# Giovanni Ricci

### Quick Info

Born
17 August 1904
Florence, Italy
Died
9 September 1973
Milan, Italy

Summary
Giovanni Ricci was an Italian mathematician who worked in number theory, differential geometry, mathematical analysis and theory of series.

### Biography

Giovanni Ricci was brought up in Florence where he underwent his school education. He went to Pisa where he studied mathematics at the Scuola Normale Superiore which is associated with the University of Pisa. He was only 21 years old when he submitted his doctoral thesis Le transformazioni de Christoffel e di Darboux per le superficie rotonde,coniche e cilindriche. Alcune generalizioni, per rotolamento del cono e del cilindro di rotazion and was awarded a doctorate on 15 December 1925.

After graduating, Ricci went to Rome where he was appointed as a lecturer. He returned to Pisa in 1928 when he was appointed as a professor at the Scuola Normale Superiore [1]:-
The time he spent at the Scuola Normale Superiore was very fruitful in scientific research and gave birth to some papers which stand out among his publications.
In fact Ricci held this position for eight years. During that time he published work on the Goldbach conjecture which concerns writing numbers as the sum of primes, and also on Hilbert's Seventh Problem which asks whether or not aı was transcendental when a and b are algebraic. Hilbert himself remarked that he expected this Seventh Problem to be harder than the solution of the Riemann conjecture. However Riemann's intuition was not too good here for it was solved independently by Gelfond and Schneider in 1934. As examples of the papers Ricci published during his eight years as a professor in Pisa we mention: Sulle funzioni simmetriche delle radici dell'unità secondo un modulo composto (1931), Sui grandi divisori primi delle coppie di interi in posti corrispondenti di due progressioni aritmetiche. Applicazione del metodo di Brun (1933), Sulle serie di potenze che rappresentano funzioni razionali a coefficienti razionali e con i poli appartenenti ad una progressione geometrica (1934), Su un teorema di Tchebychef-Nagel (1934), and Sui teoremi Tauberiani (1934). Let us also note that, in 1951, Ricci wrote a paper on the school of mathematics in Pisa La scuola matematica pisana dal 1848 al 1948 . This covered a period of one hundred years, including the time that he both studied there and worked there as a professor.

In 1936 Ricci moved to Milan where he was appointed as Professor of Mathematical Analysis at the University. Of course this was a difficult period in Italy. In 1935 the League of Nations had imposed sanctions against Italy following its invasion of Abyssinia. The following year Mussolini and Hitler declared the Rome-Berlin Axis and in 1937 Italy left the League of Nations. Although World War II began September 1939, it was not until June 1940 that Italy declared war on Britain and France. Ricci was unable to continue his research with the vigour that he had in Pisa and was more involved in teaching and administration. Between his taking up the position in Milan and the end of World War II, a period of nine years, he published only two works. Sull'irrazionalità del rapporto della circonferenza al diametro (1942) was the published version of a lecture he gave at the Second Italian Mathematical Congress in Bologna in 1940. The other paper Problemi secolari e risposte recenti nel campo dell'aritmetica (1945) was again the published version of a lecture. In it Ricci discussed the estimation of exponential sums, the distribution of $f (x)$ mod 1, Waring's problem and Goldbach's problem.

Ricci held the chair of Mathematical Analysis in Milan for over 36 years. He published many excellent expository and historical articles from 1948 onwards, and also did some interesting research on the distribution of primes. However his research never returned to the output of his days in Pisa, either in quantity nor depth. We will look briefly, however, at some of his interesting expository articles: Figure, reticoli e computo di nodi (1948) discussed classical lattice-point problems and puts these in historical perspective; La differenza di numeri primi consecutivi (1952) looks at the the history of the various problems concerned with the difference between consecutive primes; and Aritmetica additiva: aspetti e problemi (1954) is an expository account of classical and modern results in the additive theory of numbers.

We mentioned Ricci's research on the distribution of primes. In the papers Sul coefficiente di Viggo Brun (1953) and Sull'andamento della differenza di numeri primi consecutivi (1954) he presented many results on the difference between consecutive primes. We should not, however, give the impression that all Ricci's work was in number theory for he also made significant contributions to the theory of functions of a complex variable.

Cugiani, who knew Ricci from 1937 onwards, writes in [1] about his abilities as a teacher:-
He was a born teacher, partly owing to his mastery of the language, since he came from Toscana, the region where the best Italian is spoken; but, most of all, he was an excellent teacher because of his extreme clarity of mind and his profound intellectual honesty, which forced him to present topics in such a way that their logical connections became absolutely obvious; those who listened to him often thought that what he had explained was a chain of trivial statements (as he used to say). Of great help was also his wide mathematical knowledge, deriving both from the school he had followed and from his continuous interest in research.
As to his character and personality, Cugiani writes:-
His profound aesthetic sense was quite an important aspect of his personality, and it seemed to be related to his fondness for library organisation, his love of books, which he always regarded as objects of high aesthetic value. He once said to me: "Opening a book neatly printed on good paper and properly bound is one of the pleasures allowed to man". To his everyday work, his aesthetic sense brought clarity and a sober and balanced language, both in research and in teaching. Of course it also had other, more evident, manifestations, for instance his love of music, which he often combined with the pleasures of mathematical studies through a subtle transposition of activities which he felt to be very close to each other.
Finally we mention that Ricci's most famous doctoral student during his time in Milan was Enrico Bombieri.

### References (show)

1. M Cugiani, Giovanni Ricci (1904-1973), Acta Arith. 46 (4) (1986), 303-311.
2. M Cugiani, Commemoration of Giovanni Ricci (Italian), Geometry of Banach spaces and related topics, Milan, 1983, Rend. Sem. Mat. Fis. Milano 53 (1983), 11-15.
3. M Cugiani, Commemorazione di Giovanni Ricci, Rend. Sem. Mat. Fis. Milano 43 (1973), 7-23.
4. F G Tricomi, Giovanni Ricci (1904-1973) (Italian), Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 108 (3-4) (1974), 585-589.