Difference between revisions of "Page theorem"
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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 | + | 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 |
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 21:03, 9 January 2015
Page's theorem on the zeros of Dirichlet -functions. Let
be a Dirichlet
-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=36162