Erdős problem #15 — wave 5e
Access and research date: 2026-07-26 UTC.
Claim labels used throughout:
- (a) elementary-rigorous — proved from elementary facts in this report.
- (b) rigorous-modulo-named-theorem — follows from the named theorem in the cited primary source.
- (c) plausible/structural-unverified — heuristic, interpretation, search miss, or cost estimate.
- (d) computational-only — a finite result established by the supplied computation, with no claim about the infinite tail.
Outcome
This run does not solve the problem.
It does provide four checkable pieces of progress:
1. (a) an exact necessary-and-sufficient Abel-summation criterion, and a precise averaged-correlation lemma that would suffice to finish the problem;
2. (a) a counterexample to one proposed intermediate target in the live discussion: anti-concentration of short-interval prime counts alone does not imply parity mixing;
3. (d) the sharp finite bound
\[ -25{,}630\leq F(x):=\sum_{n\leq x}(-1)^{\pi(n)} \leq23{,}758\qquad(0\leq x\leq10^8), \]
including the first points attaining both extrema;
4. (d) exact parity-correlation tables for
\(\pi(n+h)-\pi(n)\), including every shift in a finite analogue of Tao's polylogarithmic range.
The standalone verifier is
0. Mandatory live-page gate
I fetched the live problem, its LaTeX view, its bibliography popups, and its discussion thread through the Bright Data cloud-browser route. Direct datacenter curl was not used for the live-page gate.
(d, live-page observation) The exact statement, copied verbatim from the live LaTeX view, is:
> Is it true that\[\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}\]converges, where $p_n$ is the sequence of primes?
(d, live-page observation) The live page displayed:
- status:
OPEN; 2 comments on this problem;0 claimed proofs for this problem;Interested in collaborating — None;Currently working on this problem — None;This problem looks difficult — None;This problem looks tractable — None;Formalised statement? Yes;- likes:
Prasannam, holyterror.
Thus none of the required stop conditions was present.
Live sources: problem #15, LaTeX view, and discussion thread.
Everything currently listed on the page
(d, live-page observation) The page cites the three original Erdős sources [Er97,p.158], [Er97e,p.535], and [Er98]. It says Erdős suggested computer exploration and saw no other attack.
(b) The page records Tao's conditional theorem: the series converges under a strong quantitative Hardy–Littlewood prime-tuples conjecture.
(d, live-page observation) The page also records three related, but logically separate, prime-gap questions from Erdős:
- convergence of
\[ \sum_{n\geq1}\frac{(-1)^n}{n(p_{n+1}-p_n)}; \]
- divergence of
\[ \sum_{n\geq1}\frac{(-1)^n}{p_{n+1}-p_n}; \]
- convergence, for every \(c>0\), of
\[ \sum_{n\geq1} \frac{(-1)^n} {n(p_{n+1}-p_n)(\log\log n)^c}. \]
The page says Zhang's bounded-gap theorem makes the terms of the second series fail to tend to zero. It also reports Weisenberg's conditional argument that this second series is unbounded in at least one direction under the prime \(k\)-tuples conjecture.
(d, live-page observation) For the third series, the page says Erdős and Nathanson proved absolute convergence for \(c>2\). It gives Sawhney's Selberg-sieve proof: large gaps contribute a convergent sum for \(c>1\), while
\[ \#\{n\leq X:\epsilon\log n\leq p_{n+1}-p_n<2\epsilon\log n\} \ll\epsilon X \]on dyadic \(\epsilon\)-ranges handles the small gaps and reduces the claim to convergence of
\[ \sum\frac1{n\log n(\log\log n)^{c-1}}. \](c, unverified live comment) Przemek Chojecki's 13 January 2026 comment recalls the reduction to
\[ \sum_{n\geq2}\frac{(-1)^{\pi(n)}}{n\log n}, \]observes via partial summation that a logarithmic power saving for
\(F(x)=\sum_{n\leq x}(-1)^{\pi(n)}\) is sufficient, and proposes short-interval parity mixing, anti-concentration, or a Poisson local limit as intermediate targets.
(c, unverified live comment) Desmond Weisenberg's 11 August 2025 comment explains his conditional argument for the separate reciprocal-gap series. Assuming prime \(k\)-tuples, widely separated admissible pairs can occur as consecutive primes and force arbitrarily large changes in the series. The conjecture does not control the parity of the starting prime index, so this does not determine the direction of unboundedness.
Neither comment claims a proof of the main series.
1. Primary-source literature audit
The direct result
(b) Tao's paper exists both as
arXiv:2308.07205 and as a published paper:
Terence Tao, “The convergence of an alternating series of Erdős, assuming the Hardy–Littlewood prime tuples conjecture”, Communications of the American Mathematical Society 4 (2024), 80–96, DOI 10.1090/cams/29.
Its Theorem 1.4 assumes a quantitative Hardy–Littlewood conjecture uniform for
\[ k\leq(\log\log x)^5,\qquad \mathcal H\subset[0,\log^2x], \]with a power-saving error, and proves convergence. The paper notes that a somewhat weaker error than a power saving would suffice, but high-order uniformity is still required.
(b) Section 2 proves the more precise equivalence
\[ \sum_{n\leq x}\frac{(-1)^n n}{p_n} =\frac12\sum_{2\leq m\leq x\log x} \frac{(-1)^{\pi(m)}}{m\log m} C+o(1), \tag{1} \]and remarks that the error can be made
\(O(\log\log x/\log x)\).
(b) Under the same conjecture, Tao proves
\[ F(x)\ll\frac{x}{(\log\log x)^{1.1}}, \tag{2} \]which is just strong enough for partial summation. His central conditional estimate is parity mixing for primes in intervals of length \(\lambda\log x\), uniformly through a polylogarithmic range of \(\lambda\).
(c) Tao reports a numerical limit near \(-0.052161\), credited to Tomás Oliveira da Silva by private communication. This is numerical guidance, not a tail bound or a theorem.
What is known unconditionally about the parity of \(\pi(n)\)
(b) Ping Ngai Chung and Shiyu Li,
“On the residue classes of \(\pi(n)\) modulo \(t\)”,
Integers 13 (2013), A79, prove that if
\[ T_{r,t}(x)=\#\{nThey improve \(16\) to \(2\) assuming Hardy–Littlewood. For \(t=2\), each parity therefore has unconditional lower density at least \(1/64\).
(b) Mihai Alboiu,
“On the parity of the prime-counting function and related problems”,
Ramanujan Journal 38 (2015), 179–187, independently proves positive proportions for every residue class of \(\pi(n)\bmod q\), and an averaged equidistribution result using the large sieve.
(a) Positive proportions do not provide the decay needed here. Chung–Li's \(t=2\) theorem gives at most a constant-factor restriction such as
\[ \limsup_{x\to\infty}\frac{|F(x)|}{x}\leq\frac{31}{32}; \]it does not imply \(F(x)=o(x)\), let alone (2). A constant bound leaves the divergent majorant
\(\sum1/(n\log n)\).
(c, honest search result) Exact-title, exact-formula, citation, arXiv, and web searches found no later primary source removing Tao's Hardy–Littlewood hypothesis or disproving convergence. OpenAlex listed zero citing works for the published Tao paper at the access date. A search miss is not a theorem, but no unconditional advance on the series was located.
For source-identity checking, the downloaded published Tao PDF had SHA-256
3b9afee998cbefc0f674500a74d980447d089515390ea12a4d67b794870f5de7,
and the Chung–Li PDF had SHA-256
7ee7e5091a8ca7546648738ffb7fa687137a6ef82f1a8e4092a320cadf1bd8d1.
2. Exact reduction and the missing lemma
Set
\[ f(n)=(-1)^{\pi(n)},\qquad F(x)=\sum_{n\leq x}f(n),\qquad g(n)=\frac1{n\log n}. \]By (1), it is enough—and is in fact equivalent—to decide convergence of
\[ \sum_{n\geq2}f(n)g(n). \tag{3} \]2.1 Exact Abel criterion
Let \(A(n)=\sum_{m=2}^n f(m)=F(n)-1\).
(a) Finite Abel summation gives the exact identity
\[ \sum_{n=2}^N f(n)g(n) =A(N)g(N)+\sum_{n=2}^{N-1}A(n)\bigl(g(n)-g(n+1)\bigr). \tag{4} \]Since \(|A(N)|\leq N\), the boundary term is
\(O(1/\log N)\to0\). Replacing \(A(n)\) by \(F(n)\) changes the right side only by a convergent telescoping series. Consequently:
(a) Exact criterion.
The series (3), and hence the original series, converges if and only if
\[ \sum_{n=2}^{\infty} F(n)\left( \frac1{n\log n}-\frac1{(n+1)\log(n+1)} \right) \tag{5} \]converges.
This is a signed criterion; absolute convergence of (5) is sufficient but not necessary.
(a) Since
\[ g(n)-g(n+1)\asymp\frac1{n^2\log n}, \]the following Dini-type estimate is sufficient:
\[ \sum_{n\geq3}\frac{|F(n)|}{n^2\log n}<\infty. \tag{6} \]In particular, for any fixed \(\eta>0\),
\[ F(x)=O\!\left(\frac{x}{(\log\log x)^{1+\eta}}\right) \tag{7} \]would settle the problem. This is weaker than the logarithmic-power saving proposed in the newest live comment and explains Tao's exponent \(1.1\).
2.2 Exact prime-gap form
Let \(p_K\leq x (a) Because \(f\) is constant on each prime-gap block, This makes the obstacle explicit: one needs cancellation in alternating prime gaps, not merely the prime number theorem. It also shows that finite extrema of \(F\) occur at prime-gap endpoints. For an integer \(x\), put \(L=x+1\) and define2.3 A precise sufficient correlation estimate
(a) Therefore the following is a clean sufficient missing lemma. If, uniformly on all large dyadic scales, there are \(H=H(x)\to\infty\) and \(\eta>0\) such that
\[ \frac1H\sum_{h=1}^{H-1}\frac{|C_x(h)|}{x} \ll\frac1{(\log\log x)^{2+2\eta}}, \tag{11} \]with \(H^{-1}\ll(\log\log x)^{-2-2\eta}\), then (10) gives
\[ \sum_{n=x}^{2x}f(n) \ll\frac{x}{(\log\log x)^{1+\eta}}. \]Dyadic decomposition then gives (7), and (4) proves convergence.
(b) Tao obtains the needed form of (11), conditionally, by proving roughly
\[ \frac1x C_x(\lfloor\lambda\log x\rfloor) \ll\lambda^{-1/2} \]through \(\lambda\ll(\log\log x)^{4.4}\), using the quantitative Hardy–Littlewood conjecture and a random sifted model.
(a) Thus the exact unconditional wall is not “show both parities occur” or “show prime counts have a spread.” It is a uniform Fourier-coefficient estimate at the parity character, averaged over enough short shifts to make (11) summable.
3. A correction to the proposed anti-concentration route
The newest live comment proposes either a local-limit theorem or a bound of the form
\[ \sup_k\Pr(\Delta_\lambda=k)\ll\lambda^{-\eta} \]as a route to parity mixing.
(a) Total-variation convergence to a Poisson law would work, because
\[ \mathbb E(-1)^{\operatorname{Poisson}(\lambda)}=e^{-2\lambda}. \]But anti-concentration alone does not work.
For any \(M\), let \(\Delta\) be uniform on
\[ \{0,2,4,\ldots,2(M-1)\}. \]Then
\[ \sup_k\Pr(\Delta=k)=\frac1M\longrightarrow0, \qquad \mathbb E(-1)^\Delta=1. \tag{12} \]So the distribution can be arbitrarily anti-concentrated while remaining entirely on one parity. The missing hypothesis must directly control parity—for example total variation to Poisson, or the Fourier coefficient \(\mathbb E(-1)^\Delta\)—not merely the largest atom.
The verifier checks (12) exactly over \(\mathbb Q\) for \(M=1000\).
4. Exact finite computation
4.1 Method
The verifier uses only Python's standard library.
1. It constructs all primes through \(10^8\) with an odd-only Eratosthenes sieve and stores them in a packed 32-bit array.
2. It independently compares every primality decision through \(100{,}000\) with trial division.
3. It scans the constant-sign blocks in (8), which proves the finite extrema without visiting every integer individually.
4. For (9), moving \(n\) to \(n+1\) removes \(n+1\) from \((n,n+h]\) and adds \(n+h+1\). A merged sweep of these two prime-event lists gives the complete exact histogram of \(\pi(n+h)-\pi(n)\).
5. A literal per-\(n\) counter independently checks the event-sweep algorithm on several smaller intervals.
6. It checks Abel summation, the normalization of (10), and the adjacent-pair identity exactly over \(\mathbb Q\).
7. It evaluates the original partial sum both directly and by adjacent pairing, in binary64 and independently at 50-digit Decimal precision.
4.2 Sharp range of \(F\) through \(10^8\)
(d) The sieve found
\[ \pi(10^8)=5{,}761{,}455,\qquad p_{\pi(10^8)}=99{,}999{,}989. \]The exact milestone values are:
| \(x\) | \(F(x)\) |
|---:|---:|
| \(10\) | \(4\) |
| \(10^2\) | \(2\) |
| \(10^3\) | \(46\) |
| \(10^4\) | \(-290\) |
| \(10^5\) | \(426\) |
| \(10^6\) | \(-3{,}990\) |
| \(10^7\) | \(-16{,}926\) |
| \(5\cdot10^7\) | \(-1{,}818\) |
| \(10^8\) | \(-15{,}950\) |
(d) The sharp finite extrema are
\[ \min_{0\leq x\leq10^8}F(x)=-25{,}630, \quad\text{first at }x=22{,}578{,}648, \]and
\[ \max_{0\leq x\leq10^8}F(x)=23{,}758, \quad\text{first at }x=76{,}085{,}520. \]These are exact integer statements, but they imply no asymptotic bound.
4.3 Exact short-interval parity table
Take the sequence \(f(n)\) on \(50{,}000{,}000\leq n\leq100{,}000{,}000\). For
\[ h=\operatorname{round}(\lambda\log(50{,}000{,}000)), \]the table gives the exact numbers of \(n\in[50{,}000{,}000,100{,}000{,}000-h]\) for which
\(\pi(n+h)-\pi(n)\) is even or odd.
| \(\lambda\) | \(h\) | intervals | even | odd | \(C_x(h)/(\text{intervals})\) |
|---:|---:|---:|---:|---:|---:|
| 1 | 18 | 49,999,983 | 26,589,188 | 23,410,795 | \(+0.063567882\) |
| 2 | 35 | 49,999,966 | 25,073,685 | 24,926,281 | \(+0.002948082\) |
| 4 | 71 | 49,999,930 | 25,019,258 | 24,980,672 | \(+0.000771721\) |
| 8 | 142 | 49,999,859 | 24,985,145 | 25,014,714 | \(-0.000591382\) |
| 16 | 284 | 49,999,717 | 24,991,207 | 25,008,510 | \(-0.000346062\) |
| 32 | 567 | 49,999,434 | 24,984,407 | 25,015,027 | \(-0.000612407\) |
| 64 | 1,135 | 49,998,866 | 24,993,455 | 25,005,411 | \(-0.000239125\) |
(d) The complete histograms, including every exact prime-count multiplicity, are printed by the verifier. Their totals, parity counts, first moments, and maximum counts are also bound into its expected-output assertions.
(d) A uniform smaller-scale scan checked every
\[ 14=\lceil\log(500{,}000)\rceil \leq h\leq \left\lfloor (\log\log500{,}000)^{4.4}\log500{,}000 \right\rfloor=841 \]on \(500{,}000\leq n\leq1{,}000{,}000-h\). It found the sharp normalized bound
\[ \max_{14\leq h\leq841} \frac{|C_{500000}(h)|}{500001-h} =\frac{16{,}685}{499{,}987} =0.0333708676\ldots, \]attained at \(h=14\). Across all 828 shifts, the exact aggregate ratio was
\[ \frac{\sum_{h=14}^{841}|C_{500000}(h)|} {\sum_{h=14}^{841}(500001-h)} =\frac{1{,}050{,}536}{413{,}646{,}858} =0.0025396929\ldots. \]This is finite evidence for the shape of the missing lemma, not uniform asymptotic control.
4.4 Original partial sum
Let \(N=\pi(10^8)=5{,}761{,}455\).
(d) The independent 50-digit summations agree through more than 40 decimal places and give
\[ \sum_{n=1}^{N}\frac{(-1)^n n}{p_n} =-0.08095846870114207157881443501821608487\ldots. \]The final index is odd, so the adjacent even partial sum is
\[ S_{N-1} =-0.02334391236354087444268275004373073768\ldots, \]and their midpoint is
\[ \frac{S_N+S_{N-1}}2 =-0.05215119053234147301074859253097341128\ldots. \](c) The midpoint's proximity to Tao's quoted \(-0.052161\) is consistent with convergence, but there is no rigorous tail bound. It cannot be used as evidence that the problem is closed.
5. Reproduction, scope, and wall
Run:
python runs/erdos15_wave5e_verify.py
The final verified run used one core for 67.1 seconds and ended with:
PI_LIMIT=5761455 LAST_PRIME=99999989
F_SHARP_RANGE min=-25630 first_at=22578648 max=23758 first_at=76085520
UNIFORM_CORRELATION ... max_abs=16685/499987 ... at_h=14
ORIGINAL_S_N_DECIMAL=-0.080958468701142071578814435018216084871872314668783
ORIGINAL_S_N_DECIMAL_PAIRED=-0.080958468701142071578814435018216084871872314668804
ORIGINAL_S_PREVIOUS_DECIMAL=-0.023343912363540874442682750043730737678484123396082
ORIGINAL_MIDPOINT_DECIMAL=-0.052151190532341473010748592530973411275178219032433
ALL_CHECKS_PASSED
(a) No finite extension of this table can settle convergence. The exact missing theorem is an asymptotic estimate such as (11), or another argument proving the signed criterion (5).
(b) Existing unconditional sieve results establish positive proportions of both parities, but not decay of the parity Fourier coefficient. Tao obtains the required decay only after imposing high-order, growing-\(k\), quantitative Hardy–Littlewood uniformity.
(c) A straightforward tenfold extension of this implementation would require roughly 0.5 GB just for the odd sieve, about 0.2 GB for packed primes, and several single-core minutes; extending to \(10^{12}\) would require hundreds of GB and at least hundreds of core-hours in this architecture. Even that would remain a finite experiment and would not supply (11), so I did not run it.
PARTIAL: exact \(F(x)\) extrema through \(10^8\), exact short-interval parity tables, a rigorous Abel/correlation reduction, and a counterexample to anti-concentration as a sufficient intermediate target; the uniform parity-correlation lemma remains open.