View source for Vinogradov method

A new method for estimating trigonometric sums (see [[Trigonometric sums, method of|Trigonometric sums, method of]]). By Vinogradov's method one may obtain very accurate estimates of a wide class of trigonometric sums, in which the summation variable runs through a sequence of integers, prime numbers, etc. In this way many classical problems in [[Analytic number theory|analytic number theory]] can be solved (the distribution of the fractional parts of a wide class of functions, the distribution of prime numbers in series of natural numbers, additive problems such as the Waring and Goldbach problems, etc.).

One distinguishes two parts in Vinogradov's method: The method of estimating Weyl sums (cf. [[Weyl sum|Weyl sum]]) and the method of estimating trigonometric sums with prime numbers. Both these parts utilize Vinogradov's basic idea — to wit, the idea of smoothing double trigonometric sums, which may be described as follows. Given the sum

\[ W=\sum_u\sum_v\psi_1(u)\psi_2(v) e^{2\pi i\alpha uv},\]

where the summation variables $u$ and $v$ run through the values of (not necessarily successive) integers in respective sets $U$ and $V$, $A<u<2A$. Let $\psi_1(u)$ and $\psi_2(v)$ be arbitrary complex-valued functions. Then

\[ \lvert W\rvert^2\leq B\sum_{A< u\leq 2A}\left\lvert \sum_v\psi_2(v)e^{2\pi i\alpha uv}\right\rvert^2,\]

where $u$ runs through the successive integers in the interval $(A,2A]$ (smoothing), and

\[ B=\sum_n\lvert \psi_1(u)\rvert^2.\]

==Vinogradov's method for estimating Weyl sums.==
The sums to be estimated are

\[ S=\sum_{1\leq x\leq P}e^{2\pi if(x)},\]

where $f(x)=\alpha_{n+1}x^{n+1}+\cdots+\alpha_1x$; here $\alpha_{n+1},\ldots,\alpha_1$ are real numbers. For $Y=[P^{1-n^2/4}]$ one finds

\begin{align*}
S&=Y^{-1}\sum_{1\leq x\leq P}\sum_{1\leq y\leq Y}e^{2\pi if(x+y)}+2\theta Y\\
&=Y^{-1}\sum_{1\leq x\leq P}\sum_{1\leq y\leq Y}e^{2\pi i\mathfrak{A}}+2\theta Y=Y^{-1}W+2\theta Y,\]

where $\mathfrak{A}=\alpha_{n+1}x^{n+1}+A_n(y)x^n+\cdots+A_1(y)x+A_0(y)$, the letter $W$ denotes the double sum over $x$ and $y$, and $\lvert \theta\rvert\leq 1$. Moreover, letting $\mathfrak{B}$ denote the expression

\[ \alpha_{n+1}x^{n+1}+(A_n(y)+B_n)x^n+\cdots+(A_1(y)+B_1)x\]

one has

\[\lvert W\rvert\leq \sum_{1\leq y\leq Y}\left\lvert \sum_{1\leq x\leq P}e^{2\pi i\mathfrak{B}}\right\rvert+Y_nP^{1-n^2/4},\]

for arbitrary $B_n,\ldots,B_1$ from the domain

\[\lvert B_n\rvert\leq L_n=P^{-n-n^2/4},\textrm{\,\,\,\,\,}\ldots\textrm{\,\,\,\,}\lvert B_1\rvert\leq L_1=P^{-1-n^2/4}.\]

For any integer $k\geq 1$:

\[ \lvert W\rvert^{2k}\leq 2^kY^{2k-1}\sum_{1\leq y\leq Y}\int_{\lvert B_n\rvert\leq L_n}\cdots \int_{\lvert B_1\rvert\leq L_1}
\left\lvert \sum_{1\leq x\leq P} e^{2\pi i\mathfrak{B}}\right\rvert\mathrm{d}B_n\cdots\mathrm{d}B_1
+2^k(Y_nP^{1-n^2/4})^{2k}\]
\[\leq 2^kY^{2k-1}G(Y)\int_0^1\cdots \int_0^1\left\lvert \sum_{1\leq x\leq P}e^{2\pi i\mathfrak{C}}\right\rvert^{2k}
\mathrm{d}\gamma_n\cdots\mathrm{d}\gamma_1+2^k(Y_nP^{1-n^2/4})^{2k},\]

where $\mathfrak{C}=\alpha_{n+1}x^{n+1}+\gamma_nx^n+\cdots+\gamma_1x$ and $G(Y)$ is the maximum number of cases of coincidence of points with coordinates

\[ \{A_n(y)+B_n\},\{A_{n-1}(y)+B_{n-1}\},\ldots,\{A_1(y)+B_1\}.\]


Here the braces denote the fractional part of the enclosed number, while $y$ varies between 1 and $Y$, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669041.png" /></td> </tr></table>

If the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669042.png" /> of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669043.png" /> have certain arithmetical properties, it is possible to obtain the estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669044.png" />. In addition, the last integral does not exceed the number of solutions of the system of equations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669045.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669046.png" /></td> </tr></table>

The number of solutions of this system is estimated using the [[Vinogradov theorem about the average|Vinogradov theorem about the average]], which is fundamental in Vinogradov's method for estimating Weyl sums (cf. [[Vinogradov estimates|Vinogradov estimates]]).

==Vinogradov's method for estimating trigonometric sums with prime numbers.==
The sums to be estimated are

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669047.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669048.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669049.png" /> are real numbers. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669051.png" />. Using the well-known property of the Möbius function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669052.png" /> is reduced to a small number of sums (this number is not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669053.png" />) of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669054.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669055.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669056.png" />. In the multiple sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669057.png" /> the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669058.png" /> run through the entire summation intervals. The sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669059.png" /> in which the summation interval over at least one variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669060.png" /> is long are estimated by Vinogradov's method for estimating Weyl sums. Otherwise the summation interval over one of the summation variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669062.png" />, will be long. In such a case one uses the following lemma of Vinogradov which, together with the idea of smoothing double sums, is fundamental to Vinogradov's method for estimating trigonometric sums with prime numbers.

Lemma. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669063.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669064.png" /> be the product of all primes not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669065.png" />; all divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669066.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669067.png" /> not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669068.png" /> may then be distributed over fewer than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669069.png" /> sets with the following properties:

1) the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669070.png" /> belonging to one of these sets have the same number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669071.png" /> of prime factors and therefore the same value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669072.png" />;

2) one of these sets — the so-called simplest set — consists of the single number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669073.png" />. For each of the remaining sets there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669074.png" /> such that all numbers of this set satisfy the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669075.png" /></td> </tr></table>

3) for each set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669076.png" /> other than the simplest set there exist, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669078.png" />, two sets of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669080.png" />, with corresponding numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669081.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669082.png" /> satisfying the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669083.png" /></td> </tr></table>

and such that, for a certain natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669084.png" />, one obtains each number in the chosen set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669085.png" /> times if, out of all the products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669086.png" /> one selects only those which satisfy the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669087.png" />.

The application of point 3) of this lemma, with a suitable value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669088.png" />, yields

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669089.png" /></td> </tr></table>

where the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669091.png" /> run through long summation intervals. It is possible to deduce Vinogradov's estimate for trigonometric sums with prime numbers from this lemma (cf. [[Vinogradov estimates|Vinogradov estimates]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669092.png" /> can be properly approximated, in a certain sense, by a polynomial, Vinogradov's method can be used to estimate sums of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669093.png" /></td> </tr></table>

(see [[#References|[2]]], ). Sums of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669094.png" /></td> </tr></table>

and other types can also be estimated by the method. It is possible in this way to solve problems on the distribution of power residues, primitive roots, etc., in sequences of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669096.png" /> is a given integer, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096690/v09669097.png" /> runs through the successive prime numbers [[#References|[3]]], [[#References|[5]]]. For the application of Vinogradov's method in analytic number theory see [[#References|[1]]], [[#References|[2]]], , [[#References|[5]]], [[#References|[6]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,   "The method of trigonometric sums in the theory of numbers" , Interscience  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,   "Selected works" , Springer  (1985)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.M. Vinogradov,   "Estimates of a sum, distribution of primes in an arithmetic series"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''30'''  (1966)  pp. 481–496  (In Russian)</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  A.A. Karatsuba,   "Estimates for trigonometric sums by Vinogradov's method, and applications"  ''Proc. Steklov Inst. Math.'' , '''112'''  (1971)  pp. 251–265  ''Trudy Mat. Inst. Steklov.'' , '''112'''  (1971)  pp. 241–255</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  A.A. Karatsuba,   "On some problems of prime number theory connected with I.M. Vinogradov's method"  ''Proc. Steklov Inst. Math.'' , '''132'''  (1973)  pp. 293–298  ''Trudy Mat. Inst. Steklov.'' , '''132'''  (1973)  pp. 257–261</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.-K. Hua,   "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , ''Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen'' , '''1''' :  2  (1959)  (Heft 13, Teil 1)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Chandrasekharan,   "Arithmetical functions" , Springer  (1970)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Vaughan,   "The Hardy–Littlewood method" , Cambridge Univ. Press  (1981)</TD></TR></table>