Music of the Primes: Chapter 5 Summary
The Mathematical Relay Race: Realising Riemann’s Revolution
Du Sautoy opens the chapter describing the “relay race” of great mathematicians
studying the primes, each generation’s insights and discoveries fueled by the cultural
outlook of their time and place, and serving as fuel for the successive generations’
accomplishments. During Riemann’s leg of the race, however, he pushed so far ahead
of the rest of the field that it took 30 years after his work was finished before another
mathematician was in place to take the next step from his results….
It was in 1885 that the unlikely victor, Dutch born Thomas Stieltjes, took
Riemann’s baton across the finish line, claiming a proof of Riemann’s Hypothesis.
Failing his university exams three times, his influential father stepped in unbeknownst
to his son and secured him a post at the Leiden Observatory to follow his passion,
celestial motion.
During his time at the Observatory, Stieltjes wrote the aging mathematician
Charles Hermite in Paris, the leading expert on Cauchy and Riemann’s work with
functions of imaginary numbers, to share some of his ideas. Though Stieltjes lacked a
degree and was merely an assistant, Hermite was enthused by Stieltjes’ letter, and
began a 12-year, 432 letter correspondence with him. In one of these letters, Stieltjes
mentioned (but did not include) his proof of the Hypothesis. As it turned out, Stieltjes
died in 1894, 9 years after his initial letter to Hermite, still claiming to have a proof and
thus becoming “the first in a long line of reputable mathematicians who have
announced proofs but failed to deliver the goods.”
Six years after Stieltjes’ erroneous claim, a major advance came from Jacques
Hadamard, a student of Hermite, who was able to prove that no Riemann zeta-zeros
existed at or beyond the real number 1. While this result didn’t prove the Riemann
hypothesis, that all non-trivial zeros be contained on the real part 1/2 line, it was a
major advance and DID prove Gauss’ Prime Number Conjecture. 100 years after it’s
inception, it was renamed the Gauss Prime Number Theory.
Though Riemann’s Hypothesis was still unproven, Hadamard’s result came only
from deep insights about zeta-landscapes gleaned from Reimann’s paper. The credit for
the proof goes jointly to Hadamard and Belgian de la Vallée-Poussin, who
simultaneously came up with a proof: “With Gauss’s Prime Number Theorem finally
claimed, it was time for Riemann’s great problem to emerge from the hidden depths of
his dense Berlin paper…as the Everest of mathematical exploration…. Hadamard and
de la Vallée-Pousson had established the base camp in preparation for the major ascent
towards Riemann’s critical line.”
Riemann had set the stage for twentieth century mathematics, and it was
Prussian born David Hilbert who first carried the torch leading twentieth century
mathematics into the pursuit of “Riemann’s principle according to which proofs should
be impelled by thought alone and not computation.” Structures and patterns became
the focus of study, not individual numbers and equations….
Before delving into the patterns and structures studied by Hilbert, the book takes
a few pages for some colorful description and anecdotes of Hilbert’s bohemian and
eccentric personality, somewhat unorthodox in the academic community of Göttingen
where he eagerly accepted professorship. Despite reservations about his character and
academic work, Hilbert continued unwavering in his academic and avocational
pursuits, and eventually won over even his most passionate critics in both realms.
While Hilbert’s early career focused on the advancement of number theory, a few
years after his university appointment, he shifted his focus to geometry. Years earlier,
Gauss had done work in secret on non-Euclidean geometries, fearing that public
exposition of his ideas might rock the mathematical boat too much. Instead of the
accepted geometry that had been used as a foundation for all prior mathematical study,
Gauss thought that perhaps other consistent geometrical conceptions could exist, and
might even reflect more accurately the world that we live in. In the non-Euclidean
geometries he considered, he challenged Euclid’s assumption that given any line and
point not on the line, there existed exactly one line through the point and parallel to the
first line. These geometries Gauss conceived, studied further by Russian mathematician
Lobachevsky and Hungarian mathematician Bolyai, contained either no parallel lines
through the point, or multiple ones.
Critics contended that non-Euclidean geometries would certainly contain
inconsistencies leading to their collapse as viable systems, and in an attempt to quiet
these critics Hilbert was able to prove that if any current geometry, Euclidean or
otherwise, contained inconsistencies, then all the systems would collapse, not just one.
He then realized that Euclidean geometry had never been rigorously proven as
consistent, and set about the task of doing just that. While no inconsistencies had been
discovered in 2000 years, Hilbert was a true mathematician and needed to find a
rigorous proof. Following in Descartes’ footsteps, he set about recasting Euclidean
geometry using algebraic equations to describe lines and Cartesian coordinates to
represent points. Since it was believed that number theory had no inherent
contradictions, Hilbert believed that if he could accurately recast all elements of
geometry in number theory, that he could prove it consistent.
This assumption of Hilbert’s eventually gave him cause for even greater concern.
In addition to no proof existing of the consistency of Euclid’s geometry, no proof had
ever been constructed of number theory’s consistency. During the nineteenth century,
mathematics had transitioned from a scientist’s mere tool into a more fundamental and
sometimes even philosophical pursuit of the inherent truths of the universe. Hilbert led
this transformation, and in 1899 was given the honor of being a speaker at the 1900
Congress of Mathematics in Paris. After much procrastination and uncertainty about
his idea to ignore the tradition of presenting a complete proof, Hilbert decided to
present a talk about “Mathematical Problems” to his peers. In his talk, Hilbert outlined
twenty-three problems that he felt should be the focus of 20th century Mathematics.
He’d selected the twenty-three problems with care, ensuring each of the wide ranging
topics “difficult in order to entice [mathematicians] yet not completely inaccessible, lest
[the problem] mock at our efforts.” When asked about the single most important
question for 20th century mathematics, Riemann highlighted his eighth problem, the
most specific of all twenty-three, to prove the Riemann Hypothesis.
Hilbert’s lecture didn’t lead directly to any advances towards a solution to the
Hypothesis, but it did change the face of mathematical focus, shifting thought of the
field’s pioneers from specifics to the abstract. Hilbert’s biggest direct contribution to the
Riemann Hypothesis might have been placing Landau in line to take the baton from
Riemann by appointing him a position at Göttingen.
Landau took up the journey from Hadamard and de la Vallée-Poussin’s
basecamp (no zeta-zeros past real part 1) and set out to prove that no zeros existed less
than real part 1/2. Along with help from Harald Bohr, Landau was able to prove that
zeros liked to “clump” near the 0.5 line, and also that zeros from 0.5 to 0.51 accounted
for a “large portion” of the overall non-trivial zeros.
In 1914, British mathematician G.H. Hardy shattered two centuries of British
mathematicians who were totally disinterested in matters concerning imaginary
numbers that so obsessed continental mathematicians. At the time, only 17 of the
infinitely many non-trivial zeros had been found along Riemann’s 1/2 line. Hardy
proved that infinitely many zeros existed along the line.
After description of Hardy’s character, the author continues to describe the
personalities involved in a great British collaboration beginning in 1910 when J.E.
Littlewood joined Hardy as a fellow at Trinity College, Cambridge. Hardy’s value of
beauty and elegance, combined with Littlewood’s “guns blazing” approach to tackle a
tough problem, produced a very productive 37 years, and represented the Britons
return to the mathematical landscape as the team combed through continental
mathematicians’ work of the past two centuries….
Thought their years of collaboration, Littlewood took a vacation each summer to
Copenhagen to meet with his friend Bohr. During his visit, all Hardy-Littlewood
collaborations were put on hold, and the one item of business for the two friends was to
work on the Riemann Hypothesis. When in their collaborative periods during the
academic year, Hardy and Littlewood spent some time working on the prime number
problem as well, their work fuelled by the arrival from Göttingen of Landau’s book
Handbook of the Theory of the Distribution of Prime Numbers. This book shoved the
connection between the zeta function and the primes to the forefront of all
mathematical thought, rather than remaining confined to the handful of top
mathematicians that had struggled with the connection earlier.
Du Sautoy states that “to prove a theorem that Gauss believed to be true but
couldn’t prove is generally regarded as a true test of a mathematician’s mettle. To
disprove such a theorem puts one in a different league altogether.” To disprove one of
Gauss conjectures is just what Littlewood did in 1912, discovering that the logarithmic
integral function, without Riemann’s refinement, would occasionally overestimate the
number of primes, despite the 10,000,000 pieces of experimental evidence to the
contrary. Littlewood also demonstrated the surprising result that after the first million
numbers or so, Riemann’s refinement to Gauss’ logarithmic integral was sometimes a
poorer prediction of the number of primes that the unrefined Gauss version. Both of
these results are particularly impressive when it is considered that Littlewood could not
predict at what point these surprising results would first manifest themselves, nor have
modern supercomputers yet counted far enough to find an example of Littlewood’s
results.
Near the close of the chapter, the author briefly mentions the phenomenon that
the Riemann Hypothesis has been so insurmountable a challenge, and has stood
without counterexample for so long, that present day mathematicians hardly hesitate to
develop proofs beginning with “suppose the Riemann Hypothesis is true….” The
danger of course lies with the possibility that someday the theory will be disproved, or
one zero discovered off the real-part equals 1/2 line, as Littlewood did with Gauss
second conjecture, and all the proofs relying upon it will crumble under the upheaval.