On a rigidity property for quadratic Gauss sums
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929726308745216 |
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| author | Mangerel, Alexander P. |
| author_facet | Mangerel, Alexander P. |
| contents | Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$ S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N} $$ satisfy the ``Gauss sum-like'' approximate dilation symmetry property $$ \frac{1}{N}\sum_{a \pmod{N}} |S_f(ap) - f(p)S_f(a)|^2 = o(N), $$ uniformly over all primes $p \leq N^c$ then $f$ coincides with a real character modulo $N$ at all but $o(N)$ integers $1 \leq n < N$. As a consequence, taking $f$ to be the Liouville function we connect this exponential sums property to the location of real zeros of $L(s,χ)$ close to $s = 1$, for $χ$ the Legendre symbol modulo $N$.
Assuming the $L$-functions of primitive Dirichlet characters modulo $N$ have a sufficiently wide zero-free region (of Littlewood type), we also show a more general result in which any $c > 0$ may be taken. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16014 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a rigidity property for quadratic Gauss sums Mangerel, Alexander P. Number Theory Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$ S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N} $$ satisfy the ``Gauss sum-like'' approximate dilation symmetry property $$ \frac{1}{N}\sum_{a \pmod{N}} |S_f(ap) - f(p)S_f(a)|^2 = o(N), $$ uniformly over all primes $p \leq N^c$ then $f$ coincides with a real character modulo $N$ at all but $o(N)$ integers $1 \leq n < N$. As a consequence, taking $f$ to be the Liouville function we connect this exponential sums property to the location of real zeros of $L(s,χ)$ close to $s = 1$, for $χ$ the Legendre symbol modulo $N$. Assuming the $L$-functions of primitive Dirichlet characters modulo $N$ have a sufficiently wide zero-free region (of Littlewood type), we also show a more general result in which any $c > 0$ may be taken. |
| title | On a rigidity property for quadratic Gauss sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.16014 |