Romanoff 1934 uber einige satze der additiven

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satz_i: Romanoff's theorem that, for fixed k, every interval (0, x) holds more than alpha x integers that are a prime plus the kth power of an integer, with alpha > 0 depending only on k.

satz_ii: Romanoff's theorem that, for a given integer a, every interval (0, x) holds more than beta x integers that are a prime plus a power of a, with beta > 0 depending only on a; the case a = 2 is the classical Romanoff theorem.

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Romanoff, N. P., Über einige Sätze der additiven Zahlentheorie. Math. Ann. 109 (1934), 668-678, doi:10.1007/BF01449161. The digitized volume read for this card prints on its cover sheet the digitizer's notice that "Publication and/or broadcast in any form (including electronic) requires prior written permission from the Goettingen State- and University Library" and on its title page "Verlag von Julius Springer 1934"; the article prints no copyright line, every other right reserved.

Romanoff proves two theorems (the paper is in German; the copy read is the full digitized Mathematische Annalen volume 109, the article starting on scan page 673). Satz I (p. 668): every interval $(0,x)$ contains more than $\alpha x$ integers that are a prime plus the $k$th power of an integer, with $\alpha>0$ depending only on $k$. Satz II (p. 668): every interval $(0,x)$ contains more than $\beta x$ integers that are a prime plus a power of a given integer $a$, with $\beta>0$ depending only on $a$. The paper restates Satz I as positive density in the sense of its footnote 1, a lower bound $N(x)/x>\alpha$ for all sufficiently large $x$, so both theorems are positive lower density statements and are to be read for large $x$.

The method is a second-moment count. For any two sequences of positive integers, inequality (1) (p. 668) bounds the number $\nu(2x)$ of integers up to $2x$ that are a sum $n_i+m_j$ with $n_i,m_j\le x$ below by $M(x)^2N(x)^2$ over $M(x)N(x)$ plus the correlation sum $\sum_{u\le x}A_1(u,x)A_2(u,x)$, where $A_1$ and $A_2$ count representations of $u$ as a difference within each sequence; it follows from the Cauchy–Schwarz inequality and the identity (2), proved on p. 669. The correlation sum is bounded with Schnirelman's generalization of Brun's sieve bound, $A_1(u,x)<c_1\frac{x}{\log^2x}\prod_{q\mid u}(1+\frac1q)$ (5, p. 670). For Satz II this reduces to the convergence of $\sum_{(k,a)=1}\mu(k)^2/(k\,l(k))$, with $l(k)$ the order of $a$ modulo $k$, which the paper proves on pp. 673--678 through the auxiliary series $\sum_l\sigma(l)/l$ and two Hilfssätze on primes $\equiv1\pmod k$ (pp. 675--676). The case $a=2$ of Satz II is the theorem the corpus cites as Romanoff's theorem on the integers $2^k+p$.

Source: https://gdz.sub.uni-goettingen.de/id/PPN235181684_0109.

Read status. Claims checked: Satz I, Satz II, inequality (1) and the proofs (pp. 668--678) were read clause by clause on the page images; the estimates the paper cites from elsewhere, Schnirelman's bound (5) and the bounds (9) and (13), were not re-derived. Nothing here is independently reviewed.

Bears on. All four rows rest on Satz II. #244: for integer $C\ge2$ the problem's integers $p+\lfloor C^k\rfloor$ are $p+C^k$, so Satz II with $a=C$ gives them positive lower density; non-integer $C$ is not covered. #851: with $a=2$, Satz II is the case of one prime divisor with a positive constant in place of the problem's $1-\epsilon$; it does not give the bound the problem asks for. #16: with $a=2$, Satz II gives positive lower density to the odd integers $2^k+p$, the fact Chen's disproof cites; it does not decide the problem. #205: the problem lists the paper as a reference; Satz II with $a=2$ gives positive lower density to integers $2^k+m$ with $m$ prime, and does not bear on the problem's answer.

Results. Satz I (p. 668); Satz II (p. 668), whose page also records inequality (1), the convergence of the auxiliary series and Hilfssätze I and II as steps of its proof.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.