Editing Riemann zeta function (section)

=== The Hardy–Littlewood conjectures ===
In 1914, [[G. H. Hardy]] proved that {{math|''ζ'' ({{sfrac|1|2}} + ''it'')}} has infinitely many real zeros.<ref>{{cite journal|first1 = G.H. |last1 = Hardy |title = Sur les zeros de la fonction ζ(s) |journal = Comptes rendus de l'Académie des Sciences | volume = 158 |publisher = [[French Academy of Sciences]]|year = 1914 |pages = 1012–1014}}</ref><ref>{{Cite journal|last1=Hardy|first1=G. H.|last2=Fekete|first2=M.|last3=Littlewood|first3=J. E.|date=1921-09-01|title=The Zeros of Riemann's Zeta-Function on the Critical Line|journal=Journal of the London Mathematical Society|pages=15–19|url=https://zenodo.org/record/1447415| volume=s1-1|  doi=10.1112/jlms/s1-1.1.15}}</ref>

Hardy and [[John Edensor Littlewood|J. E. Littlewood]] formulated two conjectures on the density and distance between the zeros of {{math|''ζ'' ({{sfrac|1|2}} + ''it'')}} on intervals of large positive real numbers. In the following, {{math|''N''(''T'')}} is the total number of real zeros and {{math|''N''<sub>0</sub>(''T'')}} the total number of zeros of odd order of the function {{math|''ζ'' ({{sfrac|1|2}} + ''it'')}} lying in the interval {{math|(0, ''T'']}}.
{{numbered list
|For any {{math|''ε'' > 0}}, there exists a {{math|''T''<sub>0</sub>(''ε'') > 0}} such that when
:<math>T \geq T_0(\varepsilon) \quad\text{ and }\quad H=T^{\frac14+\varepsilon},</math>
the interval {{math|(''T'', ''T'' + ''H'']}} contains a zero of odd order.
|For any {{math|''ε'' > 0}}, there exists a {{math|''T''<sub>0</sub>(''ε'') > 0}} and {{math|''c<sub>ε</sub>'' > 0}} such that the inequality
:<math>N_0(T+H)-N_0(T) \geq c_\varepsilon H</math>
holds when
:<math>T \geq T_0(\varepsilon) \quad\text{ and }\quad H=T^{\frac12+\varepsilon}.</math>
}}
These two conjectures opened up new directions in the investigation of the Riemann zeta function.