E. K. Fogels

ON PRIME NUMBERS IN SHORT ARITHMETIC PROGRESSIONS

(Presented by Academician I. M. Vinogradov, 9 IV 1960)

Yu. V. Linnik proved \((^{1})\) the existence of a constant \(c>1\), independent of \(k,l\), for which in the interval \(1<x<k^c\) there is at least one prime number belonging to any arithmetic progression \(ku+l\), \((k,l)=1\), \(u=0,1,2,\ldots\). K. A. Rodosskii found \((^{2})\) a simpler proof, in which Linnik’s fundamental lemma was replaced by a weaker estimate. By the same means I proved \((^{3})\) the existence of a prime number \(p\equiv l \pmod k\) \(((k,l)=1)\) in the interval

\[ (x,xk^c) \tag{1} \]

for any \(x\geqslant 1\).

In the present note a scheme is given for the proof of the following theorem, which improves (1).

Theorem. There exist positive constants \(c,c'\), independent of \(k\) and \(l\), such that for any positive \(\varepsilon\leqslant c\), for all \(k\geqslant k_0(\varepsilon)\) and all \(x\geqslant k^{c'\log(c/\varepsilon)}\), in the interval \((x,xk^\varepsilon)\) there is a prime number \(p\equiv l \pmod k\) \(((k,l)=1)\).

The function \(k_0(\varepsilon)\) depends on Siegel’s constant and therefore is still not known in explicit form.

In what follows \(A,B,C,c_0,\ldots,c_4,\alpha,\eta_0,a,b\) denote positive constants independent of \(k,l,\varepsilon\). In the proof of the theorem the following lemmas play an important role.

Lemma 1. Let \(N(\delta,T)\) be the number of zeros of all Dirichlet \(L\)-functions \(L(s,\chi)\) with characters \(\chi\) modulo \(k\), belonging to the rectangle \((\sigma\geqslant 1-\delta,\ |t|\leqslant T)\) of the plane of complex numbers \(s=\sigma+it\). Then for all \(\lambda\in[0,\log k]\) we have

\[ N(\lambda\mid \log k,\ e^\lambda\mid \log k)<e^{C\lambda}, \tag{2} \]

where \(C\) does not depend on \(\lambda\).

Lemma 2. For a suitable \(c_0\) the half-plane \(G\bigl[\sigma\geqslant 1-c_0/\log k(|t|+2)\bigr]\) is free of zeros of any function \(L(s,\chi)\), with the possible exception of one function \(L(s,\chi_1)\) (with real non-principal character \(\chi_1\)), which has in \(G\) a simple zero \(\beta_1=1-\delta_1<1\). According to Siegel’s theorem, \(\delta_1\geqslant c_1(\varepsilon')k^{-\varepsilon'}\) for any \(\varepsilon'>0\).

For suitable \(A\) and

\[ \lambda_0=A\log\frac{eA}{\delta_0\log k},\quad \text{where}\quad \delta_0= \begin{cases} \delta_1, & \text{if } \delta_1\leqslant A/\log k,\\ A/\log k, & \text{if } \delta_1>A/\log k, \end{cases} \]

the rectangle \((1-\lambda_0/\log k\leqslant\sigma\leqslant 1,\ |t|\leqslant k)\) contains no zeros \(\rho\ne\beta_1\) of any \(L\)-function with characters \(\bmod k\).

The proof of the theorem is, in its general outline, similar to Rodosskii’s proof (²). The identity used is

\[ \sum_{n=2}^{\infty}\chi(n)\Lambda(n)n^{-s} \exp\left(-\frac{\log^{2} n/x}{4y}\right) = i\sqrt{\frac{y}{\pi}}\int_{2-i\infty}^{2+i\infty} \frac{L'}{L}(w,\chi)x^{w-s}e^{(w-s)^{2}y}\,dw, \tag{3} \]

where \(x,y\) are arbitrary positive numbers; \(\Lambda(n)=\log p\) if \(n\) is a power of the prime \(p\), and is equal to 0 for the remaining \(n\); in what follows it is assumed that \(s=-1/2\). After multiplying (3) by \(\overline{\chi}(l)\) and summing over \(\chi\), the contour of integration is shifted to \(\operatorname{Re} w=-1/2\). We obtain

\[ \Phi(x,y;k,l)=\varphi(k) \sum_{\substack{n=2\\ n\equiv l(\operatorname{mod}k)}}^{\infty} \Lambda(n)\sqrt{n}\exp\left(-\frac{\log^{2} n/x}{4y}\right) = \]

\[ =2\sqrt{\pi y}\,x^{3/2}e^{9y/4}(1-S)+O(k\log k), \tag{4} \]

where \(\varphi(k)\) is Euler’s function, expressing the number of natural numbers \(l\le k\) with \((l,k)=1\); here

\[ S=\sum_{\chi}\overline{\chi}(l)\sum_{\rho_\chi} x^{-\delta}\exp\{[-\delta(3-\delta)-\tau^{2}+i\tau(3-2\delta)]y+i\tau\log x\} \]

and \(\rho_\chi=1-\delta+i\tau\) runs through all zeros of the functions \(L(s,\chi)\) with \(\delta\ge 0\). Putting in (4)

\[ x=k^{\xi},\qquad \xi\ge 0,\qquad y=\frac{\eta}{\nu}\log k \tag{5} \]

and dividing by \(2\sqrt{\pi y}\,x^{3/2}e^{9y/4}\), we get

\[ U=I_B\frac{\Phi(x,y;k,l)}{2\sqrt{\pi y}\,x^{3/2}e^{9y/4}} = 1-I_BS+I_B\frac{O(k\log k)}{2\sqrt{\pi y}\,x^{3/2}e^{9y/4}}, \tag{6} \]

where \(I_B\) denotes integration with respect to \(\eta\) over the limits \(\eta,\eta+1\), repeated \(B\ge \max(2,C+1)\) times.

In (5) \(\nu\) is restricted by the conditions \(1\le \nu\le \log k\), \(\nu\le e^{\alpha\xi}\), \(\alpha\le \min(1/10,A/3B)\).

In (6) \(I_BS\) is split into three parts corresponding to the conditions:

1) \(|\tau|\ge \log k\);

2) \(\delta\ge \lambda_0/\log k,\quad |\tau|\le d(\delta)=\min(\log k,k^{\delta}/\log k)\);

3) \(\delta\ge \lambda/\log k,\quad d(\delta)<|\tau|<\log k\).

The term corresponding to \(\rho=\beta_1\) is considered separately. By means of (2), using Abel’s identity expressing a sum through an integral, it is proved that, if the initial value \(\eta=\eta_0\ge \max(4,Ce^{3\alpha C},1/A)\) is sufficiently large, then

\[ U>\frac{c_2}{\nu}\delta_0\log k. \tag{7} \]

Let \(z=xe^{4y}\), and let \(V\) be the part of the sum \(U\) corresponding to the values \(n=p\in(x,z)\). With the aid of Abel’s identity it is proved that

\[ U-V<c_3e^{-(\eta_0/4\nu)\log k}. \tag{8} \]

Using Siegel’s theorem and putting \(c'=1/\alpha,\ c=4(\eta_0+B)\), one can show that, under the conditions of our theorem, the right-hand side of (8) is smaller than the right-hand side of (7). Hence \(V>0\), from which the assertion immediately follows.

A simple consequence of the inequalities proved is the estimate

\[ \pi(x;k,l)>x/\varphi(k)k^{3\varepsilon} \quad \text{for } k>k_1(\varepsilon),\quad x\in\bigl(k^{c'\log(c/\varepsilon)},\, e^{k^\varepsilon}\bigr), \]

where \(\pi(x;k,l)\) denotes the number of primes \(p\leq x\), \(p\equiv l\pmod k\) \(((k,l)=1)\).

Let \(\lambda(n)=(-1)^v\), where \(v\) is the number of all prime factors of \(n\), counting multiple factors separately. Further, let \(\mu(n)=\lambda(n)\) if \(n\) is not divisible by a square \(>1\), and let \(\mu(n)=0\) for all other \(n\). An obvious consequence of the theorem is the existence of such \(a\) and \(b\) \((a\leq 2c',\, b\leq 2c)\) that, for any positive \(\varepsilon\leq b\), for all \(k>k_2(\varepsilon)\) and all \(x\geq k^{a\log(b/\varepsilon)}\), the functions \(\lambda(m)\) and \(\mu(m)\) change sign in the interval \((x,xk^\varepsilon)\), when \(m\) runs over the numbers \(\equiv l\pmod k\), and in the same interval there is a prime number \(\equiv l\pmod k\) \(((k,l)=1)\).

The theorem is of interest for \(x<\exp k^{\varepsilon_1}\) (\(\varepsilon_1\) an arbitrarily small positive constant). For \(x\geq \exp k^{\varepsilon_1}\), by the same means one can prove that at least one \(p\equiv l\pmod k\) \(((a,l)=1)\) is found in the interval \((x,xc_4)\) for some \(c_4>1\). Using stronger results on the distribution of zeros of Dirichlet \(L\)-functions, N. G. Chudakov proved \({}^{4}\) the existence of a prime number \(p\equiv l\pmod k\) \(((k,l)=1)\) in the interval

\[ (x,\; x(1+x^{-\vartheta})),\qquad \vartheta=\frac14 \quad (x\geq \exp k^{\varepsilon_1}), \tag{9} \]

but for \(x<\exp k^{\varepsilon_1}\) his method is inapplicable. I proved \({}^{5}\) that for some \(\vartheta>1/3\), in the interval (9) the functions \(\Lambda(m)\), \(\lambda(m)\), \(\mu(m)\) \((m\equiv l\pmod k)\) preserve their mean values. A weak analogue of the latter for \(x<\exp k^{\varepsilon_1}\) is provided by the results of the present note.

Latvian State University
named after P. Stučka

Received
6 IV 1960

CITED LITERATURE

\({}^{1}\) Yu. V. Linnik, Matem. sborn., 15 (57), 139, 347 (1944).
\({}^{2}\) K. A. Rodosskii, Matem. sborn., 34 (76), 331 (1954).
\({}^{3}\) E. K. Fogels, DAN, 102, 455 (1955).
\({}^{4}\) N. G. Chudakov, Izv. AN SSSR, ser. matem., 12, 31 (1948).
\({}^{5}\) E. K. Fogels, Izv. AN LatvSSR, 4 (9), 109 (1948).