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Page's theorem on the zeros of Dirichlet $L$-functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710703.png" /> be a [[Dirichlet L-function]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710705.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710706.png" /> a [[Dirichlet character|Dirichlet character]] modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710708.png" />. There are absolute positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710709.png" /> such that
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{{TEX|part}}
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107010.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107012.png" />;
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Page's theorem on the zeros of Dirichlet $L$-functions.
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107015.png" />;
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Let $L(s,\chi)$ be a [[Dirichlet L-function]], $s = \sigma + i t$, with $\chi$ a [[Dirichlet character]] modulo $d$, $d \ge 3$. There are absolute positive constants $c_1,\ldots,c_8$ such that
  
c) for complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107016.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107017.png" />,
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a) $L(s,\chi) \ne 0$ for $\sigma > 1 - c_1/\log(dt)$, $t \ge 3$;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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b) $L(s,\chi) \ne 0$ for $\sigma > 1 - c_2/\log(d)$, $0 < t < 5$;
  
d) for real primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107019.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107020.png" />,
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c) for complex $\chi$  modulo $d$,
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$$
 +
L(s,\chi) \ne 0\ \ \text{for}\ \ \sigma > 1 - \frac{c_3}{\log d}\,,\ |t| \le 5\,;
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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d) for real primitive $\chi$  modulo $d$,
 +
$$
 +
L(s,\chi) \ne 0\ \ \text{for}\ \ \sigma > 1 - \frac{c_4}{\sqrt{d}\log^2 d}\,;
 +
$$
  
e) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107022.png" /> there exists at most one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107024.png" /> and at most one real primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107025.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107027.png" /> can have a real zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107029.png" /> is a simple zero; and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107032.png" /> with a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107033.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107034.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107036.png" />).
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e) for $2 \le d \le D$ there exists at most one $d=d_0$, $d_0 \ge (\log^2 D)/(\log\log^8 D)$ and at most one real primitive $\psi$ modulo $d$ for which $L(s,\psi$ can have a real zero $\beta_1 > 1- c_6/\log D$, where $\beta_1$ is a simple zero; and for all $\beta$ such that $L(\beta,\psi) =0$, $\beta > 1 - c_6/\log D$ with a real $\psi$ modulo $d$, one has $d \equiv 0 \pmod {d_0}$.
  
 
Page's theorem on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107037.png" />, the number of prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107040.png" />) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107043.png" /> are relatively prime numbers. With the symbols and conditions of Section 1, on account of a)–c) and e) one has
 
Page's theorem on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107037.png" />, the number of prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107040.png" />) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107043.png" /> are relatively prime numbers. With the symbols and conditions of Section 1, on account of a)–c) and e) one has

Revision as of 14:47, 3 July 2020


Page's theorem on the zeros of Dirichlet $L$-functions.

Let $L(s,\chi)$ be a Dirichlet L-function, $s = \sigma + i t$, with $\chi$ a Dirichlet character modulo $d$, $d \ge 3$. There are absolute positive constants $c_1,\ldots,c_8$ such that

a) $L(s,\chi) \ne 0$ for $\sigma > 1 - c_1/\log(dt)$, $t \ge 3$;

b) $L(s,\chi) \ne 0$ for $\sigma > 1 - c_2/\log(d)$, $0 < t < 5$;

c) for complex $\chi$ modulo $d$, $$ L(s,\chi) \ne 0\ \ \text{for}\ \ \sigma > 1 - \frac{c_3}{\log d}\,,\ |t| \le 5\,; $$

d) for real primitive $\chi$ modulo $d$, $$ L(s,\chi) \ne 0\ \ \text{for}\ \ \sigma > 1 - \frac{c_4}{\sqrt{d}\log^2 d}\,; $$

e) for $2 \le d \le D$ there exists at most one $d=d_0$, $d_0 \ge (\log^2 D)/(\log\log^8 D)$ and at most one real primitive $\psi$ modulo $d$ for which $L(s,\psi$ can have a real zero $\beta_1 > 1- c_6/\log D$, where $\beta_1$ is a simple zero; and for all $\beta$ such that $L(\beta,\psi) =0$, $\beta > 1 - c_6/\log D$ with a real $\psi$ modulo $d$, one has $d \equiv 0 \pmod {d_0}$.

Page's theorem on , the number of prime numbers , () for , where and are relatively prime numbers. With the symbols and conditions of Section 1, on account of a)–c) and e) one has

where or in accordance with whether exists or not for a given ; because of , for any one has for a given ,

(*)

This result is the only one (1983) that is effective in the sense that if is given, then one can state numerical values of and the constant appearing in the symbol . Replacement of the bound in

by the Siegel bound: for , , extends the range of (*) to essentially larger , for any fixed , but the effectiveness of the bound in (*) is lost, since for a given it is impossible to estimate and .

A. Page established these theorems in [1].

References

[1] A. Page, "On the number of primes in an arithmetic progression" Proc. London Math. Soc. Ser. 2 , 39 : 2 (1935) pp. 116–141
[2] A.A. Karatsuba, "Fundamentals of analytic number theory" , Moscow (1975) (In Russian)
[3] K. Prachar, "Primzahlverteilung" , Springer (1957)
How to Cite This Entry:
Page theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Page_theorem&oldid=50873
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article