New estimates for some functions defined over primes

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Christian Axler, New estimates for some functions defined over primes, Integers 18 (2018), Paper A52. The first page records receipt on 16 May 2017, revision on 22 December 2017, acceptance on 31 May 2018 and publication on 5 June 2018. The abstract announces explicit estimates for Chebyshev's $\vartheta$-function, derived bounds for the prime counting function $\pi(x)$, and two results on primes in short intervals.

Source artifact and reading coverage

The canonical local artifact is axler_2018_new_estimates_some_functions_defined_over_primes.pdf, 536,871 bytes, downloaded from the journal's copy on 2026-09-07. Printed and PDF page numbers coincide. Pages 1, 2 and 13–16 were read visually at filing; the two consumed statements below sit on pp. 13 and 16. No license line or Zenodo DOI is printed in the file; the journal's site states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02): the Creative Commons Attribution 4.0 license, by the journal's undated site-wide statement, not confirmed at the record level for the 2018 volume.

Consumed statements

Both consumed statements enter the corpus through Wang–Crapis, Lemma 4.1, the external-premise interface of the all-$k$ route for Problem 690. Neither is a theorem of Axler's own.

• p. 13, §4 (On the Existence of Prime Numbers in Short Intervals). In its survey of improvements of Bertrand's postulate, the section credits Dusart's thesis [9, Théorème 1] with a prime $p$, for each $x\ge3275$, satisfying

$$x<p\le x\left(1+\frac1{2\log^2x}\right),$$

and records that Dusart later shrank the interval to $1/(25\log^2x)$ for $x\ge396738$ in [10, Proposition 6.8]. Lemma 4.1, item 1, imports the first statement at exactly this range and cites it through Axler; the page is a locator and quotation, not a proof of the thesis result. The second statement is Proposition 6.8 of Dusart 2010 and is not used by Wang–Crapis.

• p. 16, equation (5.2). The Meissel–Mertens constant is displayed as

$$B=\gamma+\sum_p\left(\log\left(1-\frac1p\right)+\frac1p\right) =0.2614972128476427837554268386\ldots$$

Certificate 4.2 takes the enclosure $0.261497212847642<B<0.261497212847643$ from these printed digits (with OEIS A077761); the digits are imported numerical information, not a locally replayed error certificate. The same page quotes Rosser–Schoenfeld's error bound for the reciprocal-prime sum (an unnumbered display) and Dusart's bounds (5.4) and (5.5); Wang–Crapis take their version of that bound from Dusart 2010, Theorem 6.10, not from here.

Axler's own results, Theorem 1 on p. 2 for $\vartheta(x)$, Theorem 4, whose proof is on p. 14, and Proposition 6 on pp. 14–15 for short intervals, and the refinements of (5.5) announced on p. 16, are not consumed by any page of this corpus.

Read status

Claims checked for the two consumed statements, against pp. 13 and 16 of the PDF. The paper's own theorems and proofs are unread here.

Bears on

• Problem 690: supplies, by quotation, the short-interval estimate and the constant digits that the Wang–Crapis all-$k$ route imports in its Lemma 4.1 and Certificate 4.2. No result of Axler's own is used, and the problem's status rests on Cambie's cited theorem, not on this paper.