On the Number of Primes Less Than a Given Magnitude
1859 · paper · Mathematics and statistics
Investigates how many prime numbers lie below a given quantity, starting from Euler's product formula. Extends the zeta function to all complex numbers, shows a symmetry relating its values at s and 1−s, and introduces a related function ξ(t).
Checked: PDF from maths.tcd.ie, fetched 5 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance In the top display on page 6, the substitutions are printed "p^{-s} by s∫_p^∞ x^{-s-1} ds, p^{-2s} by s∫_{p^2}^∞ x^{-s-1} ds"; the integration variable is x, so the differential should be dx.
- p. 6 top display, substitution for p^{-s} and p^{-2s}
p−s by s ∫ p ∞ x−s−1 ds
- p. 6 top display, substitution for p^{-s} and p^{-2s}
p−s by s ∞ Z p x−s−1 ds
Judge's note: The page 6 image shows 'ds' in both integrals. Since s∫_p^∞ x^{-s-1} dx = p^{-s}, the differential must be dx. The next display, log ζ(s)/s = ∫ f(x)x^{-s-1} dx, correctly uses dx.
Second check: another model agreed
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Substance In the Fourier inversion passage on page 6, the text prints "πh(y)y^{−α}" and "multiplies them by iy^α", where the real part a of s (s = a + bi, x^{-a}) is meant; α is reserved for the roots of ξ.
- p. 6 paragraph after the g1, g2 equations
then one obtains πh(y)y−α on the
- p. 6 same paragraph
multiplies them by iyα
- p. 6 same paragraph
multiplies them by iyα, one obtains
- p. 6 p.5, Fourier inversion passage
πh(y)y−α on the
Judge's note: The crops show Greek α in both places, while the surrounding formulas use the Latin a (x^{-a}, integration limits a±∞i). The inversion yields y^{-a}, so the printed symbol conflicts with the definition of α as a root of ξ.
Second check: another model agreed · checked by hand
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3 minor (typos and the like; most checked by the judge only)
Typo Page 3 prints "the integral taken though values of infinitely large modulus"; 'though' should be 'through'.
- p. 3 paragraph beginning 'If the real part of s is negative'
the integral taken though values of infinitely large modulus
- p. 3 paragraph beginning 'If the real part of s is negative'
since the integral taken though values of infinitely large modulus
Judge's note: The crop of page 3 shows 'though'. The sense requires 'through'.
Second check: none (judge only)
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Typo Duplicated preposition on PDF p.5: "the number of roots of ξ(t) = 0 lying within in this region" should read "lying within this region".
- p. 5 first paragraph
lying within in this
- p. 5
of ξ(t) = 0 lying within in this region
Judge's note: I checked both crops of PDF page 5. Each shows the printed text "lying within in this region, multiplied by 2πi". Both "within" and "in" appear in the image, so this is not an extraction artefact. The repeated preposition is a copy-editing slip in the translation. It is cosmetic.
Second check: none (judge only)
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Typo Page 9 prints "the density of the prime number +", singular, while the parallel phrases read 'squares of the prime numbers' and 'cubes of the prime numbers'.
- p. 9 last paragraph
density of the prime number +
Judge's note: The crop of page 9 shows the singular form. It is a minor grammatical inconsistency; the singular could be read generically.
Second check: none (judge only)