# Albert Ingham

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Albert Ingham
Born Albert Edward Ingham
3 April 1900
Northampton
Died 6 September 1967 (aged 67)
Institutions University of Cambridge
Alma mater Trinity College, Cambridge
Doctoral students Wolfgang Fuchs
C. Haselgrove
Christopher Hooley
William Pennington
Robert Rankin[1]
Influences John Edensor Littlewood[2]
Notable awards Smith's Prize (1921)[2]
Fellow of the Royal Society[3]
Notes
Erdős Number: 1

Albert Edward Ingham FRS (3 April 1900 – 6 September 1967) was an English mathematician.[4]

## Education

Ingham was born in Northampton. He went to Stafford Grammar School and Trinity College, Cambridge.[2]

## Research

Ingham supervised the Ph.D.s of C. Brian Haselgrove, Wolfgang Fuchs and Christopher Hooley.[1] Ingham died in Chamonix, France.

Ingham proved in 1937[5] that if

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

for some positive constant c, then

$\pi\left(x+x^\theta\right)-\pi(x)\sim\frac{x^\theta}{\log x},$

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 and gn = pn+1pn denotes the n-th prime gap.

## References

1. ^ a b
2. ^ a b c
3. ^ Burkill, J. C. (1968). "Albert Edward Ingham 1900-1967". Biographical Memoirs of Fellows of the Royal Society 14: 271–226. doi:10.1098/rsbm.1968.0012. edit
4. ^ The Distribution of Prime Numbers, Cambridge University Press, 1934 (Reissued with a foreword by R. C. Vaughan in 1990)
5. ^ Ingham, A. E. (1937). "On the Difference Between Consecutive Primes". The Quarterly Journal of Mathematics: 255. doi:10.1093/qmath/os-8.1.255. edit