Tang 2025 average first passage times character sums

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theorem_1_2: Tang and Zhang's 2025 theorem that for every epsilon > 0 the sum over odd primes p up to x of the first time the Legendre-symbol partial sum S_l(p) drops below epsilon l is asymptotic to c_epsilon x/log x; the first-passage variant of Erdős's eventual-time Problem 981, not the problem itself.

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Quanyu Tang, Hao Zhang, Average first-passage times for character sums. arXiv:2512.24631 (2025). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2512.24631), every other right reserved.

The copy read for this card is arXiv:2512.24631v2 (18 January 2026; 9 pages). The arXiv listing read shows v1 (31 December 2025) and v2 ("added a reference and corrected several typos") and no journal reference; locators here are the preprint's. Read status: claims checked for Conjecture 1.1 with footnote 1 (p. 1), Theorem 1.2 with the remark that f_eps(p) <= F_eps(p) does not transfer the asymptotic (p. 2) and the trivial case eps > 1 (p. 2), each read clause by clause on the page images; the proof (Sections 2-3) was read for its outline on p. 2 and not checked.

For an odd prime p let S_l(p) be the sum of the Legendre symbols (n/p) for n <= l, and for eps > 0 define the first-passage time f_eps(p) = min{l >= 1 : S_l(p) < eps l}. Theorem 1.2 proves that for every eps > 0 there is a constant c_eps in (0,infinity) with sum_{p <= x} f_eps(p) ~ c_eps x / log x as x tends to infinity. The method writes the average as a sum of tail densities a_m(x) = #{p <= x : f_eps(p) > m}/pi_odd(x); each event depends only on the finite vector of characters (chi_p(q)) for q <= m, so the prime number theorem for Dirichlet characters gives pointwise limits, and the interchange of limit and infinite sum is justified by a uniform tail bound obtained from high-moment estimates for S_m(p) using the quadratic large sieve for m <= x^{1/6-kappa} and quadratic reciprocity together with Heath-Brown's large sieve and Polya-Vinogradov for larger m. This settles the first-passage formulation of Erdos's eventual-time conjecture as it had been stated on the Erdos problems site as problem 981; the authors note that an earlier version of that page misstated the problem in the first-passage form rather than the eventual-time threshold F_eps(p), whose asymptotic sum_{p<=x} F_eps(p) ~ C_eps x/log x was proved by Elliott, and that the F asymptotic does not by itself imply the f asymptotic.

Source: https://arxiv.org/abs/2512.24631.

Bears on. #981 (Theorem 1.2, p. 2, the first-passage variant f_eps of the problem's eventual-time threshold F_eps; the introduction's attestation that Elliott proved Erdős's display (80), the problem's statement; footnote 1, the site's earlier misstatement; theorem_1_2)

Results to transcribe.

• Theorem 1.2: For every eps > 0 there is c_eps in (0,infinity) with sum_{p <= x} f_eps(p) ~ c_eps x/log x, where f_eps(p) = min{l : S_l(p) < eps l}.
• Conjecture 1.1 (Erdos): Erdos's eventual-time problem (problem 981): sum_{p<=x} F_eps(p) ~ C_eps x/log x for the least F with S_l(p) < eps l for all l >= F; the paper reports it proved by Elliott (p. 2).
• Comparison: f_eps(p) <= F_eps(p) for each p, but the asymptotic for the sum of F_eps does not by itself imply the asymptotic for the sum of f_eps.
• Method: Tail densities a_m(x) are evaluated by the prime number theorem for Dirichlet characters and dominated uniformly via sixth-moment bounds from the quadratic large sieve, plus Heath-Brown's large sieve and Polya-Vinogradov for large m.
• Trivial case: For eps > 1 one has f_eps(p) = 1 for every odd prime, since S_1(p) = 1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.