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Page's theorem on the zeros of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710702.png" />-functions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710703.png" /> be a [[Dirichlet L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p0710704.png" />-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|done}}{{MSC|11M06|11N13}}
  
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" />;
+
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" />;
+
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" />,
+
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>
+
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" />,
+
c) for complex $\chi$  modulo $d$,
 +
\begin{equation}\label{1}
 +
L(s,\chi) \ne 0\ \ \text{for}\ \ \sigma > 1 - \frac{c_3}{\log d}\,,\ |t| \le 5\,;
 +
\end{equation}
  
<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>
+
d) for real primitive $\chi$  modulo $d$,
 +
\begin{equation}\label{2}
 +
L(s,\chi) \ne 0\ \ \text{for}\ \ \sigma > 1 - \frac{c_4}{\sqrt{d}\log^2 d}\,;
 +
\end{equation}
  
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 $\pi(x;d,l)$, the number of prime numbers $p \le x$, $p \equiv l \pmod d$ for $0 < l \le d$, where $l$ and $d$ are relatively prime numbers. With the symbols and conditions of Section 1, on account of a)–c) and e) one has
 +
$$
 +
\pi(x;d,l) = \frac{\mathrm{li}(x)}{\phi(d)} - E \frac{\chi(l)}{\phi(d)}\sum_{n \le x} \frac{n^{\beta_1 - 1}}{\log n} + O\left({x \exp\left({-c_7 \sqrt{\log x}}\right)}\right) \ ,
 +
$$
  
<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/p07107044.png" /></td> </tr></table>
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where $E=1$ or $0$ in accordance with whether $\beta_1$ exists or not for a given $d$; because of (2), for any $d \le (\log x)^{1-\delta}$ one has for a given $\delta>0$,
 +
\begin{equation}\label{3}
 +
\pi(x;d,l) = \frac{\mathrm{li}(x)}{\phi(d)}  + O\left({x \exp(-c_8 \sqrt{\log x})}\right) \ .
 +
\end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107045.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107046.png" /> in accordance with whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107047.png" /> exists or not for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107048.png" />; because of , for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107049.png" /> one has for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107050.png" />,
+
This result is the only one (1983) that is effective in the sense that if $\delta$ is given, then one can state numerical values of $c_8$ and the constant appearing in the symbol $O$. Replacement of the bound in (2)
 
+
by the Siegel bound: $L(\sigma,\chi) \ne 0$ for $\sigma > 1-c(\epsilon)d^{-\epsilon}$, $\epsilon > 0$, extends the range of (*) to essentially larger $d$, $d \le (\log x)^A$ for any fixed $A$, but the effectiveness of the bound in (3) is lost, since for a given $\epsilon > 0$ it is impossible to estimate $c_8(\epsilon)$ and $O_\epsilon$.
<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/p07107051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
 
This result is the only one (1983) that is effective in the sense that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107052.png" /> is given, then one can state numerical values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107053.png" /> and the constant appearing in the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107054.png" />. Replacement of the bound in
 
 
 
by the Siegel bound: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107057.png" />, extends the range of (*) to essentially larger <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107059.png" /> for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107060.png" />, but the effectiveness of the bound in (*) is lost, since for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107061.png" /> it is impossible to estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071070/p07107063.png" />.
 
  
 
A. Page established these theorems in [[#References|[1]]].
 
A. Page established these theorems in [[#References|[1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Page,  "On the number of primes in an arithmetic progression"  ''Proc. London Math. Soc. Ser. 2'' , '''39''' :  2  (1935)  pp. 116–141</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Karatsuba,  "Fundamentals of analytic number theory" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Page,  "On the number of primes in an arithmetic progression"  ''Proc. London Math. Soc. Ser. 2'' , '''39''' :  2  (1935)  pp. 116–141</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Karatsuba,  "Fundamentals of analytic number theory" , Moscow  (1975)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR>
 +
</table>

Latest revision as of 15:13, 3 July 2020

2010 Mathematics Subject Classification: Primary: 11M06 Secondary: 11N13 [MSN][ZBL]

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 $d$, \begin{equation}\label{1} L(s,\chi) \ne 0\ \ \text{for}\ \ \sigma > 1 - \frac{c_3}{\log d}\,,\ |t| \le 5\,; \end{equation}

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

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 $\pi(x;d,l)$, the number of prime numbers $p \le x$, $p \equiv l \pmod d$ for $0 < l \le d$, where $l$ and $d$ are relatively prime numbers. With the symbols and conditions of Section 1, on account of a)–c) and e) one has $$ \pi(x;d,l) = \frac{\mathrm{li}(x)}{\phi(d)} - E \frac{\chi(l)}{\phi(d)}\sum_{n \le x} \frac{n^{\beta_1 - 1}}{\log n} + O\left({x \exp\left({-c_7 \sqrt{\log x}}\right)}\right) \ , $$

where $E=1$ or $0$ in accordance with whether $\beta_1$ exists or not for a given $d$; because of (2), for any $d \le (\log x)^{1-\delta}$ one has for a given $\delta>0$, \begin{equation}\label{3} \pi(x;d,l) = \frac{\mathrm{li}(x)}{\phi(d)} + O\left({x \exp(-c_8 \sqrt{\log x})}\right) \ . \end{equation}

This result is the only one (1983) that is effective in the sense that if $\delta$ is given, then one can state numerical values of $c_8$ and the constant appearing in the symbol $O$. Replacement of the bound in (2) by the Siegel bound: $L(\sigma,\chi) \ne 0$ for $\sigma > 1-c(\epsilon)d^{-\epsilon}$, $\epsilon > 0$, extends the range of (*) to essentially larger $d$, $d \le (\log x)^A$ for any fixed $A$, but the effectiveness of the bound in (3) is lost, since for a given $\epsilon > 0$ it is impossible to estimate $c_8(\epsilon)$ and $O_\epsilon$.

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=36162
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article