Pafnuty Chebyshev

How to use this course

Pafnuty Chebyshev’s name is attached to an inequality, a family of polynomials, a pair of prime-number bounds, a linkage and a bias among primes. Students meet these in separate courses and rarely learn that one person found them all. What unites them is a habit of mind. Chebyshev liked questions with a practical origin and answers in the form of sharp inequalities: how far a curve can stray from a line, how far an average can stray from its mean, how far the count of primes can stray from a smooth function.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Part II follows the main areas of his work rather than strict chronology: primes (1848–1852), polynomials and approximation (1854–1859), integration (1847–1853), probability (1846–1887) and mechanisms (1850s–1890s).

Timeline

Okatovo and Moscow, 1821–1846

Pafnuty Lvovich Chebyshev was born on 16 May 1821 on the family estate of Okatovo, in the Borovsk district of Kaluga province, south-west of Moscow. His father, Lev Pavlovich, was a landowning nobleman and a former army officer who had fought against Napoleon. His mother, Agrafena Ivanovna Pozniakova, taught him to read and write; a cousin, Avdotia Sukhareva, taught him French and arithmetic. He was one of nine children.

One of his legs was shorter than the other, and he walked with a limp and later a cane. A military career like his father’s was out of the question. Biographers connect his childhood fondness for building mechanical toys with the later inventor of linkages and calculating machines, though such links are easy to overstate.

Moscow University

In 1832 the family moved to Moscow so that the older children could prepare for university. Chebyshev was tutored by P. N. Pogorelski, a well-known Moscow teacher of elementary mathematics. In 1837 he entered the department of physics and mathematics at Moscow University. There the decisive influence was Nikolai Brashman, professor of applied mathematics, who combined pure analysis with mechanics and engineering. Chebyshev kept a lasting respect for him.

In 1841 he graduated, receiving a silver medal for an essay on computing the roots of equations. The same year a famine and the collapse of his family’s finances ended their support. He chose to continue with mathematics on very little money. He passed the master’s examinations in 1843 and published his first paper, on multiple integrals, in Joseph Liouville’s Journal de mathématiques pures et appliquées. A paper on the convergence of Taylor series followed in Crelle’s journal in 1844.

His master’s thesis of 1846, An Essay on the Elementary Analysis of the Theory of Probability, aimed to derive the main results of probability with elementary means and careful error bounds. It included a rigorous treatment of Poisson’s form of the law of large numbers. The theme returned twenty years later (Lesson 09).

St Petersburg and the theses, 1847–1852

With few openings in Moscow, Chebyshev moved to St Petersburg in 1847. To gain the right to lecture he defended a thesis On integration by means of logarithms, on a problem going back to Abel and taken up by Mikhail Ostrogradsky: when can the integral of an algebraic function be expressed through logarithms? He became an adjunct at the university and taught algebra and number theory.

Euler and number theory

Viktor Bunyakovsky invited him to help prepare an edition of Euler’s papers on number theory, which appeared in two volumes in 1849. Reading Euler closely shaped his next work. His doctoral thesis, Teoriya sravnenii (Theory of Congruences), defended in May 1849, was a systematic textbook of elementary number theory. It won the Demidov Prize of the Academy of Sciences and served Russian students for decades. A German translation appeared in 1889.

Two memoirs on primes from these years made his name in Western Europe. The first (1848) proved that if the ratio \(\pi(x)\ln x/x\) has a limit, the limit is 1. The second (1850) gave explicit upper and lower bounds of the right order for the number of primes up to \(x\) and proved Bertrand’s postulate. Both are treated in Lessons 04 and 05. In 1850 he became extraordinary professor.

The 1852 journey and the engineer’s mathematics

From July to November 1852 Chebyshev toured France, England and Germany. His report to the ministry describes two parallel programmes. He met mathematicians: Liouville, Charles Hermite, Irénée-Jules Bienaymé, Joseph Serret and Jean-Victor Poncelet in Paris, Arthur Cayley and James Joseph Sylvester in London, and Peter Gustav Lejeune Dirichlet in Berlin. He also visited mills, factories and steam-engine works, and the collections of machines in Paris and London.

The machines gave him his central problem. James Watt’s “parallel motion” guides the piston rod of a steam engine along a nearly straight line using only hinged bars. Watt chose its proportions by experience. Chebyshev asked how to choose them so that the greatest deviation from straightness is as small as possible. That question, minimizing a maximum, led to the polynomials and the approximation theory of Lessons 06 and 07, and to the linkages of Lesson 10.

From 1852 to 1858 he taught practical mechanics at the Alexander Lyceum in Tsarskoe Selo. From the late 1850s he also served on the Artillery Committee, working on the ballistics of rifled guns and shells. His practical problems were various: maps of least distortion, the cutting of cloth for garments, calculating machines. In each case he looked for the extremal principle behind the engineering.

Cartography

He stated a theorem on conformal maps: among conformal projections of a region of the sphere, the one whose scale varies least is the one whose scale is constant along the boundary of the region. Chebyshev gave no proof; his student Dmitry Grave later supplied one.

The 1848 memoir: how primes thin out

For the general reader. Primes become rarer as numbers grow. Legendre and Gauss guessed, from tables, that the number of primes up to \(x\) is close to \(x/\ln x\). Chebyshev was the first to prove anything precise about this guess. He showed that no rival formula of Legendre’s type can do better, and that if the ratio settles down at all, it settles at exactly 1.

Let \(\pi(x)\) be the number of primes \(p\le x\), and let \(\operatorname{Li}(x)=\int_2^x \frac{dt}{\ln t}\). Legendre proposed in 1808 that \(\pi(x)\approx x/(\ln x - 1.08366)\). Gauss preferred \(\operatorname{Li}(x)\).

Proof sketch

Chebyshev worked with the Euler product for real \(s\gt1\), never with complex \(s\). Taking logarithms,

and since \((s-1)\zeta(s)\to1\), the left side is \(-\ln(s-1)+O(1)\). On the other hand the smooth comparison integral \(\int_2^\infty\frac{dt}{t^s\ln t}\) is also \(-\ln(s-1)+O(1)\) (Exercise 4.1). So the difference

stays bounded as \(s\to1^+\). Chebyshev then differentiated \(m\) times in \(s\), which inserts factors \(\ln^m t\), and used the known behaviour of the derivatives of \((s-1)\zeta(s)\). If \(\pi(x)-\operatorname{Li}(x)\) stayed above \(\alpha x/\ln^m x\) for all large \(x\), the differentiated integral would blow up as \(s\to1^+\), contradicting the bounds. The same argument rules out staying below \(-\alpha x/\ln^m x\).

The method shows that \(\operatorname{Li}(x)\) is the right comparison function to every order in \(1/\ln x\). It cannot prove that the limit exists: that is the prime number theorem, proved in 1896 by Jacques Hadamard and Charles de la Vallée Poussin using complex values of \(s\), as Riemann had proposed in 1859.

Chebyshev’s bounds and Bertrand’s postulate

For the general reader. Is there always a prime between a number and its double? Joseph Bertrand checked this up to three million in 1845 and assumed it. Chebyshev proved it in 1850 with an elementary trick: comparing factorials. The same trick showed that the number of primes up to \(x\) is always within about ten percent of \(x/\ln x\).

Chebyshev introduced the weighted counting functions

where \(\Lambda(n)=\ln p\) if \(n=p^k\) and 0 otherwise. They differ from \(\pi(x)\ln x\) by lower-order terms, and they are easier to handle because of the identity

which follows from Legendre’s formula for the exponent of \(p\) in \(N!\). The left side is known precisely by Stirling’s formula: \(\ln\lfloor x\rfloor!=x\ln x-x+O(\ln x)\).

The idea

Write \(T(x)=\ln\lfloor x\rfloor!=\sum_{n\le x}\psi(x/n)\). Form the combination

where \(c_n=1\) if \(n\) is coprime to 30, and \(c_n\) otherwise takes the values \(0\) or \(-1\) in a pattern of period 30. Grouping terms shows the right side lies between \(\psi(x)-\psi(x/6)\) and \(\psi(x)\) (Exercise 5.1). By Stirling, the left side equals \(Ax+O(\ln x)\), because the coefficients \(1-\frac12-\frac13-\frac15+\frac1{30}\) sum to zero and the \(x\ln x\) terms cancel. The lower bound follows at once, and the upper bound by summing \(\psi(x)-\psi(x/6)\lt Ax+O(\ln x)\) over \(x, x/6, x/36,\dots\), which gives the factor \(\frac65\).

For large \(n\), the bounds give \(\psi(2n)-\psi(n)\gt(2A-\tfrac65A)n-O(\ln^2n)\gt0\), with room to spare; the subtraction of prime powers is harmless. Chebyshev then checked the small cases by hand. Srinivasa Ramanujan (1919) and Paul Erdős (1932) later gave short proofs using only the central binomial coefficient \(\binom{2n}{n}\). Sylvester sharpened Chebyshev’s constants in the 1880s, but the method could never reach the constant 1 on both sides.

Chebyshev’s bias

In an 1853 letter to Paul Heinrich von Fuss, Chebyshev remarked that primes congruent to 3 mod 4 seem to outnumber those congruent to 1 mod 4. The race is not uniform: the lead changes infinitely often (Littlewood, 1914), but the bias is real in a logarithmic-density sense, as Michael Rubinstein and Peter Sarnak showed in 1994 under standard hypotheses.

Chebyshev polynomials

For the general reader. Among all polynomials of a given degree with leading coefficient 1, which one stays closest to zero across an interval? Chebyshev found the answer in 1854 while studying steam-engine linkages. It is a cosine in disguise, and it now sits at the heart of numerical computation.

For \(n\ge0\), the Chebyshev polynomial of the first kind is defined by

From \(\cos(n+1)\vartheta+\cos(n-1)\vartheta=2\cos\vartheta\cos n\vartheta\) one gets \(T_0=1\), \(T_1=x\) and

so \(T_n\) is a polynomial of degree \(n\) with leading coefficient \(2^{n-1}\) for \(n\ge1\). On \([-1,1]\) it satisfies \(|T_n|\le1\). Its zeros are \(x_k=\cos\frac{(2k-1)\pi}{2n}\), \(k=1,\dots,n\), and it reaches \(\pm1\) at the \(n+1\) points \(y_k=\cos\frac{k\pi}{n}\), \(k=0,\dots,n\), with \(T_n(y_k)=(-1)^k\).

Proof

Suppose \(p\) is monic of degree \(n\) with \(\max|p|\lt2^{1-n}\). Let \(q=2^{1-n}T_n-p\). The leading terms cancel, so \(\deg q\le n-1\). At the extremal points \(y_k\), \(2^{1-n}T_n(y_k)=(-1)^k2^{1-n}\) while \(|p(y_k)|\lt2^{1-n}\), so \(q(y_k)\) has the sign of \((-1)^k\). Thus \(q\) changes sign \(n\) times and has \(n\) zeros, forcing \(q=0\), a contradiction. Equality needs a slightly finer count of zeros, including double zeros at interior points.

Chebyshev also introduced the second kind \(U_n(\cos\vartheta)=\sin(n+1)\vartheta/\sin\vartheta\), and in the 1850s developed a general theory of orthogonal polynomials through continued fractions and least-squares interpolation. The \(T_n\) are orthogonal for the weight \((1-x^2)^{-1/2}\):

The German transliteration Tschebyscheff explains the conventional letter \(T\).

Best uniform approximation

For the general reader. To replace a complicated function by a polynomial, one can ask for the polynomial whose worst error is as small as possible. Chebyshev discovered that the best polynomial is recognised by a simple pattern: its error swings to the maximum, alternately above and below, more times than the polynomial has free coefficients.

Let \(f\in C[a,b]\), let \(\mathcal P_n\) be the real polynomials of degree at most \(n\), and let \(\|g\|=\max_{[a,b]}|g|\). Set \(E_n(f)=\inf_{p\in\mathcal P_n}\|f-p\|\). A minimizer exists because \(\mathcal P_n\) is finite-dimensional: the minimization may be restricted to the compact set \(\{p:\|p\|\le2\|f\|\}\).

Chebyshev stated the alternation property in his memoirs of 1854 and 1859 on minimum problems in the approximate representation of functions. Complete modern proofs were given around 1900, by Paul Kirchberger (1902) and Émile Borel (1905) among others.

Proof of sufficiency

Suppose \(f-p^*\) alternates with amplitude \(E\) at \(x_0,\dots,x_{n+1}\), and some \(p\in\mathcal P_n\) has \(\|f-p\|\lt E\). Then \(p-p^*=(f-p^*)-(f-p)\) has the sign of \(f-p^*\) at every \(x_i\), so it alternates in sign at \(n+2\) points and has \(n+1\) zeros. Being of degree \(\le n\), it is zero, a contradiction. Necessity is the harder direction: if there are fewer alternations, one constructs a small polynomial correction that lowers the error at all the extreme points simultaneously. Uniqueness follows because the average of two best approximations is best, and the alternation set forces them to agree at \(n+2\) points.

The minimal-deviation theorem of Lesson 06 is the case \(f(x)=x^{n}\), approximated by \(\mathcal P_{n-1}\) on \([-1,1]\): the error \(2^{1-n}T_n\) alternates at \(n+1\) points. In 1934 Evgeny Remez turned the theorem into an algorithm, still used to design digital filters and the polynomial kernels inside mathematical libraries.

Integration in finite terms

For the general reader. Some integrals, such as that of \(\sqrt{1+x^2}\), can be written with familiar functions; others, such as that of \(\sqrt{1+x^4}\), cannot. Calculus students learn tricks for the first kind. Chebyshev proved that for a whole classical family the tricks are complete: when they fail, no formula exists.

A binomial differential is \(x^m(a+bx^n)^p\,dx\) with \(a,b\ne0\) real and \(m,n,p\) rational, \(n\ne0\). An elementary function is one built from rational functions by finitely many algebraic operations, exponentials and logarithms.

The three cases

Substituting \(u=x^n\) turns the integral into \(\frac1n\int u^{q-1}(a+bu)^p\,du\) with \(q=\frac{m+1}{n}\). Write \(p=r/s\) and \(q=r'/s'\) in lowest terms.

• If \(p\in\mathbb Z\), put \(u=t^{s'}\): the integrand becomes rational in \(t\).
• If \(q\in\mathbb Z\), put \(a+bu=t^{s}\).
• If \(q+p\in\mathbb Z\), write the integrand as \(u^{q+p-1}\big(\tfrac{a+bu}{u}\big)^p\) and put \(\tfrac{a+bu}{u}=t^{s}\).

These substitutions were known to Euler. Chebyshev’s contribution was the converse: in every other case the integral is not elementary. His proof built on the method of Liouville, who had shown in the 1830s that an elementary antiderivative of an algebraic function must have a restricted shape (a sum of an algebraic function and constant multiples of logarithms of algebraic functions). Chebyshev’s earlier thesis of 1847 treated the logarithmic part of this problem; he later returned to pseudo-elliptic integrals, elliptic-looking integrals that turn out to be elementary.

Mean values and the laws of chance

For the general reader. Toss a fair coin many times and the fraction of heads approaches one half. Jacob Bernoulli proved this for coins in 1713. Chebyshev proved it for any sequence of independent quantities with bounded spread, using a two-line inequality, and then began the attack on the bell curve that his students Andrei Markov and Aleksandr Lyapunov completed.

Proof. \(\sigma^2=\mathbb E(X-\mu)^2\ge\mathbb E\big[(X-\mu)^2\mathbf 1_{|X-\mu|\ge k\sigma}\big]\ge k^2\sigma^2\,\mathbb P(|X-\mu|\ge k\sigma)\). Irénée-Jules Bienaymé had found the inequality in 1853 in a dispute with Cauchy over least squares. Chebyshev found it independently and made it the engine of a general theorem, in Des valeurs moyennes (1867, in Russian and in French in Liouville’s journal).

Proof. The average has variance \(\frac1{n^2}\sum\sigma_i^2\le C/n\). Apply the inequality: the probability of a deviation \(\ge\varepsilon\) is at most \(C/(n\varepsilon^2)\to0\). Both Bernoulli’s and Poisson’s laws are special cases. Andrey Kolmogorov later credited Chebyshev with being the first to reason clearly in terms of random quantities and their means.

The central limit theorem, 1887

In On two theorems concerning probability (1887; French version in Acta Mathematica, 1890–91), Chebyshev proposed the method of moments. If the moments of a normalized sum \(S_n/\sqrt{\operatorname{Var}S_n}\) converge to those of the standard normal law,

then its distribution converges to the normal distribution. He supported this with inequalities bounding a distribution function by its moments (the Chebyshev–Markov–Stieltjes inequalities), and sketched conditions under which the moments of the sum converge. The argument had gaps. Markov filled them in 1898 with the same method; Lyapunov proved a stronger theorem in 1900–1901 with characteristic functions.

Linkages and machines

For the general reader. Before precision slides, engineers guided pistons with jointed bars. No arrangement of a few bars obviously traces a straight line. Chebyshev designed linkages whose guiding point stays straight to within a fraction of a percent, built a walking machine and a calculating machine, and turned the design problem into mathematics.

Chebyshev’s linkage

Fix pivots \(A=(0,0)\) and \(D=(2,0)\). Two cranks \(AB\) and \(DC\) of length \(\tfrac52\) cross each other, and a coupler \(BC\) of length 1 joins their ends. The midpoint \(P\) of \(BC\) moves along a curve that, as \(P\) passes from \(x=0\) to \(x=2\), stays within about \(0.005\) of the line \(y=2\), never below it: a deviation of a quarter of a percent over the working stroke.

Watt’s linkage matches the straight line to high order at one point, like a Taylor expansion. Chebyshev’s criterion spreads the error evenly, like the alternation theorem of Lesson 07. His 1854 memoir analysed Watt’s parallelogram this way, and the Chebyshev polynomials appeared there as the extremal solutions.

Machines

Chebyshev designed dozens of mechanisms: the straight-line linkage above, the “lambda” linkage, a walking machine imitating the gait of an animal, a self-propelled chair and a rowing mechanism. In 1876 he described a calculating machine for addition and subtraction, and around 1882 a version that also multiplied and divided. Seven of his mechanisms were exhibited at the World’s Columbian Exposition in Chicago in 1893, and several survive in museums in Moscow, St Petersburg and Paris.

Exact straight lines

Chebyshev is said to have doubted that a linkage could trace an exact straight line. In 1864 the French engineer Charles-Nicolas Peaucellier found one, and in 1871 Lipmann Lipkin in St Petersburg found it independently. Their cell realises an inversion in a circle, which maps a circle through the centre of inversion onto a straight line.

The Academy and the St Petersburg school

Chebyshev rose steadily through the institutions of imperial Russia. The St Petersburg Academy of Sciences elected him adjunct in applied mathematics in 1853, extraordinary academician in 1856 and ordinary academician in 1859. At the university he became ordinary professor in 1860 and merited professor in 1872. He served on the scientific committee of the Ministry of Education, which set mathematics programmes for schools, and on the Artillery Committee.

Recognition abroad came early. The Paris Academy elected him a corresponding member in 1860 and a foreign associate in 1874. He became a corresponding member of the Berlin Academy in 1871 and of the Bologna Academy in 1873, a Fellow of the Royal Society of London in 1877, and a member of the Swedish Academy of Sciences in 1893. France also made him a member of the Légion d’honneur. He spent most summers in Western Europe, published largely in French, and from 1873 to 1882 spoke regularly at the French Association for the Advancement of Science. He preferred to be seen as a mathematician of the whole world rather than of one nation.

Teacher and school

Former students described him as a superb lecturer: punctual, rigorous, concise, and fond of digressions about mathematicians he had met abroad. He set his students concrete problems that called for sharp, explicit answers. Around him grew the St Petersburg school, the first lasting school of research mathematics in Russia. Its members included Aleksandr Korkin and Yegor Zolotarev (quadratic forms; Zolotarev’s work on algebraic integers and on extremal polynomials), Andrei Markov (probability and Markov chains; number theory), Aleksandr Lyapunov (stability of motion; the central limit theorem), Konstantin Posse, Vladimir Markov (extremal problems for polynomials), Dmitry Grave, who attended his lectures in the 1880s, and Lipmann Lipkin. The later Russian schools of probability, approximation and number theory, through Steklov, Bernstein and Vinogradov, descend from it.

Last years and legacy

In 1882 Chebyshev retired from teaching at the university to concentrate on research at the Academy. He continued to publish, to travel to Western Europe and to build mechanisms. He never married. He died in St Petersburg on 8 December 1894 (26 November Old Style), aged 73. His collected works, edited by Andrei Markov and Nikolai Sonin, appeared in French and Russian in two volumes (1899 and 1907).

Two years after his death, Hadamard and de la Vallée Poussin proved the prime number theorem. Chebyshev’s 1848 result fixed in advance what the limit had to be, and his functions \(\theta\) and \(\psi\) are the ones in which the theorem is usually stated: \(\psi(x)\sim x\).

Legacy

• Number theory: the Chebyshev functions, elementary bounds of the right order, Bertrand’s postulate and Chebyshev’s bias, a theme of modern research on prime races.
• Approximation theory: best uniform approximation, equioscillation and the Remez algorithm; Chebyshev polynomials as the basis of spectral methods and of software such as Chebfun.
• Probability: the inequality, the general law of large numbers, and the method of moments, carried forward by Markov and Lyapunov into the twentieth century.
• Mechanism design: the minimax criterion for linkages and the analytic study of mechanisms.
• Geometry: Chebyshev nets, from his 1878 paper on cutting cloth, are now used in the study of woven and gridshell surfaces.
• Honours: a lunar crater and the asteroid 2010 Chebyshev bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Pafnuty Lvovich Chebyshev, MacTutor History of Mathematics.
2. Pafnuty Chebyshev, Wikipedia (with references).
3. Pafnuty Lvovich Chebyshev, Mathematics Genealogy Project.
4. Pafnuty Chebyshev (1821–1894), Bhavana.
5. A. P. Youschkevitch, Chebyshev, Pafnuty Lvovich, Dictionary of Scientific Biography, vol. 3, Scribner, 1971.
6. P. Butzer and F. Jongmans, P. L. Chebyshev (1821–1894): a guide to his life and work, J. Approx. Theory 96 (1999), 111–138.

Mathematics

1. P. L. Tchebychef, Oeuvres, ed. A. Markoff and N. Sonin, 2 vols., St Petersburg, 1899–1907.
2. P. L. Tchebychef, Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée; Mémoire sur les nombres premiers, J. Math. Pures Appl. 17 (1852).
3. P. L. Tchebychef, Sur l’intégration des différentielles irrationnelles, J. Math. Pures Appl. 18 (1853).
4. P. L. Tchebychef, Des valeurs moyennes, J. Math. Pures Appl. (2) 12 (1867).
5. T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, 1990.
6. L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, 2013; and Chebfun.
7. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, ch. 22 (Chebyshev’s functions and Bertrand’s postulate).
8. M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Math. 3 (1994), 173–197.
9. E. Seneta, A tricentenary history of the law of large numbers, Bernoulli 19 (2013); arXiv:1309.6488.