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I thought of utilizing this lockdown period to study research papers in number theory by myself.

I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all Even Positive Integers are sum of two primes". I have read and understood all the proof except one equation and that too due to reason that it uses a equation as a reference from German Book " E Landau, Vorlesungen über Zahlentheorie I ( Liepzig, 1927) . I don't understand German.

I thought of posting this question here in hope that someone must have studied this research paper and also in case a researcher understanding German language wishes to help.

Image of the equation I don't understand ( equation 49 on Page 7 ) ! Equation 49]1

The problem-> 1.I am unable to understand what ( 2|m) means in (49) ?(Notation (2|m) is not used in research paper before) 2.what is the statement of theorem is used to derive (49)? . . Everything except this theorem is Understood in paper.

Images from Original German book->Image page 226 Image pg 227

What i did to resolve the issue -> I tried to find English Edition but couldn't. 2.Then I tried to translate whole pdf book which didn't materalized due to book being a bit long. 3.When I split the pdf I wanted to translate theb it Shows Error 403. 4. I changed the screenshot images( from archive.org) to pdf but again error in translation.

It is my humble request if anybody can tell me meaning of notation (2|m) ?

And it will be very much more helpful if someone tell what is the statement of theorem / result used to derive equation 49.

I shall be really thankful for any help offered.

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    $\begingroup$ $m \mid n$ means that $m$ divides $n$. In particular, $2 \mid m$ means that $m$ is even. "m gerade" in the original German means "m is even". $\endgroup$ – Carl-Fredrik Nyberg Brodda May 3 at 16:21
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    $\begingroup$ Type it in google translator... the words only, not the math. I have done this with some success. $\endgroup$ – EFinat-S May 3 at 16:22
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    $\begingroup$ You should better use "deepl.com", which in the opinion of everybody I know of is lightyears ahead of google translate for translations between English, German, and French. $\endgroup$ – Klaus Niederkrüger May 3 at 16:48
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    $\begingroup$ Estermann refers to formula (239) which is just a formula, no words (except the assumption that $m$ is even). Of course you also need to understand how he proves it. $\endgroup$ – François Brunault May 3 at 17:38
  • $\begingroup$ @KlausNiederkrüger thank you very much for telling about deepl. I am using it now. It is good $\endgroup$ – User May 4 at 5:05
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Look at the first page of this paper -->

Daniel A. Goldston, Julian Ziegler Hunts, Timothy Ngotiaoco, The Tail of the Singular Series for the Prime Pair and Goldbach Problems, Funct. Approx. Comment. Math. 56, Number 1 (2017) pp 117–141, doi:10.7169/facm/1602, arXiv:1409.2151.

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    $\begingroup$ While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review $\endgroup$ – Stefan Kohl May 3 at 20:04
  • $\begingroup$ @user155294 currently I am reading the German edition of this book by translating line by line using deepl. com. I have a question does on page 226 ( please see image 1) D (m, n) > $ c_{4, 3} $ means D( m, n) is greater than some constant? $\endgroup$ – User May 4 at 7:45
  • $\begingroup$ @Stefan well, not sure what the published version of the paper looks like, but given that the authors haven't updated the arXiv version in years, I think that's pretty stable. $\endgroup$ – David Roberts May 4 at 8:47
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    $\begingroup$ @Tim Green Yes. Moreover $S(m)\approx D(m,n)$ for $n>c_{4,2}$ and $m$ is even, with some big value for $m$ and $c_{4,2}$ and $0<m\leq n$. $\endgroup$ – user155294 May 4 at 9:26
  • $\begingroup$ @StefanKohl It's not a link-only answer since anyone can find the paper with the citation without using the link. And somehow I have more faith in the long-term viability of arXiv and DOI's than MO. $\endgroup$ – Najib Idrissi May 4 at 10:01

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