# Matyáš Lerch

### Quick Info

Born
20 February 1860
Milínov (near Sušice), South Bohemia (now Czech Republic)
Died
3 August 1922
Sušice, Czechoslovakia (now Czech Republic)

Summary
Mathias Lerch was a Czech mathematician who worked in analysis and number theory.

### Biography

Mathias Lerch's name is the first thing we have to mention. Like most people from that part of the world in that period he has both a German version of his name, Mathias, and a Czech version, Matyáš. However, things are not quite so simple since his given name (the one that appears on his birth certificate) is Matěj. However there is no evidence that he ever used the name Matěj and he always used Matyáš. His parents were Vojtěch Lerch and Barbora Adamovská, who worked on a farm, and had five children who survived past the baby stage, two boys and three girls. However, after the two eldest died as babies, a third boy, Vojtěch, died when a young child and a girl, Marie, drowned in a water tank on the farm when 3 or 4 years old. The remaining children, Matyáš (the subject of this biography), Ružena and Marie all grew to adulthood.

Lerch should have begun his education at an elementary school at the age of six. However, although he had been a bright agile child up to that age, sadly when he was six years old he had a serious accident which left him badly crippled. Even after quite a while he was only able to walk with crutches, and could not attend school. When Lerch was eight years old the family moved to Sušice, about 5 km from the small village of Milínov. Although now more able to get around, but still badly handicapped with a bent knee, he spent his days in the fields and meadows on the outskirts of the town. It was there that the burgomaster of Sušice spotted him and thought that he was fit enough to attend the elementary school. He was nine years old when he entered the elementary school in Sušice, three years older than the official starting age. However, he soon showed his abilities [17]:-
Lerch later often mentioned a mathematics class episode from the elementary school when he pointed out a slip in a teacher's solution of a problem on the blackboard. The episode made waves in the town.
After the elementary school, which he left older than other pupils because of his late start, Lerch progressed to the lower secondary burgher school of Sušice. There he was fortunate to have Emil Seifert as his mathematics teacher. Seifert was only four years older than his talented pupil and just starting out on his teaching career. They developed both a teacher-pupil relationship and also, given the closeness in age, a real friendship. After studying at the burgher school, Lerch reached the end of his compulsory schooling in 1877. At this time his teachers tried to persuade him to continue his education but the family were poor and his parents were keen that he should bring in a wage. He became clerk at František Scheinhost's match factory in Sušice. However, it only took him a few weeks to decide that this was not for him and, despite knowing that he would face an extremely difficult financial position, he decided to continue his education. He left his job at František Scheinhost's match factory at the end of summer 1877 and enrolled in the Realschule in Pilsen, entering into the 5th grade, thereby making up one of the years he had lost. He corresponded with his former teacher Emil Seifert and we see from these letters that he was reading advanced mathematics texts and often criticising the methods used in them and suggesting better ones.

From Lech's letters to Seifert we know that in 1878 he was reading Emil Weyr and Eduard Weyr's Základy vyšší geometrie , František Josef Studnička's Základy vyšší matematiky and Friedrich Autenheimer's Elementarbuch der Differential- und Integralrechnung . It seems that a religious dispute with one of his teachers led to Lerch leaving the school in Pilsen and attending a secondary school at Rakovník, due west of Prague, in 1879. He graduated from this school on 13 July 1880. His final certificate gives his examination results: mathematics, outstanding; descriptive geometry, excellent; physics, excellent; Czech, excellent; French, good; history, good; geography, good; chemistry, good; natural history, good; religion, satisfactory; German, satisfactory; and drawing, satisfactory. He had spent three years at the schools in Pilsen and Rakovník having to survive on very little money. He had been able to make a little from tutoring but life at this stage was incredibly hard.

At this time in his education, Lerch was aiming at becoming a school teacher and, in the autumn of 1880, he enrolled in the Czech Polytechnic in Prague. However, not long after he began his studies he was told that becoming a teacher was impossible since with his crippled leg he would never obtain the necessary health certificate. With most career paths now closed to him, Lerch decided at this time to aim at becoming a university teacher. In his first year 1880-81 he took courses on mathematics and physics, while later he took courses on algebra, geometry, mechanics, statics, dynamics, and elasticity. In addition to attending courses at the Czech Polytechnic, he also attended courses at the German Polytechnic. Among his teachers were Anton Karl Grünwald (1838-1920), František Josef Studnička (1836-1903) and Eduard Weyr. In the year 1883-84 Lerch wrote six papers, his first paper Beitrag zur Theorie der Kegelschnitte (1881) having been written in his first year at university.

Although Lerch had been interested in geometry during his early studies, he moved to analysis and studied the work being published by Hermann Laurent, Otto Stolz and Johannes Thomae. Finding their approach unsatisfactory he decided he had to build his own approach so, to gain the necessary expertise, he decided that he should study in Germany. Therefore, he went to the University of Berlin where he studied during 1884-85 and was taught by Karl Weierstrass, Leopold Kronecker, Carl Runge and Lazaras Fuchs. He attended courses on the theory of elliptic functions by Weierstrass, and courses on the theory of algebraic equations and on simple and multiple integrals by Kronecker. He also attended several lecture courses by Fuchs, namely (i) Introduction to the theory of infinite series, (ii) Integration of differential equations, (iii) The theory of linear differential equations, and (iv) Invariant theory. He also took Runge's courses on the solution of equations, on convergence and continuity, and on the differentiation of analytic expressions. In some of these lectures he sat beside Sofia Kovalevskaya. Although he had gone to Berlin expecting to move to research in areas covered by Weierstrass, he left having been more influenced by Kronecker. This set the direction of his future research to be on special functions, infinite series and analytic functions.

In 1886 Lerch habilitated at the Czech Technical Institute in Prague and on 14 September of that year joined the teaching staff there. Over the next few years Lerch produced, on average, about one paper per month, a quite stunning achievement. However as the years went by it became clear that he was not going to be promoted to a professorship in Prague. In fact during these years he only held positions of assistant to Eduard Weyr from 1885 to 1888 and assistant to Gabriel Blažek (1842-1910), from 1888 to 1896, who, in addition to working on pure mathematics, was a politician and a banker. Lerch lectured on analytical functions and on the geometry of rational curves while an assistant to Eduard Weyr, but, after becoming an assistant to Blažek, he gave lectures on potential theory, higher algebra, the theory of numbers, analytic geometry, and the theory of functions.

In [5] Otakar Boruvka (1899-1995), a student of Lerch's, gives what he believes are the reasons for Lerch's failure to be promoted:-
The faculty of the Czech universities and polytechnics, which, moreover, were concentrated in Prague, were often enlightened men, but their wishes and actions were muted by the views held by the Imperial and Royal dignities, and their actions were subject to the approval of governing bodies in Vienna. Therefore, it is not surprising that thirty-six year old Matyáš Lerch, who it appears had directed his criticism and sharp wit against the mentioned dignities, despite being the author of about 120 scientific papers published in many international journals and having lectured on the results in these at the University of Paris, despite being an associate member of the Royal Czech Academy of Sciences and of the Czech Academy since 1893, was not able to find a reasonable position in his Czech homeland after ten years of being an instructor in Prague, and therefore had to move abroad.
In 1896 he was appointed to a chair when he accepted a professorship at the University of Fribourg in Switzerland. Charles Hermite wrote to him about his move to the University of Fribourg.
I have noted with great satisfaction the major change which happened to you in the last few days and the happy result of your appointment as a full professor at the University of Fribourg in Switzerland. You have reasons to believe in Providence, which interferes in our lives in order to secure and save those with the courage to go straight ahead crowning with success all their actions, and whose aim is science and not success. I am prejudiced against Bohemia, which should have retained you for its honour and should have recognized long ago the great importance of the papers which you wrote and which placed you in the top rank of contemporary mathematicians. This leads me to a question, Sir, whether you will publish your discoveries in the 'Transactions of the Bohemian Academy' which you have enriched with a large number of beautiful mathematical memoirs, and whose mathematical section would be much poorer from your absence. Rather, I am inclined to believe that you will remain firmly attached to your fatherland.
At the University of Fribourg, Lerch lectured in French and German on elliptic functions and similar topics. Between 1900 and 1907 he did not publish any papers in the Czech language. In general he was unhappy with the students at Fribourg, except for a couple, especially Michel Plancherel whom he found to be an outstanding Ph.D. student. Plancherel's thesis was Sur les congruences (mod 2m) relatives au nombre des classes des formes quadratiques binaires aux coefficients entiers et à discriminant négatif (1907). Lerch was Dean of the Faculty of Science in the academic year 1900-01. He returned to the Czech Republic in 1906 where he was appointed professor of mathematics at the Czech Technical Institute in Brno. Over the following fourteen years he wrote 31 papers on infinite series, geometry, special functions, and number theory.

Tomás Masaryk was a political leader who liberated the Czechs and Slovaks from Austrian rule. In 1918 Masaryk was elected president of Czechoslovakia. A year after the founding of Czechoslovakia, a new university, named the Masaryk University after the first president, was founded in Brno and Lerch became the first professor of mathematics there in 1920. His inaugural lecture was held on 19 October 1920 in the lecture hall of the Czech Technical College in Gorkého Street. Lerch married Roůžena Sejpková on 13 January 1921.

Lerch wrote 238 papers, listed in [20], mostly on analysis (about 150 papers) and number theory (about 40 papers). Some of his work is fundamental in modern operator calculus. He also wrote on geometry and numerical methods. Matti Jutila, in his review of [2], describes Lerch's work in number theory as follows:-
Matyáš Lerch (1860-1922) was a remarkable Czech mathematician who published about 250 papers, some fifty of which were devoted to number theory. His favourite topics in number theory included binary quadratic forms, quadratic residues, Gauss sums and Fermat quotients. Also, the Lerch zeta-function is a well-known generalization of Riemann's zeta-function due to him ... this contribution to analytic number theory [is] published in 'Acta Mathematica' in 1887...
In [2] there is a short biography of Lerch and the reviews of his number-theoretic papers given in the Jahrbuch über die Fortschritte der Mathematik are reproduced. Also quoted in full in [2] is the first paper Lerch wrote on number theory and two other papers which he wrote on Fermat quotients. It also contains a summary of a paper on quadratic residues and forms which was published after his death.

The authors of [19] write:-
It is necessary to mention the fact that M Lerch very much liked dealing with factual problems and avoided general proofs. He considered the solution of a given problem finished if he could present its numerical calculation. For this purpose he even bought a 20-digit calculator for his institute at Brno Technical University. He did not like formulas inaccessible to numerical calculation. That is why among his papers there are many dealing with new methods of calculation of problems solved before by other mathematicians. This is also linked to his preference of the theory of numbers where he achieved eminent success.
As we mentioned above, Lerch won the Grand Prize of the Paris Academy of Sciences in 1900 with a work on number theory, a great honour for any mathematician and an even greater achievement for a mathematician from outside France. He was also honoured with honorary membership of the Union of Czech Mathematicians and Physicists in 1907 and an honorary degree from the Czech University in Prague in 1909. Today he is honoured with a grammar school in Brno being named for him as well as an elementary school in Sušice.

He is also well known, however, for his work in analysis. In that topic he studied infinite series, and the gamma function as well as other special functions. He also studied elliptic functions and integral equations. Often the importance of his work is in the methods which he introduced rather than the specific results themselves. He introduced an auxiliary parameter for meromorphic functions. He also studied the principle of most rapid convergence of a series. He is remembered today for his solution of integral equations in operator calculus and for the 'Lerch formula' for the derivative of Kummer's trigonometric expansion for $\log G(v)$.

Lerch suffered from diabetes which cased his health to progressively worsen over the years although staying at health resorts in the vacations improved things temporarily. In July 1922 he went to Sušice and several times bathed in the river. The day after his 31 July swim, he felt ill and on 2 August the doctors diagnosed pneumonia. With his health already greatly weakened with diabetes, he died on the morning of 3 August.

### References (show)

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14. Lerch, Matyas (1860-1922), Mathematiker, Österreichisches Biographisches Lexikon. http://www.biographien.ac.at/oebl/oebl_L/Lerch_Matyas_1860_1922.xml
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19. P Šarmanová and P Šišma, Matyáš Lerch, Masaryk University, Faculty of Science, Department of Mathematics and Statistics. http://www.math.muni.cz/~sisma/prace/fhlerch.html
20. J Škrášek, List of works of Prof Mathias Lerch (Russian), Czechoslovak. Mat. Z. 3 (78), (1953), 111-122.
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24. J Škrášek, Le centenaire de la naissance de Matyás Lerch, Czechoslovak Math. J. 10 (85) (1960), 631-635.