That the answer comes out right is no evidence that the middle was.
Euler obtained the sum of the reciprocal squares as pi squared over six in 1735. The answer is right. The argument, as written, is invalid, because it uses same zeros therefore same function, and that is false. NO NEW MATHEMATICAL THEOREM IS CLAIMED: the Basel problem, Hadamard’s factorisation theorem, the functional equation of the zeta function and Apery’s theorem are all standard. Nor is Euler criticised, since the notion of order did not exist in 1735 and both facts he used are true. On the overlap with Paper 31: that paper treated pi as a period carrying no dimension, while this one asks where pi appears at all and what determines a function.
First, Euler had the number before he had an argument. The naive partial sums err by about one over N, giving 0.0951663 at ten, 0.00995017 at a hundred, 0.0009995 at a thousand and still 9.99999 times ten to the minus seven at a million, with the error times N running 0.95166336, 0.99501667, 0.99950017 and 0.9999995. Twenty digits would need something like ten to the twentieth terms. Expanding the tail instead, as one over N minus one over twice N squared plus one over six N cubed and so on, gives about 8.9 correct digits at N equal to five, 12.1 at ten and 15.4 at twenty. Ten terms beat a million, so the number was in hand long before any proof and was recognised as pi squared over six. What 1735 supplied was not the value but a reason.
Second, and this is the core. Euler’s step runs: the sine cardinal vanishes exactly at multiples of pi, therefore it equals the product over those zeros. But the sine cardinal multiplied by the exponential has exactly the same zeros, at plus and minus pi, two pi, three pi and so on, and since the exponential never vanishes it has no others. Both are entire of order one and both equal one at the origin. And still they are different functions: the coefficient of the first power is zero in one and one in the other, the coefficient of the square is minus 0.16666666666667 against 0.33333333333333, and the values at one are 0.8414709848078965 against 2.287355287178842. If same zeros and same value at the origin forced equality, then the exponential would equal one everywhere. It does not. Euler’s inference, as written, is invalid. The answer is right and the argument is not, and a correct answer is not evidence for an argument, the same shape as Paper 170’s closing that working is not the same as meaning something.
Third, what repaired it was Hadamard in 1893. An entire function of order one taking the value one at the origin factors as an exponential of a linear term times a product over its zeros, so the freedom left once the zeros are given is exactly that exponential factor, no more and no less. What then kills the remaining constant? Measuring the parity defect, the difference of the values at plus and minus 0.7, gives zero when the constant is zero, 1.3962658 when it is plus or minus one, and 0.38937788 when it is 0.3. Only the value zero is even. Euler was using two facts he neither stated nor proved: that the sine cardinal has order one, and that it is even. Both are true, and neither is in the argument. So the correct statement is not that the zeros determine the function but that the zeros and the growth and the parity determine it, and two of those three went unnamed for some hundred and forty years.
Fourth, with the hypotheses supplied the product converges correctly. At the argument one, taking five, fifty, five hundred and five thousand factors gives 0.857081935312307703, 0.84316092931847857, 0.84164134933145336 and 0.841488035042818401, approaching the sine of one, which is 0.841470984807896507, and not approaching 2.2874. Reading off the coefficient of the square gives the sum of the reciprocal squares over pi squared equal to one sixth.
Fifth, what decides whether pi appears at all is parity. Dividing the zeta values by the corresponding power of pi gives 0.1666666666666667, 0.01111111111111111, 0.001058201058201058 and 0.0001058201058201058 at the arguments two, four, six and eight, that is one sixth, one ninetieth, one nine hundred and forty-fifth and one nine thousand four hundred and fiftieth. At three, five and seven the same ratios are 0.0387681796029168, 0.003388434818730467 and 0.0003338580731527709, and no such form is known. The reason is the factor of a sine in the functional equation, which vanishes at every even argument, numerically at the level of ten to the minus forty-three, while at three, five and seven it equals minus one, plus one and minus one. What Apery proved in 1978 is the irrationality of the value at three, not a closed form. What must be moved to make pi appear is the parity of the argument, not its size.
A census. The set of zeros alone does not determine the function, since an exponential factor remains free. Zeros together with growth nearly determine it by Hadamard, with that exponential factor still remaining. Zeros together with growth and parity do determine it, forcing the constant to zero, and that triple is what Euler used tacitly. An accelerated sum gives the answer first and is not a proof. At even arguments the functional equation makes a power of pi appear and at odd ones it does not. The phrase determined by its zeros hides two silent assumptions, and the answer stayed right while they were hidden, which is exactly why being right is no substitute for checking.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
⚠️ Scope ── what is not claimed
a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem, none is claimed; any criticism of Euler, none is offered; any closed form for the zeta value at three, none is asserted. What is added: exhibiting two functions with identical zeros and measuring their coefficients, putting a number on the parity defect, showing with an accelerated sum that the answer preceded the proof, and setting out that the appearance of pi is decided by parity. What remains: Euler reached a right answer by a road that was not right, which is no fault of his, since the notion of order lay a hundred and forty years in his future. The lesson is that the answer coming out right is no evidence that the middle was, and that a correct answer does not excuse the checking.
Reference
Yamagishi, Y. (2026). Zeros Do Not Determine a Function [Paper 172]. Zenodo.
DOI: 10.5281/zenodo.22094374
ORCID: 0009-0006-8587-3450