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Statement

Lemma 4 (p. 4). Let $p$ run over primes, $\gamma$ be Euler's constant, and

$$\eta(x)=\sum_{p\le x}\frac{\log p}{p-1}-\log x+\gamma, \tag{4}$$ $$\delta(x)=\sum_{p\le x}\frac{\log p}{p}-\log x+\gamma +\sum_{p\ge2}\frac{\log p}{p(p-1)} =\eta(x)+\sum_{p>x}\frac{\log p}{p(p-1)}. \tag{5}$$

Then $|\eta(x)|\le M_k$ and $|\delta(x)|\le M_k$ for $x\ge2^k$, where $M_k$ is given by Table 1:

The caption of Table 1 (p. 4) says the values of $M_k$ are best possible apart from rounding.

Proof pointer

P. 5. Only $k=38$ needs an argument; the other entries come from computer calculation. For $2^{38}\le x\le2^{39}$ the bound is checked by computer. Beyond that, the paper combines the Rosser--Schoenfeld identity for $\delta(y)-\delta(x)$, display (6), with Büthe's bounds for $\vartheta(x)-x$ on $1423\le x\le10^{19}$, display (7), with Dusart's explicit bounds for $\psi(x)-x$ and Rosser and Schoenfeld's bound for $|\psi(x)-\vartheta(x)|$, which give display (8) on $10^{19}\le x\le y\le e^{600}$, and with Axler's bound for $|\delta(y)|$ when $y\ge e^{600}$, display (9); the bound for $\eta$ follows since $0<\delta(x)-\eta(x)<10^{-10}$ for $x\ge2^{39}$, display (10).

Read depth

Claims checked: the definitions (4), (5), the statement and Table 1 were read on the page image of p. 4 of arXiv version 3, and the proof on p. 5 was followed for structure. The computer calculations were not repeated. Nothing here is independently reviewed.

Dependencies

J. B. Rosser and L. Schoenfeld, Illinois J. Math. 6 (1962), Eq. (4.21) and Theorem 13; J. Büthe, Math. Comp. 87 (2018), Theorem 2; P. Dusart, Ramanujan J. 45 (2018), Proposition 3.2 and Table 1; C. Axler, Integers 18 (2018), Proposition 8. The lemma is used in Theorem 1 with $k=32$.

Source. Andreas Weingartner, The constant factor in the asymptotic for practical numbers, arXiv:1906.07819; the edition read is named on the source card.

Bears on

No Erdős problem directly; the lemma is a tool for Theorem 1.