• Title/Summary/Keyword: Weierstrass ${\wp}(x)$ functions

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CONGRUENCES OF THE WEIERSTRASS ${\wp}(x)$ AND ${\wp}^{{\prime}{\prime}}(x)$($x=\frac{1}{2}$, $\frac{\tau}{2}$, $\frac{\tau+1}{2}$)-FUNCTIONS ON DIVISORS

  • Kim, Daeyeoul;Kim, Aeran;Park, Hwasin
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.241-261
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    • 2013
  • In this paper, we find the coefficients for the Weierstrass ${\wp}(x)$ and ${\wp}^{{\prime}{\prime}}(x)$($x=\frac{1}{2}$, $\frac{\tau}{2}$, $\frac{\tau+1}{2}$)-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ([23]). Finally, we reprove congruences for the functions ${\mu}(n)$ and ${\nu}(n)$ in Hahn's article [11, Theorems 6.1 and 6.2].

CONVOLUTION SUMS ARISING FROM DIVISOR FUNCTIONS

  • Kim, Aeran;Kim, Daeyeoul;Yan, Li
    • Journal of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.331-360
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    • 2013
  • Let ${\sigma}_s(N)$ denote the sum of the sth powers of the positive divisors of a positive integer N and let $\tilde{\sigma}_s(N)={\sum}_{d|N}(-1)^{d-1}d^s$ with $d$, N, and s positive integers. Hahn [12] proved that $$16\sum_{k. In this paper, we give a generalization of Hahn's result. Furthermore, we find the formula ${\sum}_{k=1}^{N-1}\tilde{\sigma}_1(2^{n-m}k)\tilde{\sigma}_3(2^nN-2^nk)$ for $m(0{\leq}m{\leq}n)$.