Vinogradov 1948 estimate trigonometric sums prime numbers

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theorem_1: Vinogradov's bound S << P^{1-rho} for the sum of e^{2 pi i l f(p)} over the primes p <= P, f a real polynomial of degree at most n without constant term, when some coefficient a_s with 2 <= s <= n has a rational approximation a/q + theta/(q tau) with P^kappa << q <= tau = P^{0.5 s}, for l up to P^{2 rho_0}, with rho explicit in n and kappa.

theorem_2: Vinogradov's bound |S| << P^{1-rho}, rho = 0.045 nu^2/(log n + 2), for the sum of e^{2 pi i l f(p)} over primes P_1 < p < P_2 with 0.5P < P_1 < P_2 <= P, when the real function f has continuous derivatives of orders n-1, n, n+1 with f^{(n)} and x f^{(n)} - (n-1) f^{(n-1)} of constant sign and of prescribed sizes, for 0 < l <= P^{2 rho}.

theorem_3: Vinogradov's distribution law for the fractional parts of a real polynomial f of degree at most n without constant term at the primes p <= P: when some coefficient of degree s, 2 <= s <= n, is a/q + theta/(q tau) with 0 < q <= tau = P^{0.5 s} and q = P^kappa, the number of p <= P with 0 <= {f(p)} < gamma is gamma pi(P) + O(P^{1-rho'}) for every 0 < gamma <= 1.

theorem_4: Vinogradov's distribution law for the fractional parts {f(p)} over primes P_1 < p <= P_2, for a real function f satisfying the derivative conditions of Theorem 2: for every 0 < gamma <= 1 the count of p with 0 <= {f(p)} < gamma is gamma(pi(P_2) - pi(P_1)) + O(P^{1-rho'}), rho' = 0.044 nu^2/(log n + 2).

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Vinogradov, I. M., On an estimate of trigonometric sums with prime numbers. Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), no. 3, 225--248.

This Russian-language paper proves two theorems giving, in very general circumstances, estimates for trigonometric (exponential) sums whose variable runs over the primes p <= P, together with some applications; they are the analogues, for sums over primes, of Theorem 1 and Theorem 2a of Chapter VI of the author's 1947 book, close to results he had announced without proof in his Doklady notes, and a footnote on p. 225 corrects the statement of cases 1 and 3 of that Theorem 1 of the book. Theorem 1 (pp. 234-235) treats S = sum_{p <= P} e^{2 pi i l f(p)} for f(x) = a_n x^n + ... + a_1 x with real coefficients and l a positive integer, where n is an integer constant at least 10, so that f has degree at most n (the paper's standing notation, pp. 225-226, under which also kappa is a positive constant at most 1 and |theta| <= 1), and shows that if some coefficient a_s (2 <= s <= n) has a rational approximation a_s = a/q + theta/(q tau) with (a,q) = 1 and P^{kappa} << q <= tau = P^{0.5 s}, then S << P^{1-rho} for l <= P^{2 rho_0}, with rho = 0.041 nu^2/(log n + 2) if q > P^{0.25} and rho = 0.37 nu^2 kappa/(log(n^2/kappa) + 4) if q <= P^{0.25}, nu = 1/n (corrected on the page images from the digest's tau = P^{0.25} and S << Z P^{-rho}, the latter an intermediate of the proof). The condition on l is printed with rho_0, the slightly larger exponent of Lemma 4 (p. 228: rho_0 = 0.0416 nu^2/(log n + 2) if q > P^{0.25} and 0.375 nu^2 kappa/(log(n^2/kappa) + 4) if q <= P^{0.25}), which the proof of Theorem 1 takes over. Theorem 2 (p. 243) is the analogous bound for a real phase f with continuous f^{(n-1)}, f^{(n)}, f^{(n+1)} on an interval (P_1, P_2], 0.5P < P_1 < P_2 <= P, where f^{(n)} and phi(x) = x f^{(n)}(x) - (n-1) f^{(n-1)}(x) keep constant signs and |f^{(n)}|, |phi| and |f^{(n+1)}| satisfy size conditions; Theorems 3 and 4 (pp. 246-248) are the applications: for every 0 < gamma <= 1, the number of primes p <= P with 0 <= {f(p)} < gamma is gamma pi(P) + O(P^{1-rho'}) under the coefficient hypothesis of Theorem 1 (with 0 < q <= tau and q = P^{kappa}), and the number of primes P_1 < p <= P_2 with 0 <= {f(p)} < gamma is gamma(pi(P_2) - pi(P_1)) + O(P^{1-rho'}) under the hypotheses of Theorem 2 on f (with the range printed 0.5 < P_1 < P_2 <= P), each with an explicit rho'. The proof is the Vinogradov method: sieving the primes into bilinear pieces, divisor-sum lemmas (Lemmas 1 and 2, the latter credited to Mardzhanishvili), and a mean-value estimate for multiple exponential-sum integrals (Lemma 3), followed by a counting argument for the number of boxes of coefficient vectors. For problem 972 the site's commentary cites the paper for the uniform distribution of {p alpha}, alpha irrational, as Erdős's 1965 lecture does; the four theorems as read here require a polynomial phase a_n x^n + ... + a_1 x in which some coefficient a_s with 2 <= s <= n has a rational approximation of the kind above, or a smooth phase with derivative conditions, and do not state the linear case, which belongs to Vinogradov's earlier work (the paper's references are his Doklady notes of 1946 and 1947 and his 1947 book on the method of trigonometric sums).

Source: https://www.mathnet.ru/eng/im3028.

The copy read for this card is a 24-page scan of the Russian text (printed pp. 225-248; printed p. n is PDF p. n-224); received 15 January 1948 (p. 248). Read status: claims checked, on the page images, for the hypotheses and conclusions of Theorems 1-4 (pp. 234-235, 243 and 246-248; PDF pp. 10-11, 19 and 22-24) and for the standing notation of p. 226, and on 2026-10-07 for the introduction and footnote of p. 225, the standing notation of pp. 225-226 and the exponent rho_0 of Lemma 4 (p. 228); no proof was checked; the other lemma descriptions above record an earlier reading that was not repeated. The statements of Theorems 1-4 were read again clause by clause on the page images on 2026-10-08 for the result pages, with the structure of the proofs of Theorem 1 (pp. 235-238) and Theorems 3 and 4 (pp. 247-248); the depth of each is recorded on its page. No copyright or license line is printed on any of the 24 pages, and the hosting site's terms of use state that its materials "are fully copyrighted by Steklov Mathematical Institute, Russian Academy of Sciences, and/or by other copyright holder" and that reproduction or republication "requires written permission of the copyright holder" (https://www.mathnet.ru/php/agreement.phtml?option_lang=eng, read 2026-10-02), every other right reserved.

Bears on. #972 (the site's [Vi48] key, and Erdős's 1965 lecture, for the uniform distribution of {p alpha}, alpha irrational, from which the problem page derives the one-prime statement; Theorems 1-4 as read here do not state the linear case, a source-identity remark recorded on the problem page, and none addresses the two-prime question; theorem_1, theorem_2, theorem_3, theorem_4)

Results to transcribe.

• Theorem 1 (pp. 234-235): For S = sum_{p<=P} e^{2 pi i l f(p)} with f(x) = a_n x^n + ... + a_1 x real and n >= 10 the paper's constant, a rational approximation a/q + theta/(q tau) to some coefficient a_s, 2 <= s <= n, with (a, q) = 1 and P^{kappa} << q <= tau = P^{0.5 s} yields S << P^{1-rho} for l <= P^{2 rho_0} (rho_0 from Lemma 4), with explicit rho.
• Theorem 2 (p. 243): For 0.5P < P_1 < P_2 <= P and a real f on (P_1, P_2] with the derivative conditions above, |S| << P^{1-rho} for the sum over P_1 < p < P_2, rho = 0.045 nu^2/(log n + 2), 0 < l <= P^{2 rho}.
• Theorem 3 (pp. 246-247): Under a coefficient hypothesis as in Theorem 1 with 0 < q <= tau in place of P^{kappa} << q, and q = P^{kappa}, the number of primes p <= P with 0 <= {f(p)} < gamma is gamma pi(P) + O(P^{1-rho'}) for every 0 < gamma <= 1, with explicit rho'.
• Theorem 4 (pp. 247-248): Under the derivative conditions of Theorem 2 (the range printed 0.5 < P_1 < P_2 <= P), the number of primes P_1 < p <= P_2 with 0 <= {f(p)} < gamma is gamma(pi(P_2) - pi(P_1)) + O(P^{1-rho'}), rho' = 0.044 nu^2/(log n + 2).
• Lemma 3 (pp. 226-227): Mean-value estimate for the integral over the unit cube of |S|^r for exponential sums with polynomial phase, the paper's main analytic tool; no page of its own, as no corpus page uses it.

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