Thinned Wallis-type prime products in residue classes modulo $2^m$
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910206798069760 |
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| author | Winkler, Mike |
| author_facet | Winkler, Mike |
| contents | For odd primes $p$ we consider the factors \[ A(p)=\frac{p-χ_4(p)}{p+χ_4(p)}, \qquad χ_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue classes modulo $2^m$. We give a simple criterion for the existence of a finite nonzero limit, prove a logarithmic asymptotic in the general case, and express the limiting constant in terms of Mertens-type constants in arithmetic progressions (hence in terms of Dirichlet $L$-values). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13371 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Thinned Wallis-type prime products in residue classes modulo $2^m$ Winkler, Mike General Mathematics 11N37, 11N13 (Primary) 11M20, 11M06 (Secondary) For odd primes $p$ we consider the factors \[ A(p)=\frac{p-χ_4(p)}{p+χ_4(p)}, \qquad χ_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] and study products of $A(p)$ restricted to unions of residue classes modulo $2^m$. We give a simple criterion for the existence of a finite nonzero limit, prove a logarithmic asymptotic in the general case, and express the limiting constant in terms of Mertens-type constants in arithmetic progressions (hence in terms of Dirichlet $L$-values). |
| title | Thinned Wallis-type prime products in residue classes modulo $2^m$ |
| topic | General Mathematics 11N37, 11N13 (Primary) 11M20, 11M06 (Secondary) |
| url | https://arxiv.org/abs/2602.13371 |