• Title/Summary/Keyword: divisor function

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THE p-PART OF DIVISOR CLASS NUMBERS FOR CYCLOTOMIC FUNCTION FIELDS

  • Daisuke Shiomi
    • Communications of the Korean Mathematical Society
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    • v.38 no.3
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    • pp.715-723
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    • 2023
  • In this paper, we construct explicitly an infinite family of primes P with h±P ≡ 0 (mod qdeg P), where h±P are the plus and minus parts of the divisor class number of the P-th cyclotomic function field over 𝔽q(T). By using this result and Dirichlet's theorem, we give a condition of A, M ∈ 𝔽q[T] such that there are infinitely many primes P satisfying with h±P ≡ 0 (mod pe) and P ≡ A (mod M).

A STUDY OF SUM OF DIVISOR FUNCTIONS AND STIRLING NUMBER OF THE FIRST KIND DERIVED FROM LIOUVILLE FUNCTIONS

  • KIM, DAEYEOUL;KIM, SO EUN;SO, JI SUK
    • Journal of applied mathematics & informatics
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    • v.36 no.5_6
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    • pp.435-446
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    • 2018
  • Using the theory of combinatoric convolution sums, we establish some arithmetic identities involving Liouville functions and restricted divisor functions. We also prove some relations involving restricted divisor functions and Stirling numbers of the first kind for divisor functions.

UNITARY ANALOGUES OF A GENERALIZED NUMBER-THEORETIC SUM

  • Traiwat Intarawong;Boonrod Yuttanan
    • Communications of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.355-364
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    • 2023
  • In this paper, we investigate the sums of the elements in the finite set $\{x^k:1{\leq}x{\leq}{\frac{n}{m}},\;gcd_u(x,n)=1\}$, where k, m and n are positive integers and gcdu(x, n) is the unitary greatest common divisor of x and n. Moreover, for some cases of k and m, we can give the explicit formulae for the sums involving some well-known arithmetic functions.

THE RELATION PROPERTY BETWEEN THE DIVISOR FUNCTION AND INFINITE PRODUCT SUMS

  • Kim, Aeran
    • Honam Mathematical Journal
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    • v.38 no.3
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    • pp.507-552
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    • 2016
  • For a complex number q and a divisor function ${\sigma}_1(n)$ we define $$C(q):=q{\prod_{n=1}^{\infty}}(1-q^n)^{16}(1-q^{2n})^4,\\D(q):=q^2{\prod_{n=1}^{\infty}}(1-q^n)^8(1-q^{2n})^4(1-q^{4n})^8,\\L(q):=1-24{\sum_{n=1}^{\infty}}{\sigma}_1(n)q^n$$ moreover we obtain the number of representations of $n{\in}{\mathbb{N}}$ as sum of 24 squares, which are possible for us to deduce $L(q^4)C(q)$ and $L(q^4)D(q)$.

CHANGING RELATIONSHIP BETWEEN SETS USING CONVOLUTION SUMS OF RESTRICTED DIVISOR FUNCTIONS

  • ISMAIL NACI CANGUL;DAEYEOUL KIM
    • Journal of applied mathematics & informatics
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    • v.41 no.3
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    • pp.553-567
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    • 2023
  • There are real life situations in our lives where the things are changing continuously or from time to time. It is a very important problem for one whether to continue the existing relationship or to form a new one after some occasions. That is, people, companies, cities, countries, etc. may change their opinion or position rapidly. In this work, we think of the problem of changing relationships from a mathematical point of view and think of an answer. In some sense, we comment these changes as power changes. Our number theoretical model will be based on this idea. Using the convolution sum of the restricted divisor function E, we obtain the answer to this problem.

DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES

  • Kim, Dae-Yeoul;Kim, Min-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.4
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    • pp.693-704
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    • 2012
  • We consider Weierstrass functions and divisor functions arising from $q$-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. In this note, however, we obtain a similar result by differentiating further. For any $n{\geq}1$, we have $$k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1;k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1;k,l}(n-j)$$. Finally, we shall give a table for $E_1(N;3)$, ${\sigma}(N)$, ${\tau}_{1;3,1}(N)$ and ${\tau}_{2;3,1}(N)$ ($1{\leq}N{\leq}50$) and state simulation results for them.

COUNTING SUBRINGS OF THE RING ℤm × ℤn

  • Toth, Laszlo
    • Journal of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1599-1611
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    • 2019
  • Let $m,n{\in}{\mathbb{N}}$. We represent the additive subgroups of the ring ${\mathbb{Z}}_m{\times}{\mathbb{Z}}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring ${\mathbb{Z}}_m{\times}{\mathbb{Z}}_n$ and its unital subrings, respectively. We show that the functions $(m,n){\mapsto}N^{u,s}(m,n)$ and $(m,n){\mapsto}N^{(us)}(m,n)$ are multiplicative, viewed as functions of two variables, and their Dirichlet series can be expressed in terms of the Riemann zeta function. We also establish an asymptotic formula for the sum $\sum_{m,n{\leq}x}N^{(s)}(m,n)$, the error term of which is closely related to the Dirichlet divisor problem.

EVALUATION OF THE CONVOLUTION SUMS Σak+bl+cm=n σ(k)σ(l)σ(m), Σal+bm=n lσ(l)σ(m) AND Σal+bm=n σ3(l)σ(m) FOR DIVISORS a, b, c OF 10

  • PARK, YOON KYUNG
    • Journal of applied mathematics & informatics
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    • v.40 no.5_6
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    • pp.813-830
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    • 2022
  • The generating functions of the divisor function σs(n) = Σ0<d|n ds are quasimodular forms. In this paper, we find the basis of the space of quasimodular forms of weight 6 on Γ0(10) consisting of Eisenstein series and η-quotients. Then we evaluate the convolution sum Σak+bl+cm=n σ(k)σ(l)σ(m) with lcm(a, b, c) = 10 and Σal+bm=n lσ(l)σ(m) and Σal+bm=n σ3(l)σ(m) with lcm(a, b) = 10.