On non-complete rational trigonometric sums Текст научной статьи по специальности «Математика»

74

Н. М. Saliba

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 19. Выпуск 3.

УДК 511.3 DOI 10.22405/2226-8383-2018-19-3-74-79

О неполных рациональных тригонометрических суммах

Салиба Холем Мансур — кандидат физико-математических наук, доцент факультета естественных и прикладных наук университета Нотр-Дам-Луэз, Ливан. e-mail: [email protected],

Аннотация

Приводится версия метода Хуа для оценки неполных рациональных тригонометрических сумм. Эти оценки не являются тривиальными для суммы с длинами, превышающими квадратный корень длины полной суммы.

Ключевые слова: метод Хуа оценки полных рациональных тригонометрических сумм, неполные рациональные тригонометрические суммы, полиномиальные сравнения, цепь показателей и корни сравнений.

Библиография: 10 названий. Для цитирования:

X. М. Салиба. О неполных рациональных тригонометрических суммах // Чебышевский сборник, 2018, т. 19, вып. 3, с. 74-79.

CHEBYSHEVSKII SBORNIK Vol. 19. No. 3.

UDC 511.3 DOI 10.22405/2226-8383-2018-19-3-74-79

On non-complete rational trigonometric sums

Saliba Holem Monsour — Ph.D. Assistant Professors of faculty of natural k, applied sciences of Notre Dame University Louaize, Lebanon. e-mail: [email protected]

Abstract

We give the version of Hua's method for the estimation of non-complete rational trigonometric sums. These estimates are non-trivial one for sums with lengths exceeding a square root of length the complete sum.

Keywords: the Hua's method of complete rational trigonometric sums estimate, non-complete rational trigonometric sums, polynomial congruencies, the chain of exponents and roots of congruencies.

Bibliography: 10 titles. For citation:

Saliba H. M., 2018, "On non-complete rational trigonometric sums" , Chebyshevskii sbornik, vol. 19, no. 3, pp. 74-79.

To the memory of Academician J. V. Linnik

1. Introduction

The purpose of this article is to give the new demonstration of the estimation of non-complete rational trigonometric sums. Early the deduction of similar estimations are realized with using the Fourier analysis [1],[2]. Here we develop the Hua's method of estimations of complete rational sums ([2], p.101-109). We follow the version of this method, proposed by V. N. Chubarikov [6]-[10].

Let n > 2,p is a prime number, f(x) = anxn + ■ ■ ■ + a1x + a0 is a polynomial with integer coefficients, (an,.. .a1,p) = 1, 0 < I < k and e(x) = e2ntx,

pk-l

S = S(pk;k - I, f) = £ e(f(x)/pk) (1)

X=1

p p^ S = £S (0, S (0= £ e(f(x)/pk), (2)

£=1 x=1

x=£ (mod p)

moreover

pk-i-i

s(o= E e(f(t+px)/pk)■

X=1

Let w = [lnn/ lnp], pT\\(nan,..., 2a2, a1), then t <w. We define, following Hua L.-K. ([2], p. 217), solutions

x = 6 + p6 + ••• + ps-16 + ... (3)

of congruence f(x) = 0 (mod pk) in the next way

p-0 f'(h) = 0 (mod p),p^g^(x) = /(6 + px) -f (£), (4)

where the coefficients of the polynomial g^ (x) and the number p have no common factor excepted 1, > 1

p-T°-1 g^.^s-MCs) = 0 (mod p), (5)

pUr9(ti,...,tr)(x) = 9(t1,...,tr-i)(tr + px) - g(t1,...,tr-i)(&), (6)

ks = ks-1 -us, ls = ls-1 -us + 1. (7)

Now we formulate statements of following theorems.

Theorem 1. Let inequalities kr-1 > 2(lr-1 +w + 1),kr < 2(lr + w + 1) be define the number

.

S(/;k-l,f)= £ e(lf- + ^ + - Ma.....«)•

Theorem 2. Let r be the smallest number over all solutions (£i,&,..., (r), defining early, and satisfying inequalities kr-1 > 2( lr-1 + w + 1), kr < 2(lr + w + 1). Then

|S(pk; k - I, f)\ < (n - 1)pk-l-r•

(8)

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H. M. Saliba

2. Lemmas

Further we have the following statement.

Lemma 1. Let ( is not a solution of the congruence p—T f'(x) = 0 (mod p), and let 0 < I < k. Then fork > I + 2(1 + w) we get S (g) = 0.

Proof. We put x = y + pk—l—T—2z, where 1 < y < pk-l-T—2, 0 < z < pT+1 - 1. It gives

pk — l — T — 2 pT + 1 —1

5(0= E E e(f(t+py+pk—l—T—1z)/pk) =

y=1 z=0

pk — l — T —2 pT+1 — 1

= E e«f(t+py)/pk)) E e(f'(£+py)z/pT+1) = 0,

y=1 z=0

as p—T f '(0 ^ 0 (mod p) and k > 2(1 + t + 1). Lemma is proved.

Lemma 2. Let £ be a solution of the congruence p—T f'(£) = 0 (mod p). Let

pug(x) = f(£ + px) - f(0, g(x) = gn(x) = bnxn + ■ ■■ + hx, (bn,...b1,p) = 1. Then we have

s(0 = e(f(0/vk) E e(g(x)/pk—u).

X=1

Proof. We find

pk I 1 pk I 1

k — u

t(y(U)/p

X=1 X=1

e(—f (o/pk)s(o= E e((f (t+px) - f №)/pk) = E <9(x)/pk-u).

Lemma is proved.

Lemma 3. The number of solutions of the congruence f '(x) = 0 (mod pk ) in the sense described (3)-(7) is at most n — 1.

Proof. See ([2], p.217, Lemma 6.1). Lemma 4. We have

n — 1 > u\ > u2 > • • • > ur > 2.

Proof. See ([2], p.219, Lemma 7.1). Lemma 5. We have

k — I + r — (u\ + • • • + ur) < [ln n/ ln p\. proof. See ([2], p.220, Lemma 7.2).

3. Proof of theorems

1. For k > 2(1 + w + 1) we get

5(pk; k — IJ) = E e(f(i)/pk)S(pk-u; k — I — 1,gs),

where ( = ^ runs all solutions of the congruence p-T f(£) = 0 (mod p), and u = u1 = u1(() is defined in the statement of the Lemma 2.

Putting k1 = k — u1, h = 1 + 1 — u1, we have

5(ph; k — I, f) = E e(f(0/Ph)S(Ph1; k1 — h, 9i).

Thus if k1 > 2(11 + w + 1) then we obtain from Lemmas 1 and 2

S(p*;k — I, f) = E + ^W1-U2 ;k1 — h — 1, g^),

where (2 is a solution of the congruence

p-^ fa) = 0 (mod p), pTl\\(nbn,..., 2b2, h), n <

w,

and

PU2 9(t- i, Ü2)(x) = 9ti (6 + px) — g^ (&), 9(£ u b)(x) = cn xn + ••• + C1 x, (cn,... C1,p) = 1. We carry on doing this procedure further. For r > 1 we put

kf — kf_1 'Uf, If — _1 \ 1 'Uf.

PUr 9(t u-t r )(x) = 9d r-i)(£ r + Px) — 9{Z r-i)( &), g(( i,...,&)(x) = c^xn + ••• + c^x, (dn),... c1\ p) = 1.

kf-1 > 2(lf-1 + w + 1),kf < 2(lf + w + 1),

we find

Stf ;k-UJ)= E « (f~¥ + ^ + ••• + ^^rfe^) -

V P P P /

XS(pkr-1-^r ■kr-1 _ lr—i — \,g{r)) =

= £ 4+ + + W-ir^&.....,)),

(tl.....tr) K P P P /

r

P-Tr-1 ^....¿r-1 )(tr ) = 0 (mod P), p^-1 Winb^,---, 2bt1], b{;-1)), Tr-1 <w.

The theorem 1 is proved. 2. Further we have

kr = kr-i — ur = kr-2 — ur-i — ur = • • • = k — ui — u1 — • • • — ur,

lr = lr-i + 1 — ur = lr-2 + 1 — ur-1 + 1 — ur-2 = • • • = I + r — u1 — ••• — ur.

Hence

kr r = k .

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Н. М. Saliba

Prom here we get

S(/;k - I, f)= £ em + ^ + - + '' W к ;k-l-r.^.....w).

(tu-tr) V P P P /

Therefore, using the Lemma 3, we find

|S(Рк;k -1, /)|< (n - i)Pk-l-r.

The theorem 2 is proved.

Conclusive notes. It's interesting to get non-trivial estimations for shorter non-complete rational trigonometric sums.

We continue recent studies of Professor V.N.Chubarikov on complete arithmetical sums. The problem of this paper due to him. I express my gratitude to Professor V.N.Chubarikov for the discussion of this problem.

СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

1. Vinogradov I. M. Selected works. / New York Inc.: Springer Verlag, 1985, P. 401.

2. HuaL.-K. Selected Papers. / New York Inc.: Springer Verlag, 1983, P. 888.

3. ArkhipovG.I. Selected Papers. / Orjol: Publ. House of Orjol State University, 2013, P. 464.

4. ArkhipovG.I., Chubarikov V. N., Karatsuba A. A. Trigonometric Sums in Number Theory and Analysis. / De Gruvter expositions in mathematics; 39. Berlin, New York, 2004. 554 P.

5. Karatsuba A. A. Distribution of fractional parts of polynomials of special form // Bull. Moscow University, Math., 1962, № 3. pp.34-38.

6. Chubarikov,V.N. Linear arithmetic sums and Gaussian multiplication theorem. // Azerbaijan — Turkey — Ukrainian Int.Conf. "Mathematical Analysis, Differential Equations and their Applications". Abstracts.(September 08-13, 2015, Baku-Azerbaijan). 2015, p.38.

7. Chubarikov V. N. Elementary of the complete rational arithmetical sums. // Chebvshevskii Sbornik. 2015;16(3) pp. 450-459.

8. Chubarikov V. N. Arithmetic sums of polynomial values // Dokl. RAN — 2016. — V.466, № 2. - pp.152-153.

9. Chubarikov V. N. Complete Rational Arithmetic Sums // Bull. Math, of Moscow Univ. Ser.I, 2015, № 1, pp.60-61.

10. Chubarikov V. N. On Complete Rational Arithmetic Sums of Polynomial Values // Proc. of the Steklov Institute of Math., 2017, V. 299, pp.50-55.

REFERENCES

1. Vinogradov I. M., 1985, Selected works. New York Inc.: Springer Verlag, P. 401.

2. HuaL.-K., 1983, Selected Papers. New York Inc.: Springer Verlag, P. 888.

3. ArkhipovG.I., 2013, Selected Papers. Orjol: Publ. House of Orjol State University, P. 464.

4. ArkhipovG. I., Chubarikov V. N., Karatsuba A. A., 2004, Trigonometric Sums in Number Theory and Analysis. De Gruvter expositions in mathematics; 39. Berlin, New York, 554 P.

5. Karatsuba A. A., 1962, "Distribution of fractional parts of polynomials of special form" , Bull. Moscow University, Math., № 3. pp. 34-38.

6. Chubarikov,V.N., 2015, "Linear arithmetic sums and Gaussian multiplication theorem" , Azerbaijan-Turkey-Ukrainian Int.Conf. "Mathematical Analysis, Differential Equations and their Applications". Abstracts.(September 08-13, 2015, Baku-Azerbaijan), p.38.

7. Chubarikov V. N., 2015, "Elementary of the complete rational arithmetical sums", Cheby-shevskii Sbornik, 16(3) pp. 450-459.

8. ChubarikovV.N., 2016, "Arithmetic sums of polynomial values", Dokl. RAN V.466, № 2. pp.152-153.

9. Chubarikov V. N., 2015, "Complete Rational Arithmetic Sums" , Bull. Math, of Moscow Univ. Ser.I, № 1, pp.60-61.

10. Chubarikov V. N., 2017, "On Complete Rational Arithmetic Sums of Polynomial Values" , Proc. of the Steklov Institute of Math., V. 299, pp.50-55.

Получено 16.09.2018 Принято к печати 10.10.2018