Albert Edward Ingham (3 April 1900 – 6 September 1967) was an English

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

Early life and education

Ingham was born in

Northampton Northampton ( ) is a town and civil parish in Northamptonshire, England. It is the county town of Northamptonshire and the administrative centre of the Unitary authorities of England, unitary authority of West Northamptonshire. The town is sit ...

. He went to Stafford Grammar School and began his studies at

Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...

in January 1919 after service in the British Army in

World War I World War I or the First World War (28 July 1914 – 11 November 1918), also known as the Great War, was a World war, global conflict between two coalitions: the Allies of World War I, Allies (or Entente) and the Central Powers. Fighting to ...

. Ingham received a distinction as a Wrangler in the

Mathematical Tripos The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics, University of Cambridge, Faculty of Mathematics at the University of Cambridge. Origin In its classical nineteenth-century form, the tripos was a di ...

at Cambridge. He was elected a fellow of Trinity in 1922. He also received an

1851 Research Fellowship The 1851 Research Fellowship is a scheme conducted by the Royal Commission for the Exhibition of 1851 to annually award a three-year research scholarship to approximately eight "young scientists or engineers of exceptional promise". The fellowship ...

Academic career

Ingham was appointed a Reader at the

University of Leeds The University of Leeds is a public research university in Leeds, West Yorkshire, England. It was established in 1874 as the Yorkshire College of Science. In 1884, it merged with the Leeds School of Medicine (established 1831) and was renamed Y ...

in 1926 and returned to Cambridge as a fellow of King's College and lecturer in 1930. Ingham was appointed after the death of Frank Ramsey. Ingham supervised the

PhDs A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

of C. Brian Haselgrove, Wolfgang Fuchs and

Christopher Hooley Christopher Hooley (7 August 1928 – 13 December 2018) was a British mathematician and professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University ...

. Ingham proved in 1937 that if :

\zeta\left(1/2+it\right)=O\left(t^c\right)

for some positive constant ''c'', then :

\pi\left(x+x^\theta\right)-\pi(x)\sim\frac,

for any θ > (1+4c)/(2+4c). Here ζ denotes the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

and π the

prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal ...

. Using the best published value for ''c'' at the time, an immediate consequence of his result was that :''g''_''n'' < ''p''_''n''^5/8, where ''p''_''n'' the ''n''-th

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

and ''g''_''n'' = ''p''_''n''+1 − ''p''_''n'' denotes the ''n''-th

prime gap A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-st and the ''n''-th prime numbers, i.e., :g_n = p_ - p_n. ...

. Ingham retired from teaching in 1959.

Honours

Ingham was elected a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

(FRS) in 1945.

Marriage and children

Ingham married Rose Marie "Jane" TupperCarey in 1932. They had two sons.

Death

Ingham died in Switzerland in 1967, aged 67.

Publications

Ingham's sole book, ''On the Distribution of Prime Numbers'', was published in 1932.

References

{{DEFAULTSORT:Ingham, Albert 1900 births 1967 deaths 20th-century English mathematicians Academics of the University of Leeds Alumni of Trinity College, Cambridge British Army personnel of World War I Fellows of King's College, Cambridge Fellows of Trinity College, Cambridge Fellows of the Royal Society British number theorists People from Northampton Military personnel from Northampton British Army soldiers