On some identities involving certain Hardy sums and Kloosterman sum
UDC 511 We give a new reciprocity theorem for the Hardy sum $s_{5}(h,p).$ Also, a hybrid mean value problem involving the Hardy sum $s_{4}(h,p)$ and Kloosterman sum is studied and two exact computational formulae are obtained.
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507093189001216 |
|---|---|
| author | Dağlı, M. C. Dağlı, M. C. |
| author_facet | Dağlı, M. C. Dağlı, M. C. |
| author_sort | Dağlı, M. C. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:35Z |
| description | UDC 511
We give a new reciprocity theorem for the Hardy sum $s_{5}(h,p).$ Also, a hybrid mean value problem involving the Hardy sum $s_{4}(h,p)$ and Kloosterman sum is studied and two exact computational formulae are obtained.
|
| doi_str_mv | 10.37863/umzh.v72i11.731 |
| first_indexed | 2026-03-24T02:03:50Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i11.731
UDC 511
M. C. Dağli (Akdeniz Univ., Turkey)
ON SOME IDENTITIES INVOLVING CERTAIN HARDY SUMS
AND KLOOSTERMAN SUM
ПРО ДЕЯКI ТОТОЖНОСТI IЗ ПЕВНИМИ СУМАМИ ГАРДI
ТА СУМОЮ КЛООСТЕРМАНА
We give a new reciprocity theorem for the Hardy sum s5(h, p). Also, a hybrid mean value problem involving the Hardy
sum s4(h, p) and Kloosterman sum is studied and two exact computational formulae are obtained.
Запропоновано нову теорему взаємностi для суми Гардi s5(h, p). Крiм цього, вивчається гiбридна задача про середнi
значення, яка мiстить суму Гардi s4(h, p) i суму Клоостермана, та отримано двi точнi обчислювальнi формули.
1. Introduction. Let
((x)) =
\left\{ x - [x] - 1/2, if x \in \BbbR \setminus \BbbZ ,
0, if x \in \BbbZ ,
with [x] being the largest integer \leq x. For positive integer p and integer h the classical Dedekind
sum s(h, p), arising in the theory of Dedekind \eta -function, were introduced by R. Dedekind in 1892
by
s(h, p) =
p\sum
a=1
\biggl( \biggl(
a
p
\biggr) \biggr) \biggl( \biggl(
ha
p
\biggr) \biggr)
.
Perhaps the most important property of Dedekind sums is the reciprocity theore
s(h, p) + s(p, h) = - 1
4
+
1
12
\biggl(
h
p
+
p
h
+
1
hp
\biggr)
, (1.1)
when (h, p) = 1 (for basic properties see [9]). The arithmetic properties of Dedekind sums were in-
vestigated by many authors (see, for example, [12, 14, 15]). J. B. Conrey et al. [4] dealt with the mean
value distribution of Dedekind sums and achieved an asymptotic formula for
\sum p
h=1
\prime | s(h, p)| 2m ,
where the dash denotes the summation over all 1 \leq h \leq p such that (h, p) = 1. Moreover,
H. Valum [11] derived a relation between the mean square value of s(h, p) and the fourth power
mean of Dirichlet L-function.
Similar arithmetic sums arise in the theory of logarithms of the classical theta functions. They
are studied by Hardy and Berndt, and for this reason they are called Hardy or Hardy – Berndt sums.
There are six such sums, two of which are [2, 6]
s4(h, p) =
p - 1\sum
a=1
( - 1)[ha/p],
s5(h, p) =
p - 1\sum
a=1
( - 1)a+[ha/p]
\biggl( \biggl(
a
p
\biggr) \biggr)
.
c\bigcirc M. C. DAĞLI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11 1495
1496 M. C. DAĞLI
Goldberg [6] showed that these sums also arise in the theory of rm(n), the number of representations
of n as a sum of m integral squares and in the study of the Fourier coefficients of the reciprocals
of the classical theta functions. Like Dedekind sums, these Hardy sums also satisfy a reciprocity
(or reciprocity-like) formula [2, 6]. R. Sitaramachandrarao [10] expressed these sums in terms of
classical Dedekind sums. For example,
s4(h, p) = - 4s(h, p) + 8s(h, 2p), if h is odd. (1.2)
Recently, Du and Zhang [5] have studied the computational problem of Dedekind sums and estab-
lished a new reciprocity formula by using analytic method and the properties of Dirichlet L-function.
That is, they gave the following theorem.
Theorem 1.1 ([5], Theorem 1). Let h and p are two positive odd numbers with (h, p) = 1.
Then
s (2\=p, h) + s
\bigl(
2\=h, p
\bigr)
=
h2 + p2 + 4
24hp
- 1
4
, (1.3)
where \=p and \=h satisfy the congruence p\=p \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d}h) and h\=h \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} p).
On the other hand, the mean value of Hardy sums or hybrid mean value involving Hardy sums
and other celebrated sums are intensively studied. For example, the authors of [13] discussed the
hybrid mean value involving certain Hardy sums and Kloosterman sum, defined for any positive
integer p > 1 and integer n by
K(n, p) =
p\sum
a=1
\prime e
\biggl(
na+ \=a
p
\biggr)
,
where \=a denotes the solution of the congruence xa \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} p), the dash denotes the summation
over all 1 \leq a \leq p such that (a, p) = 1 and e(x) = e2\pi ix. They obtained exact computational
formulas
p - 1\sum
m=1
p - 1\sum
n=1
K(m, p)K(n, p)S (2m\=n, p)
and
p - 1\sum
m=1
p - 1\sum
n=1
| K(m, p)| 2 | K(n, p)| 2 S (2m\=n, p) ,
where S(h, p) is one of the Hardy sums. Some elementary properties of K(n, p) can be found in
[3, 7]. Peng and Zhang [8] investigated the hybrid mean value involving s5(h, p) in order to help to
achieve several identities between Hardy sums and Kloosterman sums.
As mentioned in [8], little about s5(h, p) is known. Thus, it is meaningful to continue to study
the properties of s5(h, p).
In this paper, firstly, we give following new reciprocity theorem for Hardy sum s5(h, p) by
applying rather elementary method.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON SOME IDENTITIES INVOLVING CERTAIN HARDY SUMS . . . 1497
Theorem 1.2. Let h and p are odd primes. Then we have
s5 (2\=p, h) + s5
\bigl(
2\=h, p
\bigr)
=
1
2
- h2 + p2
4hp
,
where \=p and \=h satisfy the congruence p\=p \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d}h) and h\=h \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} p).
Secondly, using the properties of Gauss sums and mean value theorems of Dirichlet L-function,
we obtain the following conclusions for Hardy sum s4(h, p) and Kloosterman sum in order to help
to obtain further relations between these sums.
Theorem 1.3. Let p be odd prime, then we have
p - 1\sum
m=1
p - 1\sum
n=1
K(m, p)K(n, p)s4 (m\=n, p) = p2(p - 1).
Theorem 1.4. Let p be odd prime, then we obtain
p - 1\sum
m=1
p - 1\sum
n=1
| K(m, p)| 2 | K(n, p)| 2 s4 (m\=n, p) =
=
\left\{
p3(p - 1), if p \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4),
p3(p - 1) - 36p2h2p, if p \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 8),
p3(p - 1) - 4p2h2p, if p \equiv 7 (\mathrm{m}\mathrm{o}\mathrm{d} 8),
where hp denotes the class number of the quadratic field \BbbQ
\bigl( \surd
- p
\bigr)
.
2. Preliminaries. In order to prove our theorems, we will need some lemmas. Hereinafter, we
shall use many properties of Gauss sums, all of which can be found in [1].
Lemma 2.1. Let p be an odd prime. Then, for any odd number h with (h, p) = 1, we have
s5(h, p) = 2s(h, p) - 4s (\=2h, p) ,
where \=2 satisfies the congruence 2\=2 \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} p).
Proof. See [8] (Lemma 2.3).
Lemma 2.2. Let p > 2 be an integer, then, for any integer h with (h, p) = 1, we obtain
s(h, p) =
1
\pi 2p
\sum
d| p
d2
\phi (d)
\sum
\chi mod d
\chi ( - 1)= - 1
\chi (h) | L(1, \chi )| 2 ,
where L(1, \chi ) is the Dirichlet L-function corresponding to the character \chi \mathrm{m}\mathrm{o}\mathrm{d} d and \phi (p) is the
Euler function.
Proof. See Lemma 2 of [14].
Lemma 2.3. Let p be an odd prime. Then, for any nonprincipal character \chi \mathrm{m}\mathrm{o}\mathrm{d} p, we have
p - 1\sum
n=1
\chi (n) | K(n, p)| 2 = \=\chi ( - 1)
\tau 3(\chi )\tau
\bigl(
\=\chi 2
\bigr)
\tau (\=\chi )
,
where \tau (\chi ) denotes the Gauss sum.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1498 M. C. DAĞLI
Proof. This is Lemma 1 of [13].
Lemma 2.4. Let p be an odd prime. Then we obtain
\sum
\chi mod p
\chi ( - 1)= - 1
| L(1, \chi )| 2 = \pi 2
12
(p - 1)2(p - 2)
p2
and \sum
\chi mod p
\chi ( - 1)= - 1
\chi (2) | L(1, \chi )| 2 = \pi 2
24
(p - 1)2(p - 5)
p2
.
Proof. See [13] (Lemma 5).
3. Proofs. This section is devoted to complete the proof of theorems.
3.1. Proof of Theorem 1.2. Employing Lemma 2.1 repeatedly, one can write
s5
\bigl(
2\=h, p
\bigr)
= 2s
\bigl(
2\=h, p
\bigr)
- 4s(h, p) (3.1)
and
s5 (2\=p, h) = 2s (2\=p, h) - 4s(p, h), (3.2)
where we have used the fact that if positive integers n and q satisfying (n, q) = 1. Then s(n, q) =
= s (\=n, q) , where \=n satisfies the congruence n\=n \equiv 1 \mathrm{m}\mathrm{o}\mathrm{d} q.
Adding (3.1) and (3.2), then applying reciprocity formulas (1.1) and (1.3) give the desired result.
3.2. Proof of Theorem 1.3. Before beginning the proof, we should prove the following relation.
Lemma 3.1. For odd prime p and any odd number h with (h, p) = 1, we have the identity
s4(h, p) = 20s(h, p) - 8s(2h, p) - 8s (\=2h, p) ,
where \=2 satisfies the congruence 2\=2 \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} p).
Proof. From (1.2) and Lemma 2.2, one has
s4(h, p) = - 4s(h, p) + 8s(h, 2p) =
= - 4s(h, p) +
4
\pi 2p
\sum
d| 2p
d2
\phi (d)
\sum
\chi mod d
\chi ( - 1)= - 1
\chi (h) | L(1, \chi )| 2 . (3.3)
Since the divisors of 2p are 1, 2, p, 2p and
s(h, p) =
1
\pi 2
p
p - 1
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (h) | L(1, \chi )| 2 ,
the right-hand side of (3.3) becomes
- 4s(h, p) +
4p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (h) | L(1, \chi )| 2+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON SOME IDENTITIES INVOLVING CERTAIN HARDY SUMS . . . 1499
+
16p
\pi 2(p - 1)
\sum
\chi mod 2p
\chi ( - 1)= - 1
\chi (h) | L(1, \chi )| 2 =
=
16p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (h)\lambda (h) | L (1, \chi \lambda )| 2 , (3.4)
where \lambda denotes the character \mathrm{m}\mathrm{o}\mathrm{d}2. Now, from the Euler product formula, we obtain
| L (1, \chi \lambda )| 2 =
\prod
p1
\bigm| \bigm| \bigm| \bigm| 1 - \chi (p1)\lambda (p1)
p1
\bigm| \bigm| \bigm| \bigm| - 2
=
=
\prod
p1>2
\bigm| \bigm| \bigm| \bigm| 1 - \chi (p1)
p1
\bigm| \bigm| \bigm| \bigm| - 2
=
\bigm| \bigm| \bigm| \bigm| 1 - \chi (2)
2
\bigm| \bigm| \bigm| \bigm| 2\prod
p1
\bigm| \bigm| \bigm| \bigm| 1 - \chi (p1)
p1
\bigm| \bigm| \bigm| \bigm| - 2
=
=
\biggl(
5
4
- \chi (2)
2
- \=\chi (2)
2
\biggr)
| L(1, \chi )| 2 . (3.5)
Thus, substituting (3.5) in (3.4) completes the proof.
We proceed to the proof of Theorem 1.3. Notice that if \chi is nonprincipal character \mathrm{m}\mathrm{o}\mathrm{d} p, then
| \tau (\chi )| = \surd
p and
p - 1\sum
m=1
\chi (m)K(m, p) =
p - 1\sum
a=1
p - 1\sum
m=1
\chi (m)e
\biggl(
ma+ \=a
p
\biggr)
=
\bigm| \bigm| \tau 2(\chi )\bigm| \bigm| = p.
So, it follows from Lemmas 2.4 and 3.1 that
p - 1\sum
m=1
p - 1\sum
n=1
K(m, p)K(n, p)s4 (m\=n, p) =
=
20p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n)K(n, p)
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 -
- 8p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (2)
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n)K(n, p)
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 -
- 8p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\=\chi (2)
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n)K(n, p)
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 =
=
20p3
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
| L(1, \chi )| 2 - 8p3
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (2) | L(1, \chi )| 2 -
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1500 M. C. DAĞLI
- 8p3
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\=\chi (2) | L(1, \chi )| 2 =
= p2(p - 1),
which completes the proof.
3.3. Proof of Theorem 1.4. If p \equiv 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4), in view of Lemmas 2.3, 2.4 and 3.1, we have
p - 1\sum
m=1
p - 1\sum
n=1
| K(m, p)| 2 | K(n, p)| 2 s4 (m\=n, p) =
=
20p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n) | K(n, p)| 2
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 -
- 8p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (2)
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n) | K(n, p)| 2
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 -
- 8p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\=\chi (2)
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n) | K(n, p)| 2
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 =
= p3(p - 1).
If p \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4) , then we have the Legendre symbol
\biggl(
- 1
p
\biggr)
= \chi 2( - 1) = - 1, L(1, \chi 2) = \pi hp/
\surd
p
and
\tau
\bigl(
\chi 2
2
\bigr)
=
p - 1\sum
a=1
\biggl(
a
p
\biggr) 2
e
\biggl(
a
p
\biggr)
= - 1.
Hence, using Lemma 2.3 and proceeding as in the proof of Theorem 1.3 yield that
p - 1\sum
m=1
p - 1\sum
n=1
| K(m, p)| 2 | K(n, p)| 2 s4 (m\=n, p) =
=
20p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n) | K(n, p)| 2
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 -
- 8p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (2)
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n) | K(n, p)| 2
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 -
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON SOME IDENTITIES INVOLVING CERTAIN HARDY SUMS . . . 1501
- 8p
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\=\chi (2)
\bigm| \bigm| \bigm| \bigm| \bigm|
p - 1\sum
n=1
\chi (n) | K(n, p)| 2
\bigm| \bigm| \bigm| \bigm| \bigm|
2
| L(1, \chi )| 2 =
=
20p4
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
| L(1, \chi )| 2 - 16p4
\pi 2(p - 1)
\sum
\chi mod p
\chi ( - 1)= - 1
\chi (2) | L(1, \chi )| 2 -
- 20p3
\pi 2
| L(1, \chi 2)| 2 +
16p3
\pi 2
\biggl(
2
p
\biggr)
| L(1, \chi )| 2 =
= p3(p - 1) - 20p2h2p + 16p2
\biggl(
2
p
\biggr)
h2p =
=
\left\{ p3(p - 1) - 36p2h2p, if p \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 8) ,
p3(p - 1) - 4p2h2p, if p \equiv 7 (\mathrm{m}\mathrm{o}\mathrm{d} 8) ,
where we have used that
\biggl(
2
p
\biggr)
= - 1 if p \equiv 3 (\mathrm{m}\mathrm{o}\mathrm{d} 8) and
\biggl(
2
p
\biggr)
= 1 if p \equiv 7(\mathrm{m}\mathrm{o}\mathrm{d} 8). So, the
proof is completed.
References
1. T. M. Apostol, Introduction to analytic number theory, Undergrad. Texts Math., Springer-Verlag, New York (1976).
2. B. C. Berndt, Analytic Eisenstein series, theta functions and series relations in the spirit of Ramanujan, J. reine und
angew. Math., 303/304, 332 – 365 (1978).
3. S. Chowla, On Kloosterman’s sum, Nor. Vidensk. Selsk. Fak. Frondheim, 40, 70 – 72 (1967).
4. J. B. Conrey, E. Fransen, R. Klein, C. Scott, Mean values of Dedekind sums, J. Number Theory, 56, № 2, 214 – 226
(1996).
5. X. Du, L. Zhang, On the Dedekind sums and its new reciprocity formula, Miskolc Math. Notes, 19, 235 – 239 (2018).
6. L. A. Goldberg, Transformations of theta-functions and analogues of Dedekind sums, Ph. D. Thesis, Univ. Illinois,
Urbana (1981).
7. A. V. Malyshev, A generalization of Kloosterman sums and their estimates, Vestn. Leningr. Univ., 15, 59 – 75 (1960)
(in Russian).
8. W. Peng, T. Zhang, Some identities involving certain Hardy sum and Kloosterman sum, J. Number Theory, 165,
355 – 362 (2016).
9. H. Rademacher, E. Grosswald, Dedekind sums, Math. Assoc. America, Washington, D.C. (1972).
10. R. Sitaramachandrarao, Dedekind and Hardy sums, Acta Arith., 48, № 4, 325 – 340 (1987).
11. H. Walum, An exact formula for an average of L-series, Ill. J. Math., 26, 1 – 3 (1982).
12. Z. Wenpeng, L. Yanni, A hybrid mean value related to the Dedekind sums and Kloosterman sums, Sci. China Math.,
53, 2543 – 2550 (2010).
13. H. Zhang, W. Zhang, On the identity involving certain Hardy sums and Kloosterman sums, J. Inequal. and Appl., 52
(2014), 9 p.
14. W. Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordeaux, 8, 429 – 442 (1996).
15. W. Zhang, A note on the mean square value of the Dedekind sums, Acta Math. Hung., 86, 119 – 135 (2000).
Received 18.04.19
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
|
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:03:50Z |
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| resource_txt_mv | umjimathkievua/eb/a3b99e943c2de2b3caadeffc7654d0eb.pdf |
| spelling | umjimathkievua-article-7312025-03-31T08:49:35Z On some identities involving certain Hardy sums and Kloosterman sum On some identities involving certain Hardy sums and Kloosterman sum Dağlı, M. C. Dağlı, M. C. Dedekind sum Hardy sum Computational problem Kloosterman sum Gauss sum Hybrid mean value Dedekind sum Hardy sum Computational problem Kloosterman sum Gauss sum Hybrid mean value UDC 511 We give a new reciprocity theorem for the Hardy sum $s_{5}(h,p).$ Also, a hybrid mean value problem involving the Hardy sum $s_{4}(h,p)$ and Kloosterman sum is studied and two exact computational formulae are obtained. UDC 511 Про деякі тотожності із певними сумами Гарді та сумою Клоостермана Запропоновано нову теорему взаємності для суми Гарді $s_{5}(h,p).$ Крім цього, вивчається гібридна задача про середні значення, яка мiстить суму Гарді $s_{4}(h,p)$ і суму Клоостермана, та отримано дві точні обчислювальні формули. Institute of Mathematics, NAS of Ukraine 2020-11-13 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/731 10.37863/umzh.v72i11.731 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 11 (2020); 1495-1501 Український математичний журнал; Том 72 № 11 (2020); 1495-1501 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/731/8778 Copyright (c) 2020 Muhammet Cihat Dağlı |
| spellingShingle | Dağlı, M. C. Dağlı, M. C. On some identities involving certain Hardy sums and Kloosterman sum |
| title | On some identities involving certain Hardy sums and Kloosterman sum |
| title_alt | On some identities involving certain Hardy sums and Kloosterman sum |
| title_full | On some identities involving certain Hardy sums and Kloosterman sum |
| title_fullStr | On some identities involving certain Hardy sums and Kloosterman sum |
| title_full_unstemmed | On some identities involving certain Hardy sums and Kloosterman sum |
| title_short | On some identities involving certain Hardy sums and Kloosterman sum |
| title_sort | on some identities involving certain hardy sums and kloosterman sum |
| topic_facet | Dedekind sum Hardy sum Computational problem Kloosterman sum Gauss sum Hybrid mean value Dedekind sum Hardy sum Computational problem Kloosterman sum Gauss sum Hybrid mean value |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/731 |
| work_keys_str_mv | AT daglımc onsomeidentitiesinvolvingcertainhardysumsandkloostermansum AT daglımc onsomeidentitiesinvolvingcertainhardysumsandkloostermansum |