Editing Riemann zeta function (section)

==Zeros, the critical line, and the Riemann hypothesis==
{{main|Riemann hypothesis}}
[[File:Riemann0xf4240.png|thumb|720px|Riemann zeta spiral along the critical line from height 999000 to a million (from red to violet)]]
[[File:Zero-free region for the Riemann zeta-function.svg|right|thumb|300px|The Riemann zeta function has no zeros to the right of {{math|''σ'' {{=}} 1}} or (apart from the trivial zeros) to the left of {{math|''σ'' {{=}} 0}} (nor can the zeros lie too close to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the line {{math|''σ'' {{=}} {{sfrac|1|2}}}} and, according to the [[Riemann hypothesis]], they all lie on the line {{math|''σ'' {{=}} {{sfrac|1|2}}}}.]]
[[Image:Zeta polar.svg|right|thumb|300px|This image shows a plot of the Riemann zeta function along the critical line for real values of {{mvar|t}} running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.]]
[[File:RiemannCriticalLine.svg|thumb|300px|The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(''s'') = 1/2. The first non-trivial zeros can be seen at Im(''s'') = ±14.135, ±21.022 and ±25.011.]]
[[File:Zeta1000 1005.webm|thumb|Animation showing the Riemann zeta function along the critical line. {{math|1=&zeta;(1/2 + ''iy'')}} for {{mvar|y}} ranging from 1000 to 1005.]]
The functional equation shows that the Riemann zeta function has zeros at {{nowrap|−2, −4,...}}. These are called the '''trivial zeros'''. They are trivial in the sense that their existence is relatively easy to prove, for example, from {{math|sin {{sfrac|π''s''|2}}}} being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip <math>\{s \in \mathbb{C} : 0 < \operatorname{Re}(s) < 1\}</math>, which is called the '''critical strip'''. The set <math>\{s \in \mathbb{C} : \operatorname{Re}(s) = 1/2\}</math> is called the '''critical line'''. The [[Riemann hypothesis]], considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.<ref>{{cite journal | first = J. B. | last = Conrey | author-link = Brian Conrey | title = More than two fifths of the zeros of the Riemann zeta function are on the critical line | journal= J. Reine Angew. Math. | volume= 1989 | year = 1989 | issue = 399 | pages = 1–26 |url = http://www.digizeitschriften.de/resolveppn/GDZPPN002206781 | mr = 1004130 | doi = 10.1515/crll.1989.399.1 | s2cid = 115910600}}</ref> This has since been improved to 41.7%.<ref>{{cite journal | url=https://link.springer.com/article/10.1007/s40687-019-0199-8 | doi=10.1007/s40687-019-0199-8 | title=More than five-twelfths of the zeros of <math>\zeta </math> are on the critical line | date=2020 | last1=Pratt | first1=Kyle | last2=Robles | first2=Nicolas | last3=Zaharescu | first3=Alexandru | last4=Zeindler | first4=Dirk | journal=Research in the Mathematical Sciences | volume=7 | arxiv=1802.10521 }}</ref>

For the Riemann zeta function on the critical line, see [[Z function|{{mvar|Z}}-function]].

{| class="wikitable"
|+ First few nontrivial zeros<ref>{{cite web|url=https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html|title=Riemann Zeta Function Zeros|author=[[Eric Weisstein]]|access-date=2021-04-24}}</ref><ref>{{cite web|url=https://www.lmfdb.org/zeros/zeta/|author=The L-functions and Modular Forms Database|title=Zeros of &zeta;(''s'')}}</ref>
|-
! Zero
|-
| 1/2 ± 14.134725... ''i''
|-
| 1/2 ± 21.022040... ''i''
|-
| 1/2 ± 25.010858... ''i''
|-
| 1/2 ± 30.424876... ''i''
|-
| 1/2 ± 32.935062... ''i''
|-
| 1/2 ± 37.586178... ''i''
|-
| 1/2 ± 40.918719... ''i''
|}

===Number of zeros in the critical strip===
Let <math>N(T)</math> be the number of zeros of <math>\zeta(s)</math> in the critical strip <math>0 < \operatorname{Re}(s) < 1</math>, whose imaginary parts are in the interval <math>0 < \operatorname{Im}(s) < T</math>.
[[Timothy Trudgian]] proved that, if <math>T > e</math>, then<ref>{{cite journal | first=Timothy S. | last=Trudgian | title = An improved upper bound for the argument of the Riemann zeta function on the critical line II | journal = J. Number Theory | date = 2014 | volume = 134 | pages = 280–292 | doi = 10.1016/j.jnt.2013.07.017 | arxiv = 1208.5846}}</ref>
:<math> \left|N(T) - \frac{T}{2\pi} \log{\frac{T}{2\pi e}}\right| \leq 0.112 \log T + 0.278 \log\log T + 3.385 + \frac{0.2}{T}</math>.

=== 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.

=== Zero-free region ===
The location of the Riemann zeta function's zeros is of great importance in number theory. The [[prime number theorem]] is equivalent to the fact that there are no zeros of the zeta function on the {{math|Re(''s'') {{=}} 1}} line.<ref name="Diamond1982">{{cite journal|first=Harold G.|last=Diamond|title=Elementary methods in the study of the distribution of prime numbers|journal=Bulletin of the American Mathematical Society|volume=7|issue=3|year=1982|pages=553–89|mr=670132|doi=10.1090/S0273-0979-1982-15057-1|doi-access=free}}</ref> It is also known that zeros do not exist in certain regions slightly to the left of the {{math|Re(''s'') {{=}} 1}} line, known as zero-free regions. For instance, Korobov<ref>{{cite journal | first1 =  Nikolai Mikhailovich| last1 = Korobov  | title = Estimates of trigonometric sums and their applications | journal = Usp. Mat. Nauk | volume = 13 | number = 4 | year = 1958 | pages =185–192  }}</ref> and Vinogradov<ref>{{cite journal | first1 = I.M.| last1 = Vinogradov  | title = Eine neue Abschätzung der Funktion <math>\zeta(1+ it)</math>| journal = Russian. Izv. Akad. Nauk SSSR, Ser. Mat | volume = 22 | year = 1958 | pages =161–164  }}</ref> independently showed via the [[Vinogradov's mean-value theorem]] that for sufficiently large <math>|t|</math>, <math>\zeta(\sigma + it) \neq 0</math> for
:<math>\sigma \geq 1 - \frac{c}{(\log|t|)^{2/3 + \varepsilon}}</math>
for any <math>\varepsilon > 0</math> and a number <math>c >0</math> depending on <math>\varepsilon</math>. Asymptotically, this is the largest known zero-free region for the zeta function.

Explicit zero-free regions are also known. Platt and Trudgian<ref>{{cite journal | first1 =  David| last1 = Platt  | first2= Timothy S. | last2= Trudgian | title = The Riemann hypothesis is true up to <math>3\cdot 10^{12}</math> | journal = Bulletin of the London Mathematical Society| volume = 53 | number = 3 | year = 2021 | pages =792–797 | doi = 10.1112/blms.12460 | arxiv = 2004.09765}}</ref>
verified computationally that <math>\zeta(\sigma + it)\neq 0</math> if <math>\sigma \neq 1/2</math> and <math>|t| \leq 3\cdot 10^{12}</math>. Mossinghoff, Trudgian and Yang proved<ref>{{cite journal | first1 = Michael J. | last1 = Mossinghoff | first2 = Timothy S. | last2 = Trudgian |first3 = Andrew | last3 = Yang | title = Explicit zero-free regions for the Riemann zeta-function | journal = Res. Number Theory | volume = 10 | year = 2024 | pages = 11 | arxiv = 2212.06867 | doi = 10.1007/s40993-023-00498-y }}</ref> that zeta has no zeros in the region
:<math>\sigma\ge 1 - \frac{1}{5.558691\log|t|}</math>
for {{math|{{abs|''t''}} ≥ 2}}, which is the largest known zero-free region in the critical strip for <math>3\cdot 10^{12} < |t| < e^{64.1} \approx 7 \cdot 10^{27}</math> (for previous results see<ref>{{cite journal | first1 = Michael J. | last1 = Mossinghoff | first2 = Timothy S. | last2 = Trudgian | title = Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function | journal = J. Number Theory | volume = 157 | year = 2015 | pages = 329–349 | arxiv = 1410.3926 | doi = 10.1016/J.JNT.2015.05.010| s2cid = 117968965 }}</ref>).
Yang<ref>{{cite journal | first1 =  Andrew| last1 = Yang | title =Explicit bounds on <math>\zeta(s)</math> in the critical strip and a zero-free region | journal = J. Math. Anal. Appl.| volume = 534 | number = 2 | year = 2024 | pages =128124  | doi=10.1016/j.jmaa.2024.128124 | arxiv = 2301.03165 }}</ref> showed that <math>\zeta(\sigma+it)\neq 0</math> if
:<math>\sigma \geq 1 - \frac{\log\log|t|}{21.233\log|t|}</math>  and  <math>|t|\geq 3</math>
which is the largest known zero-free region for <math>e^{170.2}< |t| < e^{4.8\cdot 10^{5}}</math>.
Bellotti proved<ref>{{cite journal | first1 =  Chiara| last1 = Bellotti | title =Explicit bounds for the Riemann zeta function and a new zero-free region | journal = J. Math. Anal. Appl.| volume = 536 | number = 2 | year = 2024 | pages =128249 | doi = 10.1016/j.jmaa.2024.128249 | arxiv = 2306.10680}}</ref> (building on the work of Ford<ref>{{cite journal | last1 = Ford | first1 = K. | year = 2002 | title = Vinogradov's integral and bounds for the Riemann zeta function | journal = Proc. London Math. Soc. | volume = 85 | issue = 3| pages = 565–633 | doi = 10.1112/S0024611502013655 | arxiv = 1910.08209 | s2cid = 121144007 }}</ref>) the zero-free region
:<math>\sigma \ge 1 - \frac{1}{53.989(\log|t|)^{2/3}(\log\log|t|)^{1/3}}</math> and <math>|t| \ge 3</math>.
This is the largest known zero-free region for fixed <math>|t| \geq \exp(4.8\cdot 10^{5}).</math> Bellotti also showed that for sufficiently large <math>|t|</math>, the following better result is known: <math>\zeta(\sigma +it) \neq 0</math> for
:<math>\sigma \geq 1 - \frac{1}{48.0718(\log|t|)^{2/3}(\log\log|t|)^{1/3}}.</math>
The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound [[Riemann hypothesis#Consequences|consequences]] in the theory of numbers.

=== Other results ===
It is known that there are infinitely many zeros on the critical line. [[John Edensor Littlewood|Littlewood]] showed that if the sequence ({{math|''γ<sub>n</sub>''}}) contains the imaginary parts of all zeros in the [[upper half-plane]] in ascending order, then

:<math>\lim_{n\rightarrow\infty}\left(\gamma_{n+1}-\gamma_n\right)=0.</math>

The [[critical line theorem]] asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is 1.)

In the critical strip, the zero with smallest non-negative imaginary part is {{math|{{sfrac|1|2}} + 14.13472514...''i''}} ({{OEIS2C|A058303}}). The fact that
:<math>\zeta(s)=\overline{\zeta(\overline{s})}</math>
for all complex {{math|''s'' ≠ 1}} implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line {{math|Re(''s'') {{=}} {{sfrac|1|2}}}}.

It is also known that no zeros lie on the line with real part 1.