MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR Yu. V. LINNIK

THE SIXTH MOMENT FOR \(L\)-SERIES AND AN ASYMPTOTIC FORMULA IN THE HARDY–LITTLEWOOD PROBLEM

The Hardy–Littlewood equation \(\left({}^{1,2}\right)\)

\[ n=p+\xi^2+\eta^2 \tag{1} \]

(\(p\) is a prime number, \(\xi,\eta\) are integers) has a solution for all sufficiently large \(n\). For the number \(Q(n)\) of solutions of equation (1), the following asymptotic formula holds*:

Theorem 1.

\[ Q(n)\sim \pi \frac{n}{\ln n}\prod_p\left(1+\frac{\chi_4(p)}{p(p-1)}\right) \prod_{p/n}\frac{\chi^2(p-1)(p-\chi_4(p))}{p^2-p+\chi_4(p)}+R(n), \tag{2} \]

where

\[ R(n)=O\left(\frac{n}{(\ln n)^{1.028}}\right). \tag{3} \]

Equation (1), as was indicated in \(\left({}^{2}\right)\), reduces to equations of the form

\[ x_1x_2\ldots x_k+\xi^2+\eta^2=n, \tag{4} \]

where \(x_i\) \((i=1,2,\ldots,k)\) run through numbers with sufficiently large prime factors. For \(k>6\) the solutions of such equations have been obtained by the “dispersion method” \(\left({}^{3}\right)\); for \(k\leqslant 6\) they reduce to equations of the form (4), where the \(x_i\) run successively through all odd numbers. The solution of all such equations (for \(k\leqslant 6\)) with the proper asymptotic for the number of solutions can be carried out with the aid of the following theorem on the “sixth moment for \(L\)-series.”

Let \(D\) run through all integers subject to the condition

\[ D_1\leqslant D\leqslant D_1+D_2, \tag{5} \]

where \(D_2\gg D_1(\ln D_1)^{-k}\); \(k>0\) is some constant. For each modulus \(D\) consider all Dirichlet characters \(\chi_D\) and the \(L\)-series \(L(s,\chi_D)\); let \(s=\frac12+it\) (\(t\) real). Form the sixth (unnormalized) moment for \(L\)-series:

\[ \sum_{D_1\leqslant D\leqslant D_1+D_2}\sum_{\chi_D} \left|L\left(\frac12+it,\chi_D\right)\right|^6. \tag{6} \]

The solution of equations of the form (4) for \(k\leqslant 6\) is achieved with the help of an estimate of the sixth (unnormalized) moment (6). The following theorem holds, for which other applications may also be indicated.

* In my note \(\left({}^{2}\right)\) a gap was admitted in the reasoning, as a result of which the remainder term (3) comes out worse than indicated in \(\left({}^{2}\right)\).

Theorem 2.

\[ \sum_{D_1 \leq D \leq D_1 + D_2}\ \sum_{\chi_D} \left|L\left(\frac{1}{2}+it,\chi_D\right)\right|^6 = BD_2D_1(|t|+1)^{l_0}\exp(\ln D_1)^{\varepsilon_0}, \tag{7} \]

where \(B\) is a bounded quantity, \(l_0>0\) is a constant; \(\varepsilon_0>0\) is an arbitrarily small constant.

Estimate (7) for the sixth moment of \(L\)-series makes it possible to derive certain new results in the additive divisor problem, and also to obtain rather good information on the number of solutions of the equation

\[ n=p+\varphi(x,y), \tag{8} \]

where \(\varphi(x,y)\) is an arbitrary integral binary quadratic form.

Received
9 V 1960

REFERENCES

¹ G. H. Hardy, J. E. Littlewood, Acta Math., 44, 1 (1923). ² Yu. V. Linnik, DAN, 124, No. 1, 29 (1959). ³ Yu. V. Linnik, DAN, 120, No. 5, 960 (1958).