Stringent bounds for the non-zero Bernoulli numbers
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917895308574720 |
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| author | Bagul, Yogesh J. |
| author_facet | Bagul, Yogesh J. |
| contents | We present new sharper lower and upper bounds for the non-zero Bernoulli numbers using Euler's formula for the Riemann zeta function. In particular, we determine the best possible constants $ α$ and $ β$ such that the double inequality $$ \frac{2\cdot (2k)!}{π^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-α)} < \vert B_{2k} \vert < \frac{2\cdot (2k)!}{π^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-β)}, $$ holds for $ k = 1, 2, 3, \cdots.$ Our main results refine the existing bounds of $ \vert B_{2k} \vert $ in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14532 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stringent bounds for the non-zero Bernoulli numbers Bagul, Yogesh J. General Mathematics We present new sharper lower and upper bounds for the non-zero Bernoulli numbers using Euler's formula for the Riemann zeta function. In particular, we determine the best possible constants $ α$ and $ β$ such that the double inequality $$ \frac{2\cdot (2k)!}{π^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-α)} < \vert B_{2k} \vert < \frac{2\cdot (2k)!}{π^{2k} (2^{2k}-1)}\frac{3^{2k}}{(3^{2k}-β)}, $$ holds for $ k = 1, 2, 3, \cdots.$ Our main results refine the existing bounds of $ \vert B_{2k} \vert $ in the literature. |
| title | Stringent bounds for the non-zero Bernoulli numbers |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2303.14532 |