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Comparative prime-number theory. I–III: Introduction. Comparison of the progressions \(\equiv 1 \bmod k\) and \(\equiv \ell\bmod k\), \(\ell\not\equiv 1 \bmod k\). Continuation of the study of comparison of the progressions \(\equiv 1 \bmod k\) and \(\equiv \ell\bmod k\). (English) Zbl 0111.04506

Acta Math. Acad. Sci. Hung. 13, 299-314 (1962); 13, 315-342 (1962); 13, 343-364 (1962).
Investigations in the theory of distribution of primes among residue classes mod \(k\) point in two directions. The first and prevailing direction attempts to discover uniformity of this distribution, as exemplified by the prime number theorem for primitive residue classes. The second direction attempts to exhibit discrepancies in this distribution.
The first indication of possible results in the latter direction was an assertion by Chebyshev in 1853 that “there are more primes \(\equiv 3 \bmod 4\) than \(\equiv 1 \bmod 4\)”. His assertion stated exactly that
\[ \lim_{r\to 0} \sum_{p> 2} (-1)^{(p-1)/2} e^{-pr} = - \infty, \tag{1} \]
which if true, would show the preponderance of primes \(\equiv 3 \bmod 4\) in the sense of Abel summation. The few papers written on this subject have not decided the falsity or truth of (1), but have proved its equivalence to the nonvanishing of
\[ \sum_0^\infty \frac{(-1)^n}{(2n+1)^s} \]
in the halfplane \(\operatorname{Re} s > \tfrac12\). However, Littlewood did obtain a direct result related to Chebyshev’s assertion, namely, that the function \(\Pi(x, 4, 3) - \Pi(x, 4, 1)\) possesses an infinity of sign changes. In addition, Phragmén proved that, for a suitable sequence of \(x\)’s, this function divided by \(\sqrt x/\log x\) tends to \(1\).
The authors generalize these problems concerned with comparison of the distribution of primes in various related forms from the point of view of discrepancies between them. Their totality is what they term “comparative prime number theory”. The authors list 10 of the most plausible such problems; for instance, for fixed \(k\), for which pairs \(l,m\) does \(\Pi(x, k, l) - \Pi(x, k, m)\) change sign infinitely often ? Or, for each permutation \(l_1, l_2, \ldots, l_{\varphi(k)}\) of the set of reduced residue classes mod \(k\), does there exist an infinity of integral \(x\)’s such that \(\Pi(x,k,l) < \Pi(x, k,l_2)< \ldots < \Pi(x, k, l_{\varphi(k)})\).
Each of these direct problems can be restated as an “Abel-type” problem involving Abel sums. These twenty problems give rise to forty more by introducing the functions
\[ \Psi(x. k, 1) \equiv \sum_{\substack{n\le x \\ n\equiv l\bmod k}} \Lambda(n)\quad\text{and}\quad \Pi(x, k, l) \equiv \sum_{\substack{n\le x \\ n\equiv l\bmod k}} \frac{\Lambda(n)}{\log n}. \]
The authors sketch results contained implicitly in the literature and announce a series of 7 additional papers under the title “comparative prime number theory”.
The second paper starts by obtaining explicit lower and upper bounds respectively for the maximum and minimum on a certain interval of \(\psi(x, k, 1) - \psi(x, k, l)\) and \(\Pi(x, k, 1) - \Pi(x, k, l)\) for \(k = 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 24\). For these values of \(k\), it is known that Hazelgrove’s condition is satisfied, namely, that there is an \(A(k)\) with \(0 < A(k) \le 1\) such that no \(L\)-function mod \(k\) vanishes for \(0 < \sigma < 1\) and \(\vert t\vert\le A(k)\). These specific theorems follow from more general theorems involving Hazelgrove’s condition.
Corresponding results are obtained for \(\Pi(x, k, 1) - \Pi(x, k, l)\) divided by \(\sqrt x/\log x\) for those \(k\) satisfying Hazelgrove’s condition. These lead to an upper bound for the first sign change of \(\Pi(x, k, 1) - \Pi(x, k, l)\) for such \(k\). Again corresponding results are found for those \(k\) satisfying Hazelgrove’s condition for
\[ D(x) \equiv \Pi(x, k, 1) - \frac1{\varphi(k) - 1} \sum_{\substack{l,\, l\ne 1 \\ (l.k)=1}}\Pi(x, k, l) \]
divided by \(\sqrt x/\log x\). The proofs are quite formidable and constitute a remarkable achievement.
The study of the previous parts is extended to more detailed results about \(W_k(T, 1, l)\), the number of sign changes of \(\Pi(x, k, 1) - \Pi(x, k, l)\) in \(0 < x\le T\). If Hazelgrove’s condition holds for \(k\), then an explicit lower bound is obtained for \(W_k,(T, 1, l)\) for \(T\) sufficiently large. This implies immediately a result about an interval containing at least one sign change of the above function.
Analogous results are obtained for the function \(D(x)\), defined above, and for its sign changes. All results to this point have been for any \(l\) with \((l, k) = 1\) and \(l\not\equiv 1 \bmod k\). However for those \(l\) for which the number of incongruent solutions of \(x^2\equiv 1\bmod k\) equals the number of incongruent solutions of \(x^2\equiv l \bmod k\), an essential improvement in the bounds in the previous cases is possible, subject to Hazelgrove’s condition. Once again ingenious and formidable methods of proof are necessary.

MSC:

11N13 Primes in congruence classes
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