Difference between revisions of "Page theorem"
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| − | Page's theorem on the zeros of Dirichlet | + | 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 |
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" />; | 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" />; | ||
Revision as of 21:05, 9 January 2015
Page's theorem on the zeros of Dirichlet $L$-functions. Let 
be a Dirichlet L-function, 
, with 
a Dirichlet character modulo 
, 
. There are absolute positive constants 
such that
a) 
for 
, 
;
b) 
for 
, 
;
c) for complex 
modulo 
,
 | (1) |
d) for real primitive 
modulo 
,
 | (2) |
e) for 
there exists at most one 
, 
and at most one real primitive 
modulo 
for which 
can have a real zero 
, where 
is a simple zero; and for all 
such that 
, 
with a real 
modulo 
, one has 
(
).
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) |
Page theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Page_theorem&oldid=36163