Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909991010566144 |
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| author | Shparlinski, Igor E. Xiao, Yixiu |
| author_facet | Shparlinski, Igor E. Xiao, Yixiu |
| contents | We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Salié sums modulo a large prime $q$. We use these bounds to study, for fixed integers $a,b\not \equiv 0 \bmod q$, the distribution ofsolutions to the congruence $x^2 \equiv ap+b \bmod q$, over primes $p\le P$. This is similar to the recently studied case of $b = 0$, however the case $b\not \equiv 0 \bmod q$ exhibits some new difficulties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10113 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes Shparlinski, Igor E. Xiao, Yixiu Number Theory We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Salié sums modulo a large prime $q$. We use these bounds to study, for fixed integers $a,b\not \equiv 0 \bmod q$, the distribution ofsolutions to the congruence $x^2 \equiv ap+b \bmod q$, over primes $p\le P$. This is similar to the recently studied case of $b = 0$, however the case $b\not \equiv 0 \bmod q$ exhibits some new difficulties. |
| title | Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.10113 |