Real zeros of $L'(s, χ_d)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914283905875968 |
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| author | Lamzouri, Youness Nath, Kunjakanan |
| author_facet | Lamzouri, Youness Nath, Kunjakanan |
| contents | In 1990, Baker and Montgomery conjectured that $L'(s,χ_d)$ has $\asymp \log\log |d|$ real zeros in the interval $[1/2,1]$ for almost all fundamental discriminants $d$. The study of these zeros was motivated by their connection to real zeros of Fekete polynomials and to sign changes of the character sums $\sum_{n\leq x}χ_d(n)$. Recent work of Klurman, Lamzouri, and Munsch shows that the number of such zeros is $\gg (\log\log |d|)/(\log\log\log\log |d|)$ for almost all $d$, thereby establishing the conjectured lower bound up to the factor $\log\log\log\log |d|$. In this paper, we prove that for almost all fundamental discriminants $d$, $L'(s,χ_d)$ has at most $(\log\log |d|)(\log\log\log |d|)$ real zeros in $[1/2,1]$, thus resolving the Baker-Montgomery conjecture up to a factor of $\log\log\log |d|$. We also give a quantitative upper bound on the exceptional set of discriminants. Furthermore, we show, conditionally on certain natural assumptions, that $100\%$ of these zeros lie away from $1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real zeros of $L'(s, χ_d)$ Lamzouri, Youness Nath, Kunjakanan Number Theory In 1990, Baker and Montgomery conjectured that $L'(s,χ_d)$ has $\asymp \log\log |d|$ real zeros in the interval $[1/2,1]$ for almost all fundamental discriminants $d$. The study of these zeros was motivated by their connection to real zeros of Fekete polynomials and to sign changes of the character sums $\sum_{n\leq x}χ_d(n)$. Recent work of Klurman, Lamzouri, and Munsch shows that the number of such zeros is $\gg (\log\log |d|)/(\log\log\log\log |d|)$ for almost all $d$, thereby establishing the conjectured lower bound up to the factor $\log\log\log\log |d|$. In this paper, we prove that for almost all fundamental discriminants $d$, $L'(s,χ_d)$ has at most $(\log\log |d|)(\log\log\log |d|)$ real zeros in $[1/2,1]$, thus resolving the Baker-Montgomery conjecture up to a factor of $\log\log\log |d|$. We also give a quantitative upper bound on the exceptional set of discriminants. Furthermore, we show, conditionally on certain natural assumptions, that $100\%$ of these zeros lie away from $1/2$. |
| title | Real zeros of $L'(s, χ_d)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.02774 |