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Albert Edward Ingham (3 April 19006 September 1967) was an Englishmarker mathematician.

Ingham was born in Northamptonmarker. He obtained his Ph.D., which was supervised by John Edensor Littlewood, from the University of Cambridgemarker. He supervised the Ph.D.s of C. Brian Haselgrove, Wolfgang Fuchs and Christopher Hooley. Ingham died in Chamonixmarker, Francemarker.

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

gn pn5/8,

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


  • The Distribution of Prime Numbers, Cambridge University Press, 1934 (Reissued with a foreword by R. C. Vaughan in 1990)


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