Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917160312373248 |
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| author | Gurram, Mahipal |
| author_facet | Gurram, Mahipal |
| contents | We investigate Dirichlet-type series generated by representation functions that count the number of ways an integer can be expressed as a sum of 'k' signed higher even powers. By combining generalized theta generating functions with a family of generalized cotangent series introduced in previous work, we derive two distinct explicit lower bounds for these series. The first estimate arises from a geometric restriction of the lattice to its diagonal, while the second utilizes Holder's inequality on the integral representation of the series. The methods presented here avoid modular techniques and offer a flexible analytic framework for higher-power representation problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions Gurram, Mahipal Number Theory Primary 11M41, Secondary 11P05, 33E20 We investigate Dirichlet-type series generated by representation functions that count the number of ways an integer can be expressed as a sum of 'k' signed higher even powers. By combining generalized theta generating functions with a family of generalized cotangent series introduced in previous work, we derive two distinct explicit lower bounds for these series. The first estimate arises from a geometric restriction of the lattice to its diagonal, while the second utilizes Holder's inequality on the integral representation of the series. The methods presented here avoid modular techniques and offer a flexible analytic framework for higher-power representation problems. |
| title | Explicit Lower Bounds for Dirichlet Series of Higher Power Representation Functions |
| topic | Number Theory Primary 11M41, Secondary 11P05, 33E20 |
| url | https://arxiv.org/abs/2512.18502 |