Albert Ingham

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

Albert Ingham FRS
Born	Albert Edward Ingham (1900-04-03)3 April 1900 Northampton, Northamptonshire, England
Died	6 September 1967(1967-09-06) (aged 67) Switzerland
Alma mater	Trinity College, Cambridge
Spouse	Rose Marie Tupper-Carey ​ ​ (m. 1932)​
Awards	Smith's Prize (1921)[1] Fellow of the Royal Society[2]
Scientific career
Institutions	King's College, Cambridge
Doctoral students	Wolfgang Fuchs C. Haselgrove Christopher Hooley Robert Rankin[3]
Notes
Erdős Number: 1

Early life and education

Ingham was born in Northampton. He went to Stafford Grammar School and began his studies at Trinity College, Cambridge in January 1919 after service in the British Army in World War I. Ingham received a distinction as a Wrangler in the Mathematical Tripos at Cambridge. He was elected a fellow of Trinity in 1922. He also received an 1851 Research Fellowship.^[1][5]

Academic career

Ingham was appointed a Reader at the University of Leeds 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 of C. Brian Haselgrove, Wolfgang Fuchs and Christopher Hooley.^[3]

Ingham proved in 1937^[6] that if

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

for some positive constant c, then

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

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.

Ingham retired from teaching in 1959.^[5]

Honours

Ingham was elected a Fellow of the Royal Society (FRS) in 1945.^[5]

Marriage and children

Ingham married Rose Marie "Jane" Tupper‑Carey in 1932. They had two sons.

Death

Ingham died in Switzerland in 1967, aged 67.^[5]

Publications

Ingham's sole book, On the Distribution of Prime Numbers, was published in 1932.^[5]

References

1. O'Connor, John J.; Robertson, Edmund F., "Albert Ingham", MacTutor History of Mathematics Archive, University of St Andrews
2. Burkill, J. C. (1968). "Albert Edward Ingham 1900-1967". Biographical Memoirs of Fellows of the Royal Society. 14: 271–286. doi:10.1098/rsbm.1968.0012. S2CID 73247345.
3. Albert Ingham at the Mathematics Genealogy Project
4. The Distribution of Prime Numbers, Cambridge University Press, 1932 (Reissued with a foreword by R. C. Vaughan in 1990)
5. "Mr A. E. Ingham". The Times. No. 57042. London. 9 September 1967. p. 12. Retrieved 21 January 2021.
6. Ingham, A. E. (1937). "On the Difference Between Consecutive Primes". The Quarterly Journal of Mathematics: 255–266. Bibcode:1937QJMat...8..255I. doi:10.1093/qmath/os-8.1.255.