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Comptes Rendus
Mathématique
Jared Duker Lichtman
Translated sums of primitive sets
Volume 360 (2022), p. 409-414
<https://doi.org/10.5802/crmath.285>
© Académie des sciences, Paris and the authors, 2022.
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Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
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Comptes Rendus
Mathématique
2022, 360, p. 409-414
https://doi.org/10.5802/crmath.285
Combinatorics, Number theory / Combinatoire, Théorie des nombres
Translated sums of primitive sets
Jared Duker Lichtman∗, a
a Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
E-mail: jared.d.lichtman@gmail.com
Abstract. The Erdős primitive set conjecture states that the sum f (A) = ∑a∈ A 1a log a , ranging over any
primitive set A of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and
Mechik proved that the translated Erdős conjecture for the sum f (A,h) = ∑a∈ A 1a(log a+h) is false starting
at h = 81, by comparison with semiprimes. In this note we prove that such falsehood occurs already at
h = 1.04 · · · , and show this translate is best possible for semiprimes. We also obtain results for translated sums
of k-almost primes with larger k.
Mathematical subject classification (2010). 11N25, 11Y60, 11A05, 11M32.
Funding. The author is supported by a Clarendon Scholarship at the University of Oxford.
Manuscript received 30th August 2021, revised 14th October 2021, accepted 19th October 2021.
1. Introduction
A set A ⊂ Z>1 of positive integers is called primitive if no member divides another (we trivially
exclude the singleton {1}). An important family of examples is the set Nk of k-almost primes,
that is, numbers with exactly k prime factors counted with multiplicity. For example, k = 1,2
correspond to the sets of primes and semiprimes, respectively.
In 1935 Erdős [3] proved that f (A) = ∑a∈ A 1a log a converges uniformly for any primitive A. In
1988 he further conjectured that the maximum of f (A) is attained by the primes A = N1. One
may directly compute f (N1) = 1.636 · · · , whereas the best known bound is f (A) < eγ = 1.781 · · ·
for any primitive A [8]. As a special case, Zhang [9] proved that the primes maximize f (Nk ), that
is, f (N1) > f (Nk ) for all k > 1.
One may pose a translated analogue of the Erdős conjecture, namely, the maximum of
f (A,h) = ∑a∈ A 1a(log a+h) is attained by the primes A =N1. Recently Laib, Derbal, and Mechik [5]
proved that this translated conjecture is false, by showing f (N1,h) < f (N2,h) for all h ≥ 81. Their
proof method is direct, by studying partial sum truncations of f (A,h). (Laib [4] very recently
announced a bound h ≥ 60, as a refinement of the same method.)
By different methods, we extend the range of such falsehood down to h > 1.04 · · · , and show
this translate is best possible for semiprimes.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
410 J. D. Lichtman
Theorem 1. Let P (s) =∑p p−s denote the prime zeta function. We have f (N1,h) < f (N2,h) if and
only if h > h2, where t = h2 = 1.04 · · · is the unique real solution to∫ ∞
1
[
P (s)− 1
2
(
P (s)2 +P (2s))]e(1−s)t ds = 0,
Moreover f (N1,h) < f (Nk ,h) for all k sufficiently large, if and only if h > 0.277 · · · .
This suggests that the Erdős conjecture, if true, is only barely so. Moreover, for the same value
h = h2 = 1.04 · · · we show the primes minimize f (Nk ,h), which may be viewed as the inverse
analogue of Zhang’s maximization result.
Theorem 2. For h2 = 1.04 · · · , we have f (N1,h2) < f (Nk ,h2) for all k > 1.
2. Proof of Theorem 1
We introduce the zeta function for k-almost primes Pk (s) =
∑
n∈Nk n
−s , whose relevance to us
arises from the identity,
f (Nk ,h) =
∑
n∈Nk
1
n(logn +h) =
∑
n∈Nk
1
n log
(
neh
)
= ∑
n∈Nk
eh
∫ ∞
1
(
neh
)−s
ds =
∫ ∞
1
Pk (s)e
(1−s)h ds .
(1)
Here the interchange of sum and integral holds by Tonelli’s theorem, since f (Nk ,h) ≤ f (Nk )
converges uniformly after Erdős. The significance of the identity (1) was first observed when k = 1,
h = 0 by H. Cohen [2, p. 6], who rapidly computed f (N1) = 1.636616 · · · to 50 digits accuracy.
By comparison, the direct approach by partial sums
∑
p ≤x 1/(p log p) converge too slowly, i.e.
O(1/log x). Similarly for k > 1, we shall see (1) leads to sharper results.
Note one has P1(s) = P (s) and P2(s) = 12 P (s)2 + 12 P (2s), as well as
P3(s) = 1
6
P (s)3 + 1
2
P (s)P (2s)+ 1
3
P (3s),
P4(s) = 1
24
P (s)4 + 1
4
P (s)2P (2s)+ 1
8
P (2s)2 + 1
3
P (s)P (3s)+ 1
4
P (4s).
In general for k ≥ 1, [6, Proposition 3.1] gives an explicit formula for Pk in terms of P ,
Pk (s) =
∑
n1 +2n2 +···=k
∏
j ≥1
1
n j !
(
P
(
j s
)
/ j
)n j . (2)
Here the above sum ranges over all partitions of k. Also see [7, Proposition 2.1].
In practice, we may rapidly compute Pk (and P
′
k ) using recursion relations.
Lemma 3. For k ≥ 1 let Pk (s) =
∑
Ω(n)=k n−s and P1(s) = P (s) =
∑
p p
−s . We have
Pk (s) =
1
k
k∑
j=1
Pk− j (s)P
(
j s
)
and P ′k (s) =
k∑
j=1
Pk− j (s)P ′
(
j s
)
. (3)
Proof. The recursion (3) for Pk is given in [6, Proposition 3.1], and is equivalent to (2).
The recursion (3) for P ′k is obtained by differentiating (2),
P ′k (s) =
∑
n1 +2n2 +···=k
∑
i ≤k
P ′(i s)
(P (i s)/i )ni−1
(ni −1)!
∏
j 6= i
1
n j !
(
P
(
j s
)
/ j
)n j
= ∑
i ≤k
P ′(i s)
∑
n1 +2n2 +···=k−i
∏
j ≥1
1
n j !
(
P
(
j s
)
/ j
)n j = ∑
i ≤k
P ′(i s)Pk−i (s).

C. R. Mathématique — 2022, 360, 409-414
J. D. Lichtman 411
As observed empirically in [1], the Dirichlet series P2(s)−P (s) = 12 [P (s)2 +P (2s)]−P (s) has a
unique root at s =σ= 1.1403 · · · , through which it passes from positive to negative. We prove this
more generally for k ≤ 20.
Lemma 4. For 2 ≤ k ≤ 20, the Dirichlet series Pk (s)−P (s) has a unique root at s =σk > 1, through
which it passes from positive to negative.
Proof. For each k ≥ 1, Pk (s) is monotonically decreasing in s > 1. As such there is a unique sk > 1
for which P (s) passes through (k !)1/(k−1) from above. Using the main term in (2), Pk (s) > P (s)k /k !
which is larger than P (s) iff P (s)k−1 > k ! iff s < sk by definition. That is,
Pk (s) > P (s) > 0 for s ∈ (1, sk ) . (4)
Also there is a unique s′k > 1 for which Pk−1(s) passes through 1 from above. Using the first term
in the recursion (3), −P ′k (s) >−P ′(s)Pk−1(s) which is larger than −P ′(s) > 0 iff Pk−1(s) > 1 iff s < s′k
by definition. That is,
P ′k (s) < P ′(s) < 0 for s ∈
(
1, s′k
)
. (5)
For c < 2k , Pk (s)c s is monotonically decreasing in s > 1, and so is Pk (s)/(2−s + 3−s ) =
{1/[Pk (s)2
s ] + 1/[Pk (s)3s ]}−1. Thus there is a unique tk > 1 for which Pk (s)/(2−s + 3−s ) passes
through 1 from above. Now by definition P (s)/(2−s+3−s ) > 1 = Pk (tk )/(2−tk +3−tk ), which is larger
than Pk (s)/(2
−s +3−s ) iff s > tk by monotonicity. That is,
0 < Pk (s) < P (s) for s ∈ (tk ,∞) . (6)
In summary Pk −P has at most one root in (1, s′k ), and no roots in (1, sk )∪ (tk ,∞). We directly
compute that sk < tk < s′k for 2 ≤ k ≤ 20, and so Pk (s)−P (s) has a unique root σk ∈ (sk , tk ) as
claimed. 
We deduce the following corollary, which gives (the first part of) Theorem 1 when k = 2.
Corollary 5. For 2 ≤ k ≤ 20, f (Nk ,h)− f (N1,h) has a unique root at hk > 0, through which it
passes from negative to positive.
Proof. For h ≥ 0, recall f (Nk ,h) =
∫ ∞
1 Pk (s)e
(1−s)h ds by (1). Now by Lemma 4, Pk (s)−P (s) passes
from positive to negative at s =σk > 1. Thus[
f
(
Nk ,h
)− f (N1,h)]e(σk−1)h = ∫ ∞
1
[
Pk (s)−P (s)
]
e(σk−s)h ds
=
∫ σk
1
[
Pk (s)−P (s)
]
e(σk−s)h ds −
∫ ∞
σk
[
P (s)−Pk (s)
]
e(σk−s)h ds
is difference of two integrals with positive integrands, which are mononotically increasing and
decreasing in h ≥ 0, respectively. Hence the difference is mononotically increasing in h ≥ 0. And
f (Nk ,0)− f (N1,0) < 0 by Zhang [9], so the result follows. 
For the second part of Theorem 1, note for any fixed h ≥ 0 we have logn +h ∼ logn for n ∈Nk
as k →∞, since n ≥ 2k . Thus f (Nk ,h) ∼ f (Nk ,0) as k →∞. Hence by [6, Theorem 2.2],
lim
k →∞
f
(
Nk ,h
)= lim
k →∞
f
(
Nk
)= 1.
Note 1− f (N1,h) passes from negative to positive at a unique root h∞ = 0.277 · · · . So for each
h > h∞, we have f (Nk ,h)− f (N1,h) > 0 for k sufficiently large (and similarly for the converse).
This completes the proof of Theorem 1.
C. R. Mathématique — 2022, 360, 409-414
412 J. D. Lichtman
k sk tk s
′
k σk hk
2 1.11313 1.40678 1.39943 1.14037 1.04466
3 1.06861 1.23367 1.25922 1.09224 0.98213
4 1.04306 1.15231 1.17696 1.06206 0.93018
5 1.02761 1.104 1.12386 1.04231 0.89038
6 1.01795 1.07259 1.08784 1.02907 0.86146
7 1.01179 1.05125 1.06272 1.02007 0.84126
8 1.00779 1.0364 1.04493 1.0139 0.8276
9 1.00518 1.02594 1.03223 1.00964 0.8186
10 1.00346 1.0185 1.02312 1.0067 0.8128
11 1.00231 1.0132 1.01658 1.00466 0.80915
12 1.00155 1.00942 1.01187 1.00325 0.80689
13 1.00105 1.00672 1.00849 1.00226 0.80551
14 1.0007 1.00479 1.00607 1.00158 0.8047
15 1.00048 1.00341 1.00433 1.0011 0.8042
16 1.00032 1.00243 1.00309 1.00077 0.80391
17 1.00022 1.00173 1.0022 1.00053 0.80374
18 1.00015 1.00123 1.00157 1.00037 0.80365
19 1.0001 1.00087 1.00112 1.00026 0.80359
20 1.00007 1.00062 1.00079 1.00018 0.80356
2.1. Computations
For k ≤ 20, we compute the unique roots σk and hk of Pk (s) − P (s) and f (Nk ,h) − f (N1,h),
respectively, as well as verify that the auxiliary parameters (as defined in Lemma 4) satisfy
sk <σk < tk < s′k . We similarly compute the root of 1− f (N1,h) as h∞ = 0.277 · · · .
In our computations, we express Pk in terms of P using the recursion in (3). In turn by Möbius
inversion P (s) = ∑m≥1(µ(m)/m) logζ(ms), so P is obtained via well-known rapid computation
of ζ. Finally, we compute f (Nk ,h) from its integral form (1). The data are displayed in the table
below, obtained using Mathematica (for technical convenience, we first compute yk = loghk ) 1.
We believe a unique of root hk exists as in Corollary 5 for all k > 20 as well. This would enable
a strengthening of Theorem 2 to f (Nk ,h) > f (N1,h) for all values of k > 1, h ≥ h2 (so far we only
establish this for {1 < k ≤ 20, h ≥ h2} or {k > 1,h = h2}). Uniqueness of hk would follow if σk is
unique, as in Lemma 4, for all k. In turn it would suffice to show tk < s′k for all k (note sk < tk holds
automatically by (4), (6)), though it is not clear how to establish such an inequality in general.
Moreover, it appears both hk and σk are monotonically decreasing in k. This may be related
to some empirical trends for f (Nk ), found in a recent disproof of a conjecture of Banks–Martin,
see [1, 6].
1P[k_Integer,s_]:= If[k==1,PrimeZetaP[s],
Expand[(Sum[P[1,j*s]*P[k-j,s],j,1,k-1]+P[1,k*s])/k]]
FindRoot[P[1, s] == (k!)ˆ(1/(k - 1)), s, 1 + 1/kˆ3]
FindRoot[P[k, t]/(2ˆ(-t) + 3ˆ(-t)) == 1, t, 1 + 1/kˆ3]
FindRoot[P[k - 1, s1] == 1, s1, 1 + 1/kˆ3]
FindRoot[P[k, sigma] == P[1, sigma], sigma, 1 + 1/kˆ3]]
FindRoot[NIntegrate[(P[1, s] - P[k, s])/yˆs,s,1,Infinity, WorkingPrecision->30,
AccuracyGoal->13, PrecisionGoal->13], y, 1 + 1/kˆ3]
C. R. Mathématique — 2022, 360, 409-414
J. D. Lichtman 413
3. Proof of Theorem 2
Proof. We have already verified the claim directly for k ≤ 20, since in this case hk ≤ h2 = 1.04 · · · .
For k > 20, the proof strategy is similar to that of [6, Theorem 5.5]. That is, the integral
f (Nk ,h) =
∫ ∞
1 Pk (s)e
(1−s)h ds has its mass concentrated near 1 as k →∞, so it suffices to truncate
the integration to [1,1.01] say, as a lower bound. Thus by (1),
f
(
Nk ,h2
)= ∫ ∞
1
Pk (s)e
(1−s)h2 ds > e−.01h2
∫ 1.01
1
Pk (s)ds . (7)
Next, we may lower bound Pk (s) by P (s)
k /k !, which constitutes the first of the terms in the
identity (2), one per partition of k. Note the terms of partitions built from small parts contribute
the most mass. So by also including the terms for the partitions k = 1 · (k − j ) + j and k =
1 · (k − j −2)+2+ j for j ≤ 6, we shall obtain a sufficiently tight lower bound to deduce the result.
Indeed, we have∫ 1.01
1
Pk (s)ds >
1
k !
∫ 1.01
1
P (s)k ds +
6∑
j=2
∫ 1.01
1 P (s)
k− j P
(
j s
)
ds
j
(
k − j )!
+
∫ 1.01
1 P (s)
k−4P (2s)2 ds
2!22(k −4)! +
6∑
j=3
∫ 1.01
1 P (s)
k− j−2P (2s)P
(
j s
)
ds
2 j
(
k − j −2)! . (8)
From [6, (5.10)], we have
0 < P (s)− log
( α
s −1
)
< 1.4(s −1), for s ∈ [1,2], (9)
where α= exp(−∑m≥2 P (m)/m)= .7292 · · · . Thus for every k ≥ 1, since log(α/.01) > 4,∫ 1.01
1
P (s)k ds >
∫ .01
0
log
(α
s
)k
ds =α
∫ ∞
log(α/.01)
uk e−u du >αΓ(k +1,4) > .729k !, (10)
where Γ(k + 1,4) the incomplete Gamma function, and noting Γ(k + 1,4)/k ! is monotonically
increasing in k. Also note∫ 1
0
s log
(α
s
)k
ds =α2
∫ ∞
0
uk e−2u du = α
2
2k+1
k !.
Using the first order Taylor approximation P ( j s) > P ( j )+P ′( j )(s −1) for j ≥ 2,∫ 1.01
1
P (s)k− j P
(
j s
)
ds > P ( j )∫ .01
0
log
(α
s
)k− j
ds +P ′ ( j )∫ 1
0
s log
(α
s
)k− j
ds
> .729(k − j )!(P ( j )+ αP ′ ( j )
2k− j
)
by (10). Similarly,∫ 1.01
1
P (s)k− j−2P (2s)P
(
j s
)
ds
> P (2)P ( j )∫ .01
0
log
(α
s
)k− j−2
ds +
[
P ′(2)P
(
j
)+P (2)P ′ ( j )]∫ 1
0
s log
(α
s
)k− j−2
ds
> .729(k − j −2)!(P (2)P ( j )+ α
2k− j−1
[
P ′(2)P
(
j
)+P (2)P ′ ( j )]) .
Hence plugging back into (8),
f
(
Nk ,h2
)
e .01h2 >
∫ 1.01
1
P (s)k ds > `k ,
C. R. Mathématique — 2022, 360, 409-414
414 J. D. Lichtman
for the explicit lower bound
`k := .729
[
1+
6∑
j=2
P ( j )+αP ′ ( j )/2k− j
j
+ 1
8
(
P (2)2 + αP (2)P
′(2)
2k−4
)
+
6∑
j=3
1
2 j
(
P (2)P
(
j
)+ α
2k− j−1
[
P ′(2)P
(
j
)+P (2)P ′ ( j )])] .
Note `k is clearly increasing in k (recall P
′(s) < 0). Hence for k > 20 we have
f
(
Nk ,h2
)> e−.01h2`k > e−.01h2`20 > .98 > 0.91 > f (N1,h2). (11)
Here we compute `20 = 0.99069 · · · and f (N1,h2) = 0.908599 · · · . This completes the proof of
Theorem 2. 
Acknowledgments
The author thanks Paul Kinlaw and Carl Pomerance for helpful discussions.
References
[1] W. D. Banks, G. Martin, “Optimal primitive sets with restricted primes”, Integers 13 (2013), article no. A69 (10 pages).
[2] H. Cohen, “High precision computation of Hardy-Littlewood constants”, https://www.math.u-bordeaux.fr/
~hecohen/.
[3] P. Erdős, “Note on sequences of integers no one of which is divisible by any other”, J. Lond. Math. Soc. 10 (1935),
p. 126-128.
[4] I. Laib, “Note on translated sum on primitive sequences”, Notes Number Theory Discrete Math. 27 (2021), no. 3, p. 39-
43.
[5] I. Laib, A. Derbal, R. Mechik, “Somme translatée sur des suites primitives et la conjecture d’Erdős”, C. R. Math. Acad.
Sci. Paris 357 (2019), no. 5, p. 413-417.
[6] J. D. Lichtman, “Almost primes and the Banks–Martin conjecture”, J. Number Theory 211 (2020), p. 513-529.
[7] ——— , “Mertens’ prime product formula, dissected”, Integers 21A (2021), article no. A17 (15 pages).
[8] J. D. Lichtman, C. Pomerance, “The Erdős conjecture for primitive sets”, Proc. Am. Math. Soc. 6 (2019), p. 1-14.
[9] Z. Zhang, “On a problem of Erdős concerning primitive sequences”, Math. Comput. 60 (1993), no. 202, p. 827-834.
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