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.

  • 4
    $\begingroup$ $m \mid n$ 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". $\endgroup$ May 3, 2020 at 16:21
  • 4
    $\begingroup$ Type it in google translator... the words only, not the math. I have done this with some success. $\endgroup$
    – efs
    May 3, 2020 at 16:22
  • 4
    $\begingroup$ 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. $\endgroup$ May 3, 2020 at 16:48
  • 2
    $\begingroup$ 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. $\endgroup$ May 3, 2020 at 17:38
  • $\begingroup$ @KlausNiederkrüger thank you very much for telling about deepl. I am using it now. It is good $\endgroup$
    – Arnold
    May 4, 2020 at 5:05

2 Answers 2


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.

  • $\begingroup$ @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? $\endgroup$
    – Arnold
    May 4, 2020 at 7:45
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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, 2020 at 9:26

I have two comments for you, putting them as an answer because I don't have enough credits for a comment:

First, the notation $2\mid m$ should be very common, and it is definitely very common in German books. As an example, I own a copy of Algebra from Siegfried Bosch, first edition from 1992, and it uses this notation as well.

Second, I think it is great that you take the extra effort to read the original text and that you are translating it line by line. So the following post on MO could be very interesting for you: Do you read the masters?


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