h(N^{-0.9j}), which depends on j only through the exponent; the argument actually gives h·Σ_{g>N_{j+1}} g^{-2} ≪_h N_{j+1}^{-1}. Please check that the intended expression is O_h(N_j^{-0.9}).
[Remark 4.3] The remark invokes an unrefereed preprint [24] (OpenAI, 2026) claiming a zero-free half-plane Re s > 7/8. The remark is not load-bearing for any theorem, but it should either be labelled as conditional on that preprint or dropped.
[§1, first page] The sentence 'Pomerance [1, 7] constructed another set' cites [1] (Bloom) and [7] (Croot's thesis); the numbers and the attribution should be checked against the intended source.

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The paper

The paper claims

The paper proves that for every fixed nonzero integer $h$, the numbers $g-h$ with $g$ a Golomb prime form a reciprocal basis for the positive rationals: every $a/b$, subject only to the congruence conditions the problem forces, is a…

The reading

This paper asks how thin a set of integers can be and still write every positive rational number as a sum of distinct reciprocals of its members, and answers with the shifted Golomb primes. Golomb primes are produced by running the sieve of Eratosthenes but striking out the residue class $1$ instead of the class $0$, giving $3, 5, 17, 23, 29, 53, \dots$; they are far sparser than the primes, making up only about a $1/\log\log x$ proportion of the primes up to $x$, with reciprocal sum growing like $\log\log\log x$. That sparsity puts them out of reach of the known theorem for shifted primes, which needs positive relative density in the primes. The author proves that for any nonzero fixed $h$ and any coprime $a,b$ satisfying the congruence conditions forced by the problem, every $a/b$ is a finite sum of distinct unit fractions $1/(g-h)$ with $g$ a Golomb prime, and every such $g$ can be taken larger than any preassigned bound. If this holds, the reciprocal-basis phenomenon survives at the very edge of what current machinery allows, and it arrives with a new equidistribution theorem for Golomb primes in short arithmetic progressions, plus a bound saying at least about three quarters of shifted Golomb primes are power-smooth.

What carries the argument

The engine is a Croot-type proposition (Proposition 6.1): for a set $A$ of shifted primes inside an interval $(M,N]$ whose elements are $N^{0.99}$-power-smooth, carry about $\log\log N$ prime factors, spread their reciprocal mass evenly over prime powers, and have $R(A)$ not too small, one can hit every target of the shape $x/Q$ (with $Q=\mathrm{lcm}(A)$) by a sub-sum of reciprocals. The obstruction is that inside a single narrow interval the shifted Golomb primes have reciprocal sum only of order $1/\log\log N$, far too little to reach a fixed fraction. The author's fix is to slice the range into $J\asymp 333\log\log N$ blocks $(N_{j+1},N_j]$ with $N_{j+1}=N_j e^{-0.003}$, each contributing $\Theta(1/\log\log N)$, prune each block to satisfy the proposition using Theorem 1.4 and Theorem 1.5, and then arrange one integer $P$, divisible by $b$ and of size about $(\log\log N)^2$, that divides the lcm of every pruned block. That shared modulus lets the fractions $y_j/P$ be represented block by block and summed to $a/b$ with all denominators distinct.

Extended reading notes

Core claim

The central claim is Theorem 1.3: let $h\neq 0$ and let $a,b\in\mathbb{Z}^+$ with $(a,b)=1$, $(b,h)=1$ and $(b,h-1)$ Golomb-prime-free; then for every $y\ge |h|$ there is a finite set $B$ of Golomb primes, all exceeding $y$, with $a/b=\sum_{g\in B} 1/(g-h)$. The two conditions on $b$ are shown in Section 6 to be necessary, not technical: if a prime $p$ divided both $b$ and $h$ then $p$ would have to equal some $g>|h|\ge p$, and if a Golomb prime $q$ divided both $b$ and $h-1$ then $q\mid g-1$, which the defining sieve forbids. The proof rests on two new distribution results: a Siegel--Walfisz-type asymptotic for $|\{g\le x : g\equiv h \pmod b\}|$ that holds for $b\le(\log x)^A$ with the expected Euler-product correction for Golomb primes dividing $b$, and an upper bound saying that among Golomb primes $g\le x$ in the class $h \bmod b$, at most a proportion $0.259+o(1)$ have $g-h$ divisible by a prime power exceeding $x^{0.99}$.

Load-bearing premise

The deduction applies Proposition 6.1 to blocks whose endpoints satisfy $M=N_{j+1}=N_j e^{-0.003}$, whereas the proposition as printed requires $M\le N/10^4$; the appendix appears to use only $M\ge N^{0.997}$, so the main theorem is formally resting on the proposition holding in that wider range rather than the stated one.

Editorial extensions

If this is right

The reciprocal sum over shifted Golomb primes inherits a basis property: for every nonzero $h$ and every target fraction, one can reach or exceed it using distinct terms $1/(g-h)$, so the associated series diverges and is unbounded on distinct terms.
Golomb primes are equidistributed among the allowed residue classes $h \bmod b$ for $b\le(\log x)^A$, with the same $1/\varphi(b)$ factor as the primes and an explicit correction $\prod_{q\mid b, q\in G}(1-1/(q-1))^{-1}$ for Golomb primes dividing the modulus.
At least a positive proportion, roughly $0.741$ of the expected count, of shifted Golomb primes in a progression remain after discarding those with a prime-power divisor above $x^{0.99}$, which is what makes the Croot-type proposition applicable at all.
A remark in the paper observes that a recently announced zero-free half-plane $\mathrm{Re}\,s>7/8$ for Dirichlet $L$-functions would push the equidistribution theorem from $b\le(\log x)^A$ to $b\le x^{1/8-2\varepsilon}$ with the same method.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Golomb primes are one instance of a family: primes sieved by forbidding one residue class modulo each earlier member. The same three ingredients used here (equidistribution in progressions, a bound on non-smooth shifts, and a shared modulus across blocks) should give reciprocal-basis theorems for other sieves of this shape, so the argument reads as a template rather than a one-off.
The author states the method does not extend to an arbitrary positive-density subset of the Golomb primes, which leaves an obvious gradient to test: whether a random subset of the Golomb primes of relative density $(\log\log x)^{-\theta}$ still represents every rational, and where the threshold sits.
The common-modulus step is the part most likely to generalise beyond this paper's setting: any sparse set whose elements' least common multiples can be forced to carry a fixed modulus of size $\mathrm{polylog}\log$ can be assembled block by block, which suggests a reusable criterion independent of the specific arithmetic of Golomb primes.
A concrete check a reader can run: take the smallest cases, such as $h=1$, $a/b=1/2$, and search for short explicit subsets of $\{g-1\}$ summing to it, to see the block structure and modulus in action at small scale.

Original abstract

We prove that every positive rational number is a sum of distinct unit fractions with denominators of the form $g-h$, where $g$ is a Golomb prime and $h$ is a fixed nonzero integer, as long as the necessary congruence conditions are satisfied. Golomb primes, introduced by Golomb in 1955, are a natural analogue of the primes. Erd\H{o}s showed that the relative density of Golomb primes among the primes up to $N$ is about $1/\log\log N$, so they form a sparse subset of the primes. Our result is an analogue of Bloom's result on unit fraction representations using shifted primes, for this sparse subset of the primes. A key ingredient in our proof is a Siegel--Walfisz-type theorem for Golomb primes in arithmetic progressions, which we also prove here.

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We prove that every positive rational number is a sum of distinct unit fractions with denominators of the form $g-h$, where $g$ is a Golomb prime and $h$ is a fixed nonzero integer, as long as the necessary congruence conditions are satisfied. Golomb primes, introduced by Golomb in 1955, are a natural analogue of the primes. Erd\H{o}s showed that the relative density of Golomb primes among the primes up to $N$ is about $1/\log\log N$, so they form a sparse subset of the primes. Our result is an analogue of Bloom's result on unit fraction representations using shifted primes, for this sparse subset of the primes. A key ingredient in our proof is a Siegel--Walfisz-type theorem for Golomb primes in arithmetic progressions, which we also prove here.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove that every positive rational number is a sum of distinct unit fractions with denominators of the form $g-h$, where $g$ is a Golomb prime and $h$ is a fixed nonzero integer, as long as the necessary congruence conditions are satisfied.	result 0.86	0.96	0.96	asserted 1.00
s1	Golomb primes, introduced by Golomb in 1955, are a natural analogue of the primes.	background 1.00	0.04	0.01	asserted 1.00
s2	Erd\H{o}s showed that the relative density of Golomb primes among the primes up to $N$ is about $1/\log\log N$, so they form a sparse subset of the primes.	background 1.00	0.03	0.02	asserted 0.99
s3	Our result is an analogue of Bloom's result on unit fraction representations using shifted primes, for this sparse subset of the primes.	interpretation 0.85	0.63	0.08	asserted 1.00
s4	A key ingredient in our proof is a Siegel--Walfisz-type theorem for Golomb primes in arithmetic progressions, which we also prove here.	method 0.68	0.96	0.03	asserted 1.00

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References

Works cited (33)

[1]

T. F. Bloom,On a density conjecture about unit fractions, Journal of the European Mathematical Society27 (2025), no. 11, 4563–4589. With an appendix by T. F. Bloom and B. Mehta

View on Pith 2025
[2]

,Unit fractions with shifted prime denominators, Proceedings of the Royal Society of Edinburgh Section A: Mathematics155(2025), no. 6, 2285–2295

View on Pith 2025
[3]

T. F. Bloom and C. Elsholtz,Egyptian fractions, Nieuw Archief voor Wiskunde. Vijfde Serie23(2022), no. 4, 237–245

View on Pith 2022
[4]

E. Bombieri,On the large sieve, Mathematika12(1965), 201–225

View on Pith 1965
[5]

R. Breusch,A special case of Egyptian fractions, American Mathematical Monthly61(1954), no. 3, 200–201. Solution to Advanced Problem 4512, proposed by E. P. Starke

View on Pith 1954
[6]

S. Butler, P. Erd˝ os, and R. Graham,Egyptian fractions with each denominator having three distinct prime divisors, Integers15(2015). Paper No. A51, 9 pp

View on Pith 2015
[7]

E. S. Croot III,Unit fractions, Ph.D. Thesis, Athens, Georgia, 2000

View on Pith 2000
[8]

,On a coloring conjecture about unit fractions, Annals of Mathematics. Second Series157(2003), no. 2, 545–556

View on Pith 2003
[9]

H. Davenport,Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised by H. L. Montgomery

View on Pith 2000
[10]

H. G. Diamond and H. Halberstam,A higher-dimensional sieve method, Cambridge Tracts in Mathematics, vol. 177, Cambridge University Press, Cambridge, 2008. With W. F. Galway

View on Pith 2008
[11]

D. Eppstein,Egyptian fractions with denominators from sequences closed under doubling, Journal of Integer Sequences24(2021), no. 8

View on Pith 2021
[12]

P. Erd˝ os,On a problem of G. Golomb, Journal of the Australian Mathematical Society2(1961), no. 1, 1–8

View on Pith 1961
[13]

P. Erd˝ os and R. L. Graham,Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Math´ ematique, vol. 28, Universit´ e de Gen` eve, 1980

View on Pith 1980
[14]

P. Erd˝ os,Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.)12(1948), 75–78

View on Pith 1948
[15]

K. Ford,Correction to: A Hardy–Ramanujan-type inequality for shifted primes and sifted sets, Lithuanian Mathematical Journal62(2022), no. 1, 133

View on Pith 2022
[16]

S. W. Golomb,Sets of primes with intermediate density, Mathematica Scandinavica3(1955), 264–274

View on Pith 1955
[17]

R. L. Graham,On finite sums of unit fractions, Proceedings of the London Mathematical Society. Third Series 14(1964), no. 2, 193–207

View on Pith 1964
[18]

R. L. Graham,Paul Erd˝ os and Egyptian fractions, Erd˝ os centennial, 2013, pp. 289–309

View on Pith 2013
[19]

H. Halberstam and H.-E. Richert,Sieve methods, London Mathematical Society Monographs, Academic Press, London–New York, 1974

View on Pith 1974
[20]

W. Hoeffding,Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association58(1963), no. 301, 13–30

View on Pith 1963
[21]

D. Koukoulopoulos,The distribution of prime numbers, Graduate Studies in Mathematics, vol. 203, American Mathematical Society, Providence, RI, 2019

View on Pith 2019
[22]

Y. P. Liu and M. Sawhney,On further questions regarding unit fractions, Int. Math. Res. Not.2026(2026), no. 2, rnaf382

View on Pith 2026
[23]

H. L. Montgomery and R. C. Vaughan,Multiplicative number theory I: Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR2378655

View on Pith 2007
[24]

OpenAI,The quasi-Riemann hypothesis: A zero-free half-plane Re(s) > 7/8, 2026. Preprint, September 30, 2026

View on Pith 2026
[25]

P. Pollack,Numbers which are orders only of cyclic groups, Proc. Amer. Math. Soc.150(2022), no. 2, 515–524

View on Pith 2022
[26]

L. E. Sigler,Fibonacci’s Liber Abaci: A translation into modern English of Leonardo Pisano’s book of calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Springer-Verlag, New York, 2002

View on Pith 2002
[27]

B. M. Stewart,Sums of distinct divisors, American Journal of Mathematics76(1954), no. 4, 779–785

View on Pith 1954
[28]

J. Stucky,The distribution ofgcd(n, φ(n)), Acta Arithmetica219(2025), no. 2, 169–197

View on Pith 2025
[29]

Z.-W. Sun,Conjectures on representations involving primes, Combinatorial and additive number theory II, 2017, pp. 279–310

View on Pith 2017
[30]

P. J. van Albada and J. H. van Lint,Reciprocal bases for the integers, American Mathematical Monthly70(1963), no. 2, 170–174

View on Pith 1963
[31]

W. van Doorn and V. Kovaˇ c,Lacunary sequences whose reciprocal sums represent all rational numbers in an interval, Acta Arithmetica223(2026), no. 3, 275–295

View on Pith 2026
[32]

A. I. Vinogradov,The density hypothesis for Dirichlet L-series, Izv. Akad. Nauk SSSR Ser. Mat.29(1965), 903–934. (Russian); Corrigendum, ibid. 30 (1966), 719–720. 34 RISHIKA AGRA W AL

View on Pith 1965
[33]

H. S. Wilf,Research problem 6: Reciprocal bases for integers, Bulletin of the American Mathematical Society67 (1961), no. 5, 456. Department of Mathematics, University of Georgia, Athens, GA 30602, USA Email address:rishika.agrawal@uga.edu

View on Pith 1961

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

T. F. Bloom,On a density conjecture about unit fractions, Journal of the European Mathematical Society27 (2025), no. 11, 4563–4589. With an appendix by T. F. Bloom and B. Mehta

View on Pith 2025

[2] [2]

,Unit fractions with shifted prime denominators, Proceedings of the Royal Society of Edinburgh Section A: Mathematics155(2025), no. 6, 2285–2295

View on Pith 2025

[3] [3]

T. F. Bloom and C. Elsholtz,Egyptian fractions, Nieuw Archief voor Wiskunde. Vijfde Serie23(2022), no. 4, 237–245

View on Pith 2022

[4] [4]

E. Bombieri,On the large sieve, Mathematika12(1965), 201–225

View on Pith 1965

[5] [5]

R. Breusch,A special case of Egyptian fractions, American Mathematical Monthly61(1954), no. 3, 200–201. Solution to Advanced Problem 4512, proposed by E. P. Starke

View on Pith 1954

[6] [6]

S. Butler, P. Erd˝ os, and R. Graham,Egyptian fractions with each denominator having three distinct prime divisors, Integers15(2015). Paper No. A51, 9 pp

View on Pith 2015

[7] [7]

E. S. Croot III,Unit fractions, Ph.D. Thesis, Athens, Georgia, 2000

View on Pith 2000

[8] [8]

,On a coloring conjecture about unit fractions, Annals of Mathematics. Second Series157(2003), no. 2, 545–556

View on Pith 2003

[9] [9]

H. Davenport,Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised by H. L. Montgomery

View on Pith 2000

[10] [10]

H. G. Diamond and H. Halberstam,A higher-dimensional sieve method, Cambridge Tracts in Mathematics, vol. 177, Cambridge University Press, Cambridge, 2008. With W. F. Galway

View on Pith 2008

[11] [11]

D. Eppstein,Egyptian fractions with denominators from sequences closed under doubling, Journal of Integer Sequences24(2021), no. 8

View on Pith 2021

[12] [12]

P. Erd˝ os,On a problem of G. Golomb, Journal of the Australian Mathematical Society2(1961), no. 1, 1–8

View on Pith 1961

[13] [13]

P. Erd˝ os and R. L. Graham,Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Math´ ematique, vol. 28, Universit´ e de Gen` eve, 1980

View on Pith 1980

[14] [14]

P. Erd˝ os,Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.)12(1948), 75–78

View on Pith 1948

[15] [15]

K. Ford,Correction to: A Hardy–Ramanujan-type inequality for shifted primes and sifted sets, Lithuanian Mathematical Journal62(2022), no. 1, 133

View on Pith 2022

[16] [16]

S. W. Golomb,Sets of primes with intermediate density, Mathematica Scandinavica3(1955), 264–274

View on Pith 1955

[17] [17]

R. L. Graham,On finite sums of unit fractions, Proceedings of the London Mathematical Society. Third Series 14(1964), no. 2, 193–207

View on Pith 1964

[18] [18]

R. L. Graham,Paul Erd˝ os and Egyptian fractions, Erd˝ os centennial, 2013, pp. 289–309

View on Pith 2013

[19] [19]

H. Halberstam and H.-E. Richert,Sieve methods, London Mathematical Society Monographs, Academic Press, London–New York, 1974

View on Pith 1974

[20] [20]

W. Hoeffding,Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association58(1963), no. 301, 13–30

View on Pith 1963

[21] [21]

D. Koukoulopoulos,The distribution of prime numbers, Graduate Studies in Mathematics, vol. 203, American Mathematical Society, Providence, RI, 2019

View on Pith 2019

[22] [22]

Y. P. Liu and M. Sawhney,On further questions regarding unit fractions, Int. Math. Res. Not.2026(2026), no. 2, rnaf382

View on Pith 2026

[23] [23]

H. L. Montgomery and R. C. Vaughan,Multiplicative number theory I: Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR2378655

View on Pith 2007

[24] [24]

OpenAI,The quasi-Riemann hypothesis: A zero-free half-plane Re(s) > 7/8, 2026. Preprint, September 30, 2026

View on Pith 2026

[25] [25]

P. Pollack,Numbers which are orders only of cyclic groups, Proc. Amer. Math. Soc.150(2022), no. 2, 515–524

View on Pith 2022

[26] [26]

L. E. Sigler,Fibonacci’s Liber Abaci: A translation into modern English of Leonardo Pisano’s book of calculation, Sources and Studies in the History of Mathematics and Physical Sciences, Springer-Verlag, New York, 2002

View on Pith 2002

[27] [27]

B. M. Stewart,Sums of distinct divisors, American Journal of Mathematics76(1954), no. 4, 779–785

View on Pith 1954

[28] [28]

J. Stucky,The distribution ofgcd(n, φ(n)), Acta Arithmetica219(2025), no. 2, 169–197

View on Pith 2025

[29] [29]

Z.-W. Sun,Conjectures on representations involving primes, Combinatorial and additive number theory II, 2017, pp. 279–310

View on Pith 2017

[30] [30]

P. J. van Albada and J. H. van Lint,Reciprocal bases for the integers, American Mathematical Monthly70(1963), no. 2, 170–174

View on Pith 1963

[31] [31]

W. van Doorn and V. Kovaˇ c,Lacunary sequences whose reciprocal sums represent all rational numbers in an interval, Acta Arithmetica223(2026), no. 3, 275–295

View on Pith 2026

[32] [32]

A. I. Vinogradov,The density hypothesis for Dirichlet L-series, Izv. Akad. Nauk SSSR Ser. Mat.29(1965), 903–934. (Russian); Corrigendum, ibid. 30 (1966), 719–720. 34 RISHIKA AGRA W AL

View on Pith 1965

[33] [33]

H. S. Wilf,Research problem 6: Reciprocal bases for integers, Bulletin of the American Mathematical Society67 (1961), no. 5, 456. Department of Mathematics, University of Georgia, Athens, GA 30602, USA Email address:rishika.agrawal@uga.edu

View on Pith 1961