Page theorem

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)