Albert Ingham

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 Wolfgang Fuchs and Christopher Hooley. Ingham died in Chamonix, France.

Ingham proved^[1] 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.

[edit] Books

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

O'Connor, John J., and Edmund F. Robertson. "Albert Ingham". MacTutor History of Mathematics archive.

^ Ingham, A. E. On the difference between consecutive primes, Quarterly Journal of Mathematics (Oxford Series), 8, pages 255–266, (1937)

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