Abstract
The book Ramanujan learned from was some five thousand results with the proofs stripped out. When a proof goes, the fence goes with it — and when the fence goes, the symbol no longer carries its type. This paper sorts the equals signs he wrote into four kinds and checks each in numbers. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, some are exactly true. His series for $1/\pi$ buys about eight digits a term, the correct digits running $7\to15\to23\to31\to39\to47$ (Section 3).
Second, some are only asymptotic. For the partitions the absolute error grows from $6.10$ to $4.71\times10^{43}$ while the relative error falls from $14.53\%$ to $0.998\%$ (Section 4).
Third, this is the core. The partial sums of $1+2+3+\cdots$ reach $5{,}000{,}000{,}050{,}000{,}000$ at $10^8$ terms — what agrees with $-1/12$ is $\zeta(-1)$, to a difference of $\mathbf{0}$, and not the sum (Section 5).
Fourth, some need a proof of convergence. The nested radical reaches $2.999999999986$ at depth $40$ — and nowhere does the symbol say that it converges (Section 6).
Fifth, one inequality no computation can refute. $\pi(x)<\mathrm{li}(x)$ holds at every $x$ out to $10^7$, the gap reaching $339.41$ — and Littlewood proved it fails infinitely often (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the character of Carr's Synopsis, Ramanujan's series for $1/\pi$, the Hardy–Ramanujan asymptotic for the partitions, $\zeta(-1)=-1/12$ and Littlewood's theorem are all standard. No biography is offered — neither Ramanujan's life nor the bibliography of Carr's book is treated. No claim about teaching is made — whether one should learn from a book without proofs is not asked. No general theory of divergent series is offered — Ramanujan summation and Abel summation are cited, not developed. Littlewood's theorem is not proved — and no first crossing is given, because none is known. Nobody is being faulted — the claim here is about the symbol and not about the man. Relation to earlier papers: Paper 16 counted six distinct roots of the equals sign — this paper applies that census to one writer's own formulas. Paper 211 showed that adding more terms takes you further from the answer — Section 4 is an instance of it. Paper 227 showed that the symbol does not settle whether a wall can be touched — Section 7 is its most extreme case. Paper 223 showed that everything can pass every test and structure still be there — $\pi(x)<\mathrm{li}(x)$ is its number-theoretic twin. Paper 31 treated Ramanujan's series from the side of what $\pi$ is — this paper takes the side of what the equals sign is, and they do not overlap. What is added is measuring the digits gained per term, showing the absolute and relative errors of the partition formula moving in opposite directions, setting the partial sums beside $\zeta(-1)$, following the nested radical by depth, and measuring $\mathrm{li}(x)-\pi(x)$ out to $10^7$.
1. Notation and Position
Paper 16 counted six distinct roots of the equals sign — a definition, a consequence of substitution, an identity, a conditional equation, an empirical law, and one that is dimensionally empty. This paper applies that census to the formulas of one writer. The question is which equals sign is of which type, not who was right.
Position: Paper 31 treated these series from the side of $\pi$ — this one takes the side of the equals sign.
2. What Carr's Book Left Out
Observation 2.1 Carr's Synopsis sets out some five thousand results with little or no proof. It was Ramanujan's principal entry into the subject.
Observation 2.2 When a proof goes, something else goes with it — the fence. Where a result holds, whether a series converges, in what sense two things are equal: these live only inside the proof. Keep the results alone and the symbols remain without their types.
This is not a complaint. Carr's book was written as a revision summary, and omitting proofs is right for that purpose. The point is not about the book but about the symbol carrying no type.
3. An Equals Sign That Is Exactly True
Check 3.1: sum Ramanujan's series for $1/\pi$ term by term.
| Terms | Error | Correct digits |
|---|---|---|
| $1$ | $7.64\times10^{-8}$ | $7$ |
| $2$ | $6.4\times10^{-16}$ | $15$ |
| $3$ | $5.68\times10^{-24}$ | $23$ |
| $4$ | $5.24\times10^{-32}$ | $31$ |
| $5$ | $4.94\times10^{-40}$ | $39$ |
| $6$ | $4.74\times10^{-48}$ | $\mathbf{47}$ |
Observation 3.2 Each term buys exactly eight digits. This is a genuine equality: the series converges and its limit is $1/\pi$ itself.
4. An Equals Sign That Is Only Asymptotic
Check 4.1: compare the leading Hardy–Ramanujan term with the exact $p(n)$.
| $n$ | Exact $p(n)$ | Absolute error | Relative error |
|---|---|---|---|
| $10$ | $42$ | $6.10$ | $14.53\%$ |
| $100$ | $190569292$ | $8.71\times10^{6}$ | $4.57\%$ |
| $1000$ | $2.41\times10^{31}$ | $3.41\times10^{29}$ | $1.42\%$ |
| $2000$ | $4.72\times10^{45}$ | $\mathbf{4.71\times10^{43}}$ | $\mathbf{0.998\%}$ |
Observation 4.2 The absolute error grows without bound and the relative error falls. Written with an equals sign the statement is false; written with a tilde it is a theorem — one symbol decides the truth value. An instance of Paper 211.
5. The Core: An Equals Sign That Is a Convention
Check 5.1: the partial sums of $1+2+3+\cdots$.
| Terms added | Partial sum |
|---|---|
| $10$ | $55$ |
| $10^4$ | $50{,}005{,}000$ |
| $10^6$ | $500{,}000{,}500{,}000$ |
| $10^8$ | $\mathbf{5{,}000{,}000{,}050{,}000{,}000}$ |
Meanwhile $\zeta(-1)=-0.083333333333333333333$ and $-1/12=-0.083333333333333333333$, differing by $\mathbf{0}$.
Observation 5.2 The partial sums on the left approach nothing. What agrees with $-1/12$ is the value of an analytically continued function, not a sum — the two are both correct and not the same statement. One equals sign is doing two different jobs.
Observation 5.3 In the census of Paper 16 this is the convention. Writing $=-1/12$ is not a discovery but the consequence of choosing one summation method — and which method was chosen is not written in the symbol.
6. An Equals Sign That Needs a Proof of Convergence
Check 6.1: evaluate the nested radical at increasing depth.
| Depth | Value |
|---|---|
| $5$ | $2.755053261329895984801018$ |
| $10$ | $2.989920360631050804024012$ |
| $20$ | $2.999987880599493711392443$ |
| $40$ | $\mathbf{2.999999999986179938154983}$ |
Observation 6.2 The value is $3$. And nowhere in the symbol is it written that the nesting converges — that must be shown separately. Ramanujan himself posed it as a problem and not as a proof.
7. An Inequality No Computation Can Ever Refute
Check 7.1: measure $\mathrm{li}(x)-\pi(x)$.
| $x$ | $\pi(x)$ | $\mathrm{li}(x)-\pi(x)$ |
|---|---|---|
| $10^2$ | $25$ | $+5.126141584$ |
| $10^4$ | $1229$ | $+17.1372159$ |
| $10^6$ | $78498$ | $+129.5491595$ |
| $10^7$ | $664579$ | $\mathbf{+339.4050486}$ |
Observation 7.2 It never once fails. And in $1914$ Littlewood proved that the sign changes infinitely often — and the first crossing has still not been located.
Observation 7.3 It passes every test that can be run, and it is false. The number-theoretic twin of Paper 223, where a test sees only what it was built to see, and the most extreme case of Paper 227, where the symbol does not settle whether the wall is touched.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates one equals sign from another is which fence was left out.
| Statement | The type of its “$=$” | The fence that is missing |
|---|---|---|
| The series for $1/\pi$ | exact | none |
| The partition formula | asymptotic | “as $n\to\infty$, in relative error” |
| $1+2+3+\cdots=-1/12$ | a convention | “under which summation” |
| The nested radical | exact | “this nesting converges” |
| $\pi(x)<\mathrm{li}(x)$ | false | “checked over a finite range” |
Observation 8.2 (the fence) The fence around this formula is correct is saying which type its equals sign has. Outside it the sentence is not even false but empty — the same shape as Papers 228, 229 and 230.
Observation 8.3 A proof is not only a guarantee of the result. It is also the vessel that carries the fence — so a summary with the proofs removed keeps the results and discards their types. Ramanujan's astonishment began from that state, and what Hardy mostly added was the fences.
9. The Census and the Close
What this paper measured:
- The series for $1/\pi$ is exact (Check 3.1): eight digits a term.
- The partition formula is asymptotic (Check 4.1): $4.71\times10^{43}$ and $0.998\%$.
- The $-1/12$ is a convention (Check 5.1): partial sums at $5\times10^{15}$, $\zeta(-1)$ agreeing to $0$.
- The nested radical needs convergence (Check 6.1): $2.999999999986$ at depth $40$.
- $\pi(x)<\mathrm{li}(x)$ is unbroken to $10^7$ and false (Check 7.1).
- In all five, the symbol does not give the type (Observation 8.1).
References
- Carr, G. S. (1880–1886). A Synopsis of Elementary Results in Pure and Applied Mathematics. London: Francis Hodgson.
- Ramanujan, S. (1914). Modular equations and approximations to $\pi$. Quarterly Journal of Mathematics, 45, 350–372.
- Hardy, G. H., Ramanujan, S. (1918). Asymptotic formulae in combinatory analysis. Proceedings of the London Mathematical Society, s2–17(1), 75–115.
- Littlewood, J. E. (1914). Sur la distribution des nombres premiers. Comptes Rendus de l'Académie des Sciences, 158, 1869–1872.
- Hardy, G. H. (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge: Cambridge University Press.
- Hardy, G. H. (1949). Divergent Series. Oxford: Clarendon Press.
- Berndt, B. C. (1985). Ramanujan's Notebooks, Part I. New York: Springer.
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). The $\pi$ That Carries No Dimension [Paper 31]. Zenodo. 10.5281/zenodo.22090430
- Yamagishi, Y. (2026). Adding More Terms Takes You Further from the Answer [Paper 211]. Zenodo. 10.5281/zenodo.22111772
- Yamagishi, Y. (2026). It Passes Every Test, and the Points Lie on Fifteen Planes [Paper 223]. Zenodo. 10.5281/zenodo.22120728
- Yamagishi, Y. (2026). The Same $\le$ Points at a Wall You Can Touch and at One You Cannot [Paper 227]. Zenodo. 10.5281/zenodo.22123129
- Yamagishi, Y. (2026). There Are Eight Plus Signs, and Seven of Them Become Real Addition in the Right Coordinate [Paper 228]. Zenodo. 10.5281/zenodo.22126254
Reference
Yamagishi, Y. (2026). Learn from a Book with the Proofs Removed and the Equals Sign Loses Its Type [Paper 232]. YOI: https://www.yamagishiyuuki.com/yoi/232
DOI: 10.5281/zenodo.22127528
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 232. This paper is a preprint and has not been peer reviewed.