# Was Vinogradov's 1937 proof of the three-prime theorem effective?

Was Vinogradov's first proof of the three-prime theorem effective?

Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. Davenport is based on the second version of the monograph, from 1947. There is no doubt that the proof in the second version is effective - Vinogradov goes to great pains to prove a "nearly-log-free" result (as opposed to a bound worse by a factor of $(\log x)^c$, which he proves with much greater ease) because he allows himself to use Siegel-Walfisz only in Page's effective version (= primes are well distributed in arithmetic progressions up to modulus $m\leq (\log n)^{2-\epsilon}$, and exceptions after that in a moderate range would all have to occur for moduli that are multiples of a single modulus of size $> (\log n)^{2-\epsilon}$).

However, I'm finding the 1937 original (in Russian) hard to get. Did he use the usual, ineffective version of Siegel-Walfisz there? Did he already have "nearly-log-free" estimates at the time?

Two more historical questions.

(a) It must have been realized at some point that one does not really need nearly-log-free results to get an effective result (= every odd integer larger than a constant $C$ is the sum of three primes, where $C$ is an enormous constant that can in principle be specified). This is so because there isn't really an exceptional modulus $q$, but, rather, there could be an exceptional character modulo $q$; the other characters modulo $q$ are fine. Thus, Vinogradov's simpler, non-log-free bound is enough for effectivity, even though it's far from optimal. When was this first remarked in the literature? (I can't find any awareness of this in Vinogradov's monograph (translation of 1947 version); he uses Siegel-Walfisz-Page as a black box.)

(b) The first explicit value for $C$ was computed by Borozdkin, who was apparently an assistant and former student of Vinogradov's. The only reference I've got for this is what looks like a mention in the proceedings of a Soviet conference. Did the full version appear anywhere?

Note: I can read Russian, but, like most people working outside Russian-speaking areas, I find many Russian-language historical materials to be hard to get. Links to electronic versions of the documents discussed above would be very welcome.

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Is your usage of "effective" to mean "computable"? IANAM – ArtB Mar 31 '14 at 17:19
"Effective" = "if you went through the proof with unbounded patience, you could get a value for C, in principle". Non-effective results in number theory often come from the Siegel zero phenomenon (as is the case here). – H A Helfgott Mar 31 '14 at 18:41
Note to self: perhaps it would be fairer to say Landau-Siegel-Walfisz in this context... – H A Helfgott Apr 1 '14 at 15:06

Concerning question (b). It seems the full version of Borozdkin's proof has never appeared as a normal article. In 1939 he got $C=e^{e^{e^{41.96}}}$ in his unpublished PhD thesis (see http://cheb.tsput.ru/attachments/451_tom13_v2_Kasimov.pdf ). The bound was further improved by him in 1956 to $C=e^{e^{16.038}}$ and a short content of his talk should be in this book (I was not able to find an electronic version) http://www.ozon.ru/context/detail/id/13616734/ on page 3 under the title K voprosu o postoyannoj I. M. Vinogradova.

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This is very interesting, in that it implies that Vinogradov (and/or Borozdkin) had an effective proof by 1939. Do you agree that the 1937 proof was ineffective? (This seems clear enough to me, but perhaps I'm missing something.) Do we have any idea of what exactly Borozdkin was doing in 1939? – H A Helfgott Apr 9 '14 at 12:56
Any answers given in the next 48 hours will be especially appreciated - I need to submit my paper to the ICM proceedings, and I want to get the history part right! – H A Helfgott Apr 9 '14 at 18:58
I agree that the 1937 proof was ineffective. Narkiewicz in "Rational Number Theory in the 20th Century: From PNT to FLT", p.231, writes "Vinogradov's result was made effective in 1956, when K.G. Borozdkin [648] showed that every odd integer exceeding exp(exp(16.038)) is the sum of three primes". I do not have any idea of what exactly Borozdkin was doing in 1939. Kasimov's paper is the only place I have seen a claim that Borozdkin had estimated Vinogradov's constant already in 1939. Interestingly, in math.ru/lib/book/djvu/istoria/1917-1947.djvu Borozdkin is not mentioned at all. – Zurab Silagadze Apr 10 '14 at 6:10
I would say Vinogradov's result was already made effective by Vinogradov himself in 1947 - in fact, some of the complications of the 1947 versions clearly have that as their intent. The result was made explicit by Borozdkin in 1956, if not before. Is Kasimov's paper really the only source for 1939? I think I saw that year once before in an online source, and took it to be an error. – H A Helfgott Apr 10 '14 at 7:39
Narkiewicz (The Development of Prime Number Theory, p. 336) states that Cudakov claimed in 1947 that Borozdkin had already proved "this" (exp(exp(16.038))? some explicit constant C? an effective bound?) in 1939. Narkiewicz also states that no proof of Borodzkin's claim ever appeared. This is fun. – H A Helfgott Apr 10 '14 at 7:42

Ke Gong kindly sent me a link to Vinogradov's work online:

http://www.mathnet.ru/php/person.phtml?&personid=26537&option_lang=rus

In summary, it seems clear that the 1937 proof was ineffective.

Questions (a) and (b) remain to be answered.

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Moreover, the 1947 version seems to be the first one to give an effective result - though Vinogradov had already given effective estimates in 1939 for exponential sums with prime support and rational parameter $\alpha$. – H A Helfgott Mar 31 '14 at 14:40