Albert Ingham
Encyclopedia
Albert Edward Ingham was an English
mathematician
.
Ingham was born in Northampton
. He went to Stafford Grammar School and Trinity College, Cambridge
. He obtained his Ph.D.
, which was supervised by John Edensor Littlewood
, from the University of Cambridge
. He supervised the Ph.D.s of C. Brian Haselgrove
, Wolfgang Fuchs
and Christopher Hooley
. Ingham died in Chamonix
, France
.
Ingham proved in 1937 that if
for some positive constant c, then
for any θ > (1+4c)/(2+4c). Here ζ denotes the Riemann zeta function and π the prime-counting function.
Using the best published value for c at the time, an immediate consequence of his result was that
where pn the n-th prime number
and gn = pn+1 − pn denotes the n-th prime gap.
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
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....
.
Ingham was born in Northampton
Northampton
Northampton is a large market town and local government district in the East Midlands region of England. Situated about north-west of London and around south-east of Birmingham, Northampton lies on the River Nene and is the county town of Northamptonshire. The demonym of Northampton is...
. He went to Stafford Grammar School and Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
. He obtained his Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
, which was supervised by John Edensor Littlewood
John Edensor Littlewood
John Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...
, from the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
. He supervised the Ph.D.s of C. Brian Haselgrove
C. Brian Haselgrove
Colin Brian Haselgrove was an English mathematician who is best known for his disproof of the Pólya conjecture in 1958....
, Wolfgang Fuchs
Wolfgang Heinrich Johannes Fuchs
Wolfgang Heinrich Johannes Fuchs was a mathematician specializing in complex analysis. His main area of research was Nevanlinna theory....
and Christopher Hooley
Christopher Hooley
Christopher Hooley FLSW FRS is a British mathematician, emeritus professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University in 1973. He was elected a Fellow of the Royal Society in 1983...
. Ingham died in Chamonix
Chamonix
Chamonix-Mont-Blanc or, more commonly, Chamonix is a commune in the Haute-Savoie département in the Rhône-Alpes region in south-eastern France. It was the site of the 1924 Winter Olympics, the first Winter Olympics...
, France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
.
Ingham proved in 1937 that if
for some positive constant c, then
for any θ > (1+4c)/(2+4c). Here ζ denotes the Riemann zeta function and π the prime-counting function.
Using the best published value for c at the time, an immediate consequence of his result was that
- gn < pn5/8,
where pn the n-th prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
and gn = pn+1 − pn denotes the n-th prime gap.
Books
- The Distribution of Prime Numbers, Cambridge University Press, 1934 (Reissued with a foreword by R. C. Vaughan in 1990)