View source for Albert Ingham

{{Short description|English mathematician}}
{{Use dmy dates|date=February 2023}}
{{Use British English|date=May 2012}}
{{Infobox scientist
| name              = Albert Ingham
| honorific_suffix  = {{postnom|country=GBR|size=100|FRS}}
| image             =
| image_size        = 
| caption           = Ingham (right) with Jane Ingham (left) in 1966
| birth_name        = Albert Edward Ingham
| birth_date        = {{birth date|1900|4|3|df=y}}
| birth_place       = [[Northampton]], [[Northamptonshire]], England
| death_date        = {{death date and age|1967|9|6|1900|4|3|df=y}}
| death_place       = Switzerland
| fields            = 
| workplaces        = [[King's College, Cambridge]]
| alma_mater        = [[Trinity College, Cambridge]]
| spouse            = {{marriage|[[Jane Ingham|Rose Marie Tupper-Carey]]|6 July 1932}}
| academic_advisors = 
| doctoral_advisor  = <!-- No advisor as Ingham didn't do a PhD. See talk page -->
| doctoral_students = [[Wolfgang Heinrich Johannes Fuchs|Wolfgang Fuchs]]<br />[[C. Brian Haselgrove|C. Haselgrove]]<br />[[Christopher Hooley]]<br />[[Robert Alexander Rankin|Robert Rankin]]<ref name="mathgene"/>
| notable_students  = 
| known_for         = 
| author_abbrev_bot = 
| author_abbrev_zoo = 
| awards            = {{Nowrap|[[Smith's Prize]] (1921)<ref name="mactutor" /><br>[[Fellow of the Royal Society]]<ref name="frs">{{Cite journal | last1 = Burkill | first1 = J. C. | doi = 10.1098/rsbm.1968.0012 | title = Albert Edward Ingham 1900-1967 | journal = [[Biographical Memoirs of Fellows of the Royal Society]] | volume = 14 | pages =  271–286| year = 1968 | s2cid = 73247345 }}</ref>}}
| signature         = <!--(filename only)-->
| footnotes         = Erdős Number: 1 
}}
'''Albert Edward Ingham''' {{postnom|country=GBR|FRS}} (3 April 1900 – 6 September 1967) was an English [[mathematician]].<ref>''The Distribution of Prime Numbers'', Cambridge University Press, 1932 ([https://books.google.com/books/about/The_Distribution_of_Prime_Numbers.html?id=BZLEFyNuGGAC Reissued with a foreword] by [[Robert Charles Vaughan (mathematician)|R. C. Vaughan]] in 1990)</ref>

==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]].<ref name="mactutor">{{MacTutor Biography|id=Ingham}}</ref><ref name="TimesObit">{{cite news|title=Mr A. E. Ingham|url=https://link.gale.com/apps/doc/CS203910441/TTDA?u=wes_ttda&sid=TTDA&xid=8ae08a6e |accessdate=21 January 2021|work=The Times|location=London|issue=57042|date=9 September 1967|page=12}}</ref>

==Academic career==
Ingham was appointed a [[Reader (academic rank)|Reader]] at the [[University of Leeds]] in 1926 and returned to Cambridge as a fellow of [[King's College, Cambridge|King's College]] and lecturer in 1930. Ingham was appointed after the death of [[Frank Ramsey (mathematician)|Frank Ramsey]].

Ingham supervised the [[Doctor of Philosophy|PhDs]] of [[C. Brian Haselgrove]], [[Wolfgang Heinrich Johannes Fuchs|Wolfgang Fuchs]] and [[Christopher Hooley]].<ref name="mathgene">{{MathGenealogy|id=30623}}</ref>

Ingham proved in 1937<ref>{{Cite journal | last1 = Ingham | first1 = A. E. | author-link = Albert Ingham| title = On the Difference Between Consecutive Primes | doi = 10.1093/qmath/os-8.1.255 | journal = The Quarterly Journal of Mathematics | pages = 255–266 | year = 1937 | bibcode = 1937QJMat...8..255I }}</ref> that if

:<math>\zeta\left(1/2+it\right)=O\left(t^c\right)</math>

for some positive constant ''c'', then

:<math>\pi\left(x+x^\theta\right)-\pi(x)\sim\frac{x^\theta}{\log x},</math>

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''<sub>''n''</sub> < ''p''<sub>''n''</sub><sup>5/8</sup>,

where ''p''<sub>''n''</sub> the ''n''-th [[prime number]] and ''g''<sub>''n''</sub> = ''p''<sub>''n''+1</sub> − ''p''<sub>''n''</sub> denotes the ''n''-th [[prime gap]].

Ingham retired from teaching in 1959.<ref name="TimesObit"/>

==Honours==
Ingham was elected a [[Fellow of the Royal Society]] (FRS) in 1945.<ref name="TimesObit"/>

==Marriage and children==
Ingham married [[Jane Ingham|Rose Marie "Jane" Tupper{{Nbh}}Carey]] in 1932.  They had two sons.

==Death==
Ingham died in Switzerland in 1967, aged 67.<ref name="TimesObit"/>

==Publications==
Ingham's sole book, ''On the Distribution of Prime Numbers'', was published in 1932.<ref name="TimesObit"/>

==References==
{{Reflist}}

{{Authority control}}

{{DEFAULTSORT:Ingham, Albert}}
[[Category:1900 births]]
[[Category:1967 deaths]]
[[Category:20th-century English mathematicians]]
[[Category:Academics of the University of Leeds]]
[[Category:Alumni of Trinity College, Cambridge]]
[[Category:British Army personnel of World War I]]
[[Category:Fellows of King's College, Cambridge]]
[[Category:Fellows of Trinity College, Cambridge]]
[[Category:Fellows of the Royal Society]]
[[Category:British number theorists]]
[[Category:People from Northampton]]
[[Category:Military personnel from Northampton]]
[[Category:British Army soldiers]]