Hans-Egon Richert
Encyclopedia
Hans-Egon Richert was a German
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 who worked primarily in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

. He is the author (with Heini Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

) of a definitive book
on sieve theory
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

.

Life and education

Hans-Egon Richert was born in 1924 in Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

. He attended the University of Hamburg
University of Hamburg
The University of Hamburg is a university in Hamburg, Germany. It was founded on 28 March 1919 by Wilhelm Stern and others. It grew out of the previous Allgemeines Vorlesungswesen and the Kolonialinstitut as well as the Akademisches Gymnasium. There are around 38,000 students as of the start of...

 and received his Ph.D under Max Deuring
Max Deuring
Max Deuring was a mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p...

 in 1950. He held a temporary chair at the University of Göttingen and then a newly-created chair at the University of Marburg. In 1972 he moved to the University of Ulm
University of Ulm
The University of Ulm is a public university in the city of Ulm, in the South German state of Baden-Württemberg. The university was founded in 1967 and focuses on natural sciences, medicine, engineering sciences, mathematics, economics and computer science...

, where he remained until his retirement in 1991. He died on November 25, 1993 in Blaustein
Blaustein
Blaustein is a municipality in the district of Alb-Donau Baden-Württemberg in Germany. It is situated on the Blau River, 6 km west of Ulm and has about 15,000 inhabitants.It was created in 1968 after the union of Ehrenstein and Klingenstein...

, near Ulm
Ulm
Ulm is a city in the federal German state of Baden-Württemberg, situated on the River Danube. The city, whose population is estimated at 120,000 , forms an urban district of its own and is the administrative seat of the Alb-Donau district. Ulm, founded around 850, is rich in history and...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

.

Work

Richert worked primarily in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

, and beginning around 1965 started a collaboration with Heini Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

 and shifted his focus to sieve theory
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

. For many years he was a chairman of the Analytic Number Theory meetings at the Mathematical Research Institute of Oberwolfach
Mathematical Research Institute of Oberwolfach
The Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany, was founded by mathematician Wilhelm Süss in 1944...

.

Analytic number theory

Richert made contributions to additive number theory
Additive number theory
In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has...

, Dirichlet series, Riesz summability
Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean...

, the multiplicative analog of the Erdős–Fuchs theorem
Erdos–Fuchs theorem
In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.The theorem...

, estimates of the number of non-isomorphic abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s, and bounds for exponential sum
Exponential sum
In mathematics, an exponential sum may be a finite Fourier series , or other finite sum formed using the exponential function, usually expressed by means of the functione = \exp.\,...

s. He proved the exponent 15/46 for the Dirichlet divisor problem, a record that stood for many years.

Sieve methods

One of Richert's notable results was the Jurkat–Richert theorem
Jurkat–Richert theorem
The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.It was proved in 1965 by Wolfgang B...

, joint work with Wolfgang B. Jurkat that improved the Selberg sieve
Selberg sieve
In mathematics, in the field of number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences...

 and is used in the proof of Chen's theorem
Chen's theorem
right|thumb|Chen JingrunChen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime . The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original...

.
Richert also produced a "readable form" of Chen's theorem (it is covered in the last chapter of Sieve Methods).

Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

 & Richert's book Sieve Methods was the first exhaustive account of the subject.

In reviewing the book in 1976, Hugh Montgomery
Hugh Montgomery
Hugh Montgomery is an American mathematician, working in the fields of analytic number theory and mathematical analysis. As a Rhodes scholar, Montgomery earned his Ph.D. from the University of Cambridge...

 wrote "In the past, researchers have generally derived the sieve bounds required for an application, but now workers will find that usually an appeal to an appropriate theorem of Sieve methods will suffice," and "For years to come, Sieve methods will be vital to those seeking to work in the subject, and also to those seeking to make applications."

External links

Hans-Egon Richert (1924, Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...

–November 25, 1993, Blaustein
Blaustein
Blaustein is a municipality in the district of Alb-Donau Baden-Württemberg in Germany. It is situated on the Blau River, 6 km west of Ulm and has about 15,000 inhabitants.It was created in 1968 after the union of Ehrenstein and Klingenstein...

) was a German
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 who worked primarily in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

. He is the author (with Heini Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

) of a definitive book
on sieve theory
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

.

Life and education

Hans-Egon Richert was born in 1924 in Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

. He attended the University of Hamburg
University of Hamburg
The University of Hamburg is a university in Hamburg, Germany. It was founded on 28 March 1919 by Wilhelm Stern and others. It grew out of the previous Allgemeines Vorlesungswesen and the Kolonialinstitut as well as the Akademisches Gymnasium. There are around 38,000 students as of the start of...

 and received his Ph.D under Max Deuring
Max Deuring
Max Deuring was a mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p...

 in 1950. He held a temporary chair at the University of Göttingen and then a newly-created chair at the University of Marburg. In 1972 he moved to the University of Ulm
University of Ulm
The University of Ulm is a public university in the city of Ulm, in the South German state of Baden-Württemberg. The university was founded in 1967 and focuses on natural sciences, medicine, engineering sciences, mathematics, economics and computer science...

, where he remained until his retirement in 1991. He died on November 25, 1993 in Blaustein
Blaustein
Blaustein is a municipality in the district of Alb-Donau Baden-Württemberg in Germany. It is situated on the Blau River, 6 km west of Ulm and has about 15,000 inhabitants.It was created in 1968 after the union of Ehrenstein and Klingenstein...

, near Ulm
Ulm
Ulm is a city in the federal German state of Baden-Württemberg, situated on the River Danube. The city, whose population is estimated at 120,000 , forms an urban district of its own and is the administrative seat of the Alb-Donau district. Ulm, founded around 850, is rich in history and...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

.

Work

Richert worked primarily in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

, and beginning around 1965 started a collaboration with Heini Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

 and shifted his focus to sieve theory
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

. For many years he was a chairman of the Analytic Number Theory meetings at the Mathematical Research Institute of Oberwolfach
Mathematical Research Institute of Oberwolfach
The Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany, was founded by mathematician Wilhelm Süss in 1944...

.

Analytic number theory

Richert made contributions to additive number theory
Additive number theory
In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has...

, Dirichlet series, Riesz summability
Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean...

, the multiplicative analog of the Erdős–Fuchs theorem
Erdos–Fuchs theorem
In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.The theorem...

, estimates of the number of non-isomorphic abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s, and bounds for exponential sum
Exponential sum
In mathematics, an exponential sum may be a finite Fourier series , or other finite sum formed using the exponential function, usually expressed by means of the functione = \exp.\,...

s. He proved the exponent 15/46 for the Dirichlet divisor problem, a record that stood for many years.

Sieve methods

One of Richert's notable results was the Jurkat–Richert theorem
Jurkat–Richert theorem
The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.It was proved in 1965 by Wolfgang B...

, joint work with Wolfgang B. Jurkat that improved the Selberg sieve
Selberg sieve
In mathematics, in the field of number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences...

 and is used in the proof of Chen's theorem
Chen's theorem
right|thumb|Chen JingrunChen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime . The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original...

.
Richert also produced a "readable form" of Chen's theorem (it is covered in the last chapter of Sieve Methods).

Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

 & Richert's book Sieve Methods was the first exhaustive account of the subject.

In reviewing the book in 1976, Hugh Montgomery
Hugh Montgomery
Hugh Montgomery is an American mathematician, working in the fields of analytic number theory and mathematical analysis. As a Rhodes scholar, Montgomery earned his Ph.D. from the University of Cambridge...

 wrote "In the past, researchers have generally derived the sieve bounds required for an application, but now workers will find that usually an appeal to an appropriate theorem of Sieve methods will suffice," and "For years to come, Sieve methods will be vital to those seeking to work in the subject, and also to those seeking to make applications."

External links

Hans-Egon Richert (1924, Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...

–November 25, 1993, Blaustein
Blaustein
Blaustein is a municipality in the district of Alb-Donau Baden-Württemberg in Germany. It is situated on the Blau River, 6 km west of Ulm and has about 15,000 inhabitants.It was created in 1968 after the union of Ehrenstein and Klingenstein...

) was a German
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 who worked primarily in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

. He is the author (with Heini Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

) of a definitive book
on sieve theory
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

.

Life and education

Hans-Egon Richert was born in 1924 in Hamburg
Hamburg
-History:The first historic name for the city was, according to Claudius Ptolemy's reports, Treva.But the city takes its modern name, Hamburg, from the first permanent building on the site, a castle whose construction was ordered by the Emperor Charlemagne in AD 808...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

. He attended the University of Hamburg
University of Hamburg
The University of Hamburg is a university in Hamburg, Germany. It was founded on 28 March 1919 by Wilhelm Stern and others. It grew out of the previous Allgemeines Vorlesungswesen and the Kolonialinstitut as well as the Akademisches Gymnasium. There are around 38,000 students as of the start of...

 and received his Ph.D under Max Deuring
Max Deuring
Max Deuring was a mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p...

 in 1950. He held a temporary chair at the University of Göttingen and then a newly-created chair at the University of Marburg. In 1972 he moved to the University of Ulm
University of Ulm
The University of Ulm is a public university in the city of Ulm, in the South German state of Baden-Württemberg. The university was founded in 1967 and focuses on natural sciences, medicine, engineering sciences, mathematics, economics and computer science...

, where he remained until his retirement in 1991. He died on November 25, 1993 in Blaustein
Blaustein
Blaustein is a municipality in the district of Alb-Donau Baden-Württemberg in Germany. It is situated on the Blau River, 6 km west of Ulm and has about 15,000 inhabitants.It was created in 1968 after the union of Ehrenstein and Klingenstein...

, near Ulm
Ulm
Ulm is a city in the federal German state of Baden-Württemberg, situated on the River Danube. The city, whose population is estimated at 120,000 , forms an urban district of its own and is the administrative seat of the Alb-Donau district. Ulm, founded around 850, is rich in history and...

, Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

.

Work

Richert worked primarily in analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

, and beginning around 1965 started a collaboration with Heini Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

 and shifted his focus to sieve theory
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

. For many years he was a chairman of the Analytic Number Theory meetings at the Mathematical Research Institute of Oberwolfach
Mathematical Research Institute of Oberwolfach
The Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany, was founded by mathematician Wilhelm Süss in 1944...

.

Analytic number theory

Richert made contributions to additive number theory
Additive number theory
In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has...

, Dirichlet series, Riesz summability
Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean...

, the multiplicative analog of the Erdős–Fuchs theorem
Erdos–Fuchs theorem
In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.The theorem...

, estimates of the number of non-isomorphic abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s, and bounds for exponential sum
Exponential sum
In mathematics, an exponential sum may be a finite Fourier series , or other finite sum formed using the exponential function, usually expressed by means of the functione = \exp.\,...

s. He proved the exponent 15/46 for the Dirichlet divisor problem, a record that stood for many years.

Sieve methods

One of Richert's notable results was the Jurkat–Richert theorem
Jurkat–Richert theorem
The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.It was proved in 1965 by Wolfgang B...

, joint work with Wolfgang B. Jurkat that improved the Selberg sieve
Selberg sieve
In mathematics, in the field of number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences...

 and is used in the proof of Chen's theorem
Chen's theorem
right|thumb|Chen JingrunChen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime . The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original...

.
Richert also produced a "readable form" of Chen's theorem (it is covered in the last chapter of Sieve Methods).

Halberstam
Heini Halberstam
Heini Halberstam is a British mathematician, working in the field of analytic number theory. He is one of the two mathematicians after whom the Elliott-Halberstam conjecture is named....

 & Richert's book Sieve Methods was the first exhaustive account of the subject.

In reviewing the book in 1976, Hugh Montgomery
Hugh Montgomery
Hugh Montgomery is an American mathematician, working in the fields of analytic number theory and mathematical analysis. As a Rhodes scholar, Montgomery earned his Ph.D. from the University of Cambridge...

wrote "In the past, researchers have generally derived the sieve bounds required for an application, but now workers will find that usually an appeal to an appropriate theorem of Sieve methods will suffice," and "For years to come, Sieve methods will be vital to those seeking to work in the subject, and also to those seeking to make applications."
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