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A new method for estimating trigonometric sums (see 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 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) 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, \end{align*}

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

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

For any integer $k\geq 1$:

\begin{align*} \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}, \end{align*}

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

$\lvert B_n\rvert\leq L_n,\ldots,\lvert B_1\rvert\leq L_1.$

If the coefficients $\alpha_{n+1},\ldots,\alpha_2$ of the polynomial $f(x)$ have certain arithmetical properties, it is possible to obtain the estimate $G(Y)\leq Y^{0.9}$. In addition, the last integral does not exceed the number of solutions of the system of equations:

\begin{align*} x_1^n+\cdots+x_k^n &= y_1^n+\cdots+y_k^n,\\ x_1^{n-1}+\cdots+x_k^{n-1} &= y_1^{n-1}+\cdots+y_k^{n-1},\\ \vdots\\ x_1+\cdots+x_k &= y_1+\cdots+y_k, \end{align*} $1\leq x_1,\ldots,y_k\leq P.$

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

## Vinogradov's method for estimating trigonometric sums with prime numbers.

The sums to be estimated are

$S' = \sum_{p\leq P}e^{2\pi i f(p) },$

where $f(p)=\alpha_np^n+\cdots+\alpha_1p$, and $\alpha_n,\ldots,\alpha_1$ are real numbers. Let $D=\prod_{p\leq H}p$, where $H\leq P^{0.25}$. Using the well-known property of the Möbius function, $S'$ is reduced to a small number of sums (this number is not larger than $\ln P/\ln H$) of the form

$W_S=\sum_{d_1\mid D}\cdots \sum_{d_s\mid D}\sum_{m_1>0}\cdots\sum_{m_s>0}\mu(d_1)\cdots\mu(d_s)e^{2\pi i f(m_1\cdots m_sd_1\cdots d_s)},$

where $m_1\cdots m_sd_1\cdots d_s\leq P$. In the multiple sum $W_S$ the variables $m_1,\ldots,m_s$ run through the entire summation intervals. The sums $W_S$ in which the summation interval over at least one variable $m$ is long are estimated by Vinogradov's method for estimating Weyl sums. Otherwise the summation interval over one of the summation variables $d$, $d\mid D$, 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 $0\leq \sigma\leq 1/3$ and let $D$ be the product of all primes not larger than $x^{\sigma}$; all divisors $d$ of $D$ not larger than $x$ may then be distributed over fewer than $x^{\epsilon}$ sets with the following properties:

1) the numbers $d$ belonging to one of these sets have the same number $\beta$ of prime factors and therefore the same value of $\mu(d)=(-1)^\beta$;

2) one of these sets — the so-called simplest set — consists of the single number $d=1$. For each of the remaining sets there is a number $\phi$ such that all numbers of this set satisfy the relation

$\phi< d\leq \phi^{1+\epsilon_1},\hspace{1cm} \epsilon_1=\epsilon_1(\epsilon);$

3) for each set of numbers $d$ other than the simplest set there exist, for any $U$, $0\leq U\leq \phi$, two sets of numbers $d'$ and $d''$, with corresponding numbers $\phi'$ and $\phi''$ satisfying the relations

$U<\phi'\leq Ux^{\sigma},\hspace{1cm}\phi'\phi''=\phi,$

and such that, for a certain natural number $B$, one obtains each number in the chosen set $B$ times if, out of all the products $d'd''$ one selects only those which satisfy the relation $(d',d'')=1$.

The application of point 3) of this lemma, with a suitable value of $U$, yields

$W_S=\sum_u\sum_v\psi_1(u)\psi_2(v)e^{2\pi i f(uv) },$

where the variables $u$ and $v$ 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).

If $F(x)$ can be properly approximated, in a certain sense, by a polynomial, Vinogradov's method can be used to estimate sums of the type

$S=\sum_{1\leq x\leq P}e^{2\pi i F(x)},\hspace{1cm} S'=\sum_{p\leq P}e^{2\pi i F(p)}$

(see , ). Sums of the type

$\sum_{p\leq P}\chi(p+a),\hspace{1cm} \sum_{1\leq n\leq N}\mu(n)\chi(n+a)$

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 $p+a$, where $a>0$ is a given integer, while $p$ runs through the successive prime numbers , . For the application of Vinogradov's method in analytic number theory see , , , , .

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