Shokichi Iyanaga was a Japanese mathematician. Iyanaga was born in Tokyo, Japan on April 2, 1906, he studied at the University of Tokyo from 1926 to 1929. He studied under Teiji Takagi; as an undergraduate, he published two papers in the Japanese Journal of Mathematics and the Proceedings of the Imperial Academy of Tokyo. Both of his papers appeared in print in 1928. After completing his undergraduate degree in 1929, he stayed at Tokyo and worked under Takagi for his doctorate, he completed his Ph. D. in mathematics 1931. In 1931, Iyanaga obtained a scholarship from the French government, he went to Hamburg, Germany where he studied with Austrian mathematician Emil Artin. In 1932, he attended the International Congress of Mathematicians in Zurich. During his time in Europe, he met with top mathematicians such as Claude Chevalley, Henri Cartan, others. Iyanaga was appointed Assistant Professor at the University of Tokyo. From 1935 to 1939, he didn't publish any research papers. According to Iyanaga, it was because of the pressure of teaching and other business to which he was not accustomed.
He managed to solve a question of Artin on generalizing the principal ideal theorem and this was published in 1939. Iyanaga did publish many papers which arose through several courses such as algebraic topology, functional analysis, geometry, which he taught, he became Professor at the University of Tokyo in 1942. It was during World War II. Towards the end of the war, many Japanese cities were bombarded and he had to find refuge in the countryside, he was busy in editing textbooks from primary and secondary schools and he continued to give courses and organise seminars. After the end of the war, he joined the Science Council of Japan in 1947, he became a member of the Executive Committee of the International Mathematical Union in 1952. He was responsible for organizing the International Congress of Mathematicians in Amsterdam in 1954, which he attended, he was President of the International Commission on Mathematical Instruction from 1957 to 1978. Iyanaga spent the year 1961-62 at the University of Chicago.
He became Dean of the faculty of Science at the University of Tokyo in 1965, a position he held until his retirement in 1967. After his retirement, he was a visiting professor during 1967-68 at the University of Nancy in France. From 1967 to 1977, he was a professor at Gakushuin University in Tokyo. Iyanaga received several awards for his work, he received the Rising Sun from Japan in 1976. He was elected a member of the Japan Academy in 1978, he received the Légion d'honneur in 1980. Iyanaga, S. Sur les Classes d'Idéaux dans les Corps Quadratiques, Actualités Scientifiques et Industrielles, No. 197, Paris: Hermann. Shokichi Iyanaga at the Mathematics Genealogy Project
World War II
World War II known as the Second World War, was a global war that lasted from 1939 to 1945. The vast majority of the world's countries—including all the great powers—eventually formed two opposing military alliances: the Allies and the Axis. A state of total war emerged, directly involving more than 100 million people from over 30 countries; the major participants threw their entire economic and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. World War II was the deadliest conflict in human history, marked by 50 to 85 million fatalities, most of whom were civilians in the Soviet Union and China, it included massacres, the genocide of the Holocaust, strategic bombing, premeditated death from starvation and disease, the only use of nuclear weapons in war. Japan, which aimed to dominate Asia and the Pacific, was at war with China by 1937, though neither side had declared war on the other. World War II is said to have begun on 1 September 1939, with the invasion of Poland by Germany and subsequent declarations of war on Germany by France and the United Kingdom.
From late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Finland and the Baltic states. Following the onset of campaigns in North Africa and East Africa, the fall of France in mid 1940, the war continued between the European Axis powers and the British Empire. War in the Balkans, the aerial Battle of Britain, the Blitz, the long Battle of the Atlantic followed. On 22 June 1941, the European Axis powers launched an invasion of the Soviet Union, opening the largest land theatre of war in history; this Eastern Front trapped most crucially the German Wehrmacht, into a war of attrition. In December 1941, Japan launched a surprise attack on the United States as well as European colonies in the Pacific. Following an immediate U. S. declaration of war against Japan, supported by one from Great Britain, the European Axis powers declared war on the U.
S. in solidarity with their Japanese ally. Rapid Japanese conquests over much of the Western Pacific ensued, perceived by many in Asia as liberation from Western dominance and resulting in the support of several armies from defeated territories; the Axis advance in the Pacific halted in 1942. Key setbacks in 1943, which included a series of German defeats on the Eastern Front, the Allied invasions of Sicily and Italy, Allied victories in the Pacific, cost the Axis its initiative and forced it into strategic retreat on all fronts. In 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained its territorial losses and turned toward Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in Central China, South China and Burma, while the Allies crippled the Japanese Navy and captured key Western Pacific islands; the war in Europe concluded with an invasion of Germany by the Western Allies and the Soviet Union, culminating in the capture of Berlin by Soviet troops, the suicide of Adolf Hitler and the German unconditional surrender on 8 May 1945.
Following the Potsdam Declaration by the Allies on 26 July 1945 and the refusal of Japan to surrender under its terms, the United States dropped atomic bombs on the Japanese cities of Hiroshima and Nagasaki on 6 and 9 August respectively. With an invasion of the Japanese archipelago imminent, the possibility of additional atomic bombings, the Soviet entry into the war against Japan and its invasion of Manchuria, Japan announced its intention to surrender on 15 August 1945, cementing total victory in Asia for the Allies. Tribunals were set up by fiat by the Allies and war crimes trials were conducted in the wake of the war both against the Germans and the Japanese. World War II changed the political social structure of the globe; the United Nations was established to foster international co-operation and prevent future conflicts. The Soviet Union and United States emerged as rival superpowers, setting the stage for the nearly half-century long Cold War. In the wake of European devastation, the influence of its great powers waned, triggering the decolonisation of Africa and Asia.
Most countries whose industries had been damaged moved towards economic expansion. Political integration in Europe, emerged as an effort to end pre-war enmities and create a common identity; the start of the war in Europe is held to be 1 September 1939, beginning with the German invasion of Poland. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred and the two wars merged in 1941; this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935; the British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the fo
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
University of Hamburg
The University of Hamburg is a comprehensive university in Hamburg, Germany. It was founded on 28 March 1919, having grown out of the previous General lecture system and the Colonial Institute of Hamburg as well as the Akademic Gymnasium. In spite of its short history, six Nobel Prize Winners and serials of scholars are affiliated to the university; the University of Hamburg is the biggest research and education institution in Northern Germany and one of the most extensive universities in Germany. The main campus is located in the central district of Rotherbaum, with affiliated institutes and research centres spread around the city state; the institution is classified as a global top 200 university by cited ranking systems such as the Times Higher Education Ranking, the Shanghai Ranking and the CWTS Leiden Ranking, placing it among the top 1% of global universities. On a national scale, U. S. News & World Report ranks UHH 7th and QS World University Rankings 14th out of a total of 426 German institutions of higher education.
At the beginning of the 20th century, wealthy individuals made several petitions to the Hamburg Senate and Parliament requesting the establishment of a university, however those were made to no avail. Although for a time, senator Werner von Melle supported the merger of existing institutions into one university, this plan failed because of the parliaments composition due to the effects of class voting. Much of the establishment wanted to see Hamburg limited to its dominant role as a trading center and shunned both the costs of a university and the social demands of the professors that would have to be employed. Progress was made however, since proponents of a university founded the Hamburg Science Foundation in 1907 and the Hamburg Colonial Institute in 1908; the former institution supported the recruitment of scholars for the chairs of the General lecture system and funding of research cruises, the latter was responsible for all education and research questions concerning overseas territories.
In the same year, the citizenry approved a construction site on the Moorweide for the establishment of a lecture building, which opened in 1911 and became the main building of the university. However, the plans for the foundation of the university itself had to be shelved, following the outbreak of the First World War. After the war, the first elected senate chose von Melle as mayor, he and Rudolf Ross made a push for education reform in Hamburg, their law establishing the university and an adult high school went through. On March 28, 1919 the University of Hamburg opened its gates; the number of full professorships in Hamburg was increased from 19 to 39. Both the Colonial Institute and the General lecture system were absorbed into the university; the first faculties created by the university were Law and Political Science, Medicine and Natural Sciences. During the Weimar Republic, the university grew into importance. Several thousand students were continuously enrolled, it drew scholars like Albrecht Mendelssohn Bartholdy, Aby Warburg and Ernst Cassirer to Hamburg.
The number of full professors had by 1931 grown to 75. Because many students were suffering due to the bad economic situation that prevailed in the early republic, the Hamburg Association of Student Aid was founded in 1922. Ernst Cassirer became principal of the university in 1929, one of the first Jewish scholars with that role in Germany; the academic situation shifted after the general election in March 1933. On May 1 of that year – the university held a ceremony to honor Adolf Hitler as its leader. Massive political influence by the Nazis followed, including the removal of books from the libraries and harassment against alleged enemies of the people. About fifty scientists, including Ernst Cassirer and William Stern, had to leave the university. At least ten students working with the White Rose in Hamburg were arrested. In the foyer of the lecture hall a design by Fritz Fleer commemorative plate was taken in 1971 in memory of the four resistance fighters. Once the Second World War was over, the university was reopened in the winter of 1945 with 17,800 employees.
Out of the 2,872 students who were enrolled at the University of Hamburg in the first postwar semester of 1945/46, 601 had been admitted at the Philosophical, 952 at the Medical and 812 to the Faculty of Law and Political Science. The smallest number joined the Faculty of Mathematics and Natural Sciences with 506 students in total; the first student association during this period was elected in 1946 under British supervision, it formed the foundation of the AStA in 1947. During the West German era, new departments were added to the university, most notably the Faculty of Theology as well as the Faculty of Economic and Social Sciences in 1954; the late 1950s and early 1960s saw a lot of construction: the Auditorium and the Philosopher's Tower where inaugurated near the Von-Melle-Park, while the Botanical Institute and Botanical Garden were relocated to Flottbeck. The university grew from 12,600 students in 1960 to 19,200 in 1970. A wave of protests during the student movements of 1968 sparked a reform of the university structure, in 1969 the faculties were dissolved in favor of more interdisciplinary departments.
Student and staff involvement in the administration was strengthened, the office of Rektor abolished in favor of a university president. However, parts of the reform were rescinded in 1979. Further construction in the 1970s built up the remaining space on the main campus of Rotherbaum quarter, with the Geomatikum
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Johannesburg is the largest city in South Africa and one of the 50 largest urban areas in the world. It is the provincial capital and largest city of Gauteng, the wealthiest province in South Africa. While Johannesburg is not one of South Africa's three capital cities, it is the seat of the Constitutional Court; the city is located in the mineral-rich Witwatersrand range of hills and is the centre of large-scale gold and diamond trade. The metropolis is an alpha global city as listed by the Globalization and World Cities Research Network. In 2011, the population of the city of Johannesburg was 4,434,827, making it the most populous city in South Africa. In the same year, the population of Johannesburg's urban agglomeration was put at 7,860,781; the land area of the municipal city is large in comparison with those of other major cities, resulting in a moderate population density of 2,364/km2. The city was established in 1886 following the discovery of gold on; the city is interpreted as the modern day El Dorado due to the large gold deposit found along the Witwatersrand.
In ten years, the population grew to 100,000 inhabitants. A separate city from the late 1970s until 1994, Soweto is now part of Johannesburg. An acronym for "South-Western Townships", Soweto originated as a collection of settlements on the outskirts of Johannesburg, populated by native African workers from the gold mining industry. Soweto, although incorporated into Johannesburg, had been separated as a residential area for Blacks, who were not permitted to live in Johannesburg proper. Lenasia is predominantly populated by English-speaking South Africans of Indian descent; these areas were designated as non-white areas in accordance with the segregationist policies of the South African government known as Apartheid. Controversy surrounds the origin of the name. There was quite a number of people with the name "Johannes" who were involved in the early history of the city. Among them are the principal clerk attached to the office of the surveyor-general Hendrik Dercksen, Christiaan Johannes Joubert, a member of the Volksraad and was Republic's chief of mining.
Another was Stephanus Johannes Paulus Kruger, president of the South African Republic from 1883 - 1900. Johannes Meyer, the first government official in the area is another possibility. Precise records for the choice of name were lost. Johannes Rissik and Johannes Joubert were members of a delegation sent to England to attain mining rights for the area. Joubert had a park in the city named after him and Rissik has his name for one of the main streets in the city where the important albeit dilapidated Rissik Street Post Office is located; the City Hall is located on Rissik Street. The region surrounding Johannesburg was inhabited by San people. By the 13th century, groups of Bantu-speaking people started moving southwards from central Africa and encroached on the indigenous San population. By the mid-18th century, the broader region was settled by various Sotho–Tswana communities, whose villages, towns and kingdoms stretched from what is now Botswana in the west, to present day Lesotho in the south, to the present day Pedi areas of the Northern Province.
More the stone-walled ruins of Sotho–Tswana towns and villages are scattered around the parts of the former Transvaal province in which Johannesburg is situated. The Sotho–Tswana practised farming and extensively mined and smelted metals that were available in the area. Moreover, from the early 1960s until his retirement, Professor Revil Mason of the University of the Witwatersrand and documented many Late Iron Age archaeological sites throughout the Johannesburg area; these sites dated from between the 12th century and 18th century, many contained the ruins of Sotho–Tswana mines and iron smelting furnaces, suggesting that the area was being exploited for its mineral wealth before the arrival of Europeans or the discovery of gold. The most prominent site within Johannesburg is Melville Koppies, which contains an iron smelting furnace. Many Sotho–Tswana towns and villages in the areas around Johannesburg were destroyed and their people driven away during the wars emanating from Zululand during the late 18th and early 19th centuries, as a result, an offshoot of the Zulu kingdom, the Ndebele, set up a kingdom to the northwest of Johannesburg around modern-day Rustenburg.
The main Witwatersrand gold reef was discovered in June 1884 on the farm Vogelstruisfontein by Jan Gerritse Bantjes that triggered the Witwatersrand Gold Rush and the founding of Johannesburg in 1886. The discovery of gold attracted people to the area, making necessary a name and governmental organisation for the area. Jan and Johannes were common male names among the Dutch of that time. Johannes Meyer, the first government official in the area is another possibility. Precise records for the choice of name were lost. Within ten years, the city of Johannesburg included 100,000 people. In September 1884, the Struben brothers discovered the Confidence Reef on the farm Wilgespruit near present-day Roodepoort, which further boosted excitement over gold prospects; the first gold to be crushed on the Witwatersrand was the gold-bearing rock from the Bantjes mine crushed using the Struben brothers stamp machine. News of t
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a Dirichlet series convergent on a half-plane, that may give rise to an L-function via analytic continuation; the theory of L-functions has become a substantial, still conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, their general properties, in most cases still out of reach of proof, are set out in a systematic way. We distinguish at the outset between the L-series, an infinite series representation, the L-function, the function in the complex plane, its analytic continuation; the general constructions start with an L-series, defined first as a Dirichlet series, by an expansion as an Euler product indexed by prime numbers. Estimates are required to prove. One asks whether the function so defined can be analytically continued to the rest of the complex plane.
It is this meromorphic continuation to the complex plane, called an L-function. In the classical cases one knows that useful information is contained in the values and behaviour of the L-function at points where the series representation does not converge; the general term L-function here includes many known types of zeta-functions. The Selberg class is an attempt to capture the core properties of L-functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions. One can list characteristics of known examples of L-functions that one would wish to see generalized: location of zeros and poles. Since the Riemann zeta-function connects through its values at positive integers to the Bernoulli numbers, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for p-adic L-functions, which describe certain Galois modules; the statistics of the zero distributions are of interest because of their connection to problems like the Generalized Riemann hypothesis, distribution of prime numbers, etc.
The connections with random matrix theory and quantum chaos are of interest. The fractal structure of the distributions has been studied using rescaled range analysis; the self-similarity of the zero distribution is quite remarkable, is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the Riemann zeta function, for the zeros of other L-functions of different orders and conductors. One of the influential examples, both for the history of the more general L-functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of the 1960s, it applies to an elliptic curve E, the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers: i.e. the number of free generators of its group of rational points. Much previous work in the area began to be unified around a better knowledge of L-functions.
This was something like a paradigm example of the nascent theory of L-functions. This development preceded the Langlands program by a few years, can be regarded as complementary to it: Langlands' work relates to Artin L-functions, like Hecke L-functions, were defined several decades earlier, to L-functions attached to general automorphic representations, it became clearer in what sense the construction of Hasse–Weil zeta-functions might be made to work to provide valid L-functions, in the analytic sense: there should be some input from analysis, which meant automorphic analysis. The general case now unifies at a conceptual level a number of different research programs. Generalized Riemann hypothesis Dirichlet L-function Automorphic L-function Modularity theorem Artin conjecture Special values of L-functions Shimizu L-function Neukirch, Jürgen. Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. "LMFDB, the database of L-functions, modular forms, related objects".
Lavrik, A. F. "L-function", in Hazewinkel, Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4Articles about a breakthrough third degree transcendental L-function "Glimpses of a new world". Mathematics. Physorg.com. American Institute of Mathematics. March 13, 2008. Rehmeyer, Julie. "Creeping Up on Riemann". Science News. "Hunting the elusive L-function". Mathematics. Physorg.com. University of Bristol. August 6, 2008