Editing Riemann zeta function (section)

=== 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.