Page theorem

2020 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)