Albert Edward Ingham
(3 April 1900–6 September 1967) was an
English mathematician.
Ingham was
born in Northampton. 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
- \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
- g_{n}
p_{n}^{5/8},
where
p_{n} the
n-th
prime number and
g_{n} =
p_{n+1}
−
p_{n} 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)
References
External links