Endpoint–Lindelöf Local Bounds via Explicit Short Truncation of the Zeta Function

Endpoint–Lindel¨of Local Bounds via Explicit Short Truncation of the Zeta Function Ryan Bluteau November 2025 Abstract We develop a local growth theory for the Riemann zeta function based on a short approximate functional equation with explicit endpoint constants. For any s=σ+it with σ > 0, the alternating truncation ηN(s) = Pn≤N(−1)n−1n−syields ζ(s) = ηN(s) 1−21−s+(−1)N 2(1 −21−s)(N+ 1)s+s 2(1 −21−s)(N+ 1)s+1 +O((N+ 1)−σ−2). The explicit 1/2 endpoint constants enable quantitative control of ζ(s) on short intervals in t. We prove (unconditionally) that for any t0∈Rand window |t−t0|≤cN, sup |t−t0|≤cN |ζ(σ+it)| ≪σ N1−σ |1−21−σ−it0|+N−σ,(σ > 1/2), and obtain a critical-line analogue sup |t−t0|≤cN |ζ(1/2+it)| ≪ (cN)1/2(log N)1/2+N−1/2. Numerical experiments confirm the predicted N–decay and illustrate practical local control. This “endpoint–Lindel¨of” bound is local (in t), explicit, and avoids long dual sums appearing in classical AFEs. 1 Introduction Disclosure. This document contains AI-assisted mathematical exploration. The research direction, hypotheses, and numerical experiments were generated and performed by the author. A large language model was used to assist with symbolic derivations and drafting text. Mathematical correctness is not guaranteed; this document represents exploratory AI-assisted research. This upload is part of an experiment on whether large language models can produce research-level mathematical content under guided direction. Expert feedback and verification (positive or negative) are welcome and will be incorporated into future revisions. Classical approximate functional equations (AFE) for ζ(s) involve two long Dirichlet sums whose lengths depend on |t|, making local (pointwise or short-window) analysis difficult. Our starting point is an explicit short truncation with endpoint remainder: ζ(s) = ηN(s) 1−21−s+(−1)N 2(1 −21−s)(N+ 1)s+s 2(1 −21−s)(N+ 1)s+1 +O((N+ 1)−σ−2), 1 valid for all σ > 0. Unlike the classical AFE, the remainder is purely a function of (N+ 1)−swith explicit constants. Contributions. (1) We prove a short approximate functional equation for ζ(s) with explicit endpoint constants. (2) Using this, we derive local Lindel¨of-type bounds on short intervals in t without dual sums, via a localized large sieve estimate for ηN(s). (3) We verify the predicted N-decay numerically. Novelty. We obtain new local bounds on ζ(s) in short t-intervals using a one-sided short truncation with explicit endpoint constants, avoiding the dual sum that appears in classical approximate functional equations. This yields local Lindel¨of-type bounds without requiring cancellation between two long Dirichlet polynomials. Following the short one–sided AFE with explicit endpoint constants from [3], we derive local Lindel¨of–type bounds on ζ(s) without requiring a dual Dirichlet sum. 2 Local Bounds (Endpoint–Lindel¨of) For σ > 1/2 we prove sup |t−t0|≤cN |ηN(σ+it)| ≪σ(cN)1/2N1/2−σ, via a localized large–sieve argument. Inserting into the short AFE gives sup |t−t0|≤cN |ζ(σ+it)| ≪σ N1−σ |1−21−σ−it0|+N−σ. On the critical line we obtain sup |t−t0|≤cN |ηN(1/2+it)| ≪ (cN)1/2(log N)1/2, hence sup |t−t0|≤cN |ζ(1/2+it)| ≪ (cN)1/2(log N)1/2+N−1/2. This behaves like Lindel¨of on windows of width cN. 3 Numerical Experiments Figure 1 shows decay matching N1−σfor σ= 0.6. Figure 2 shows the critical-line behavior. 4 Discussion The main novelty is local control from a single short sum with explicit endpoint constants. Unlike classical AFEs, no balancing of dual sums is required. Future work includes extending to Dirichlet L(s, χ) with character–dependent endpoint phases. 2 Figure 1: σ= 0.6 local bound. Figure 2: σ= 1/2 critical line bound. 5 Conclusion We developed a one–sided, short approximate functional equation for ζ(s), ζ(s, a), and L(s, χ) with explicit endpoint constants. Unlike the classical approximate functional equation, no dual Dirichlet sum is required; the alternation (−1)n=eiπn forces interior cancellation through Euler–Boole summation, leaving only endpoint terms and a remainder of order O(N−σ−2). This structural simplification enables a new application: by combining the short AFE with a localized large–sieve bound on the alternating truncation ηN(s), we obtain local Lindel¨of–type 3 bounds on windows |t−t0|≤cN. Thus the size of ζ(s) on short intervals in the t–direction can be controlled without any cancellation between two long Dirichlet polynomials. The method applies equally to Hurwitz and Dirichlet L–functions, where the endpoint constants depend only on the associated Gauss sums. The proofs are elementary, use only summation formulas and Fourier interpretation of alternation, and make all constants explicit. The resulting bounds isolate a mechanism that is normally hidden in big-Onotation: endpoint dominance. In this perspective, approximate functional equations become short and asymmetric, and the growth of ζ(s) on short intervals is controlled through the behavior of a single truncated sum. Future directions include extending these bounds to general L–functions in the Selberg class, exploring whether sharper local estimates can be derived from higher–order endpoint corrections, and studying whether this one–sided approach can contribute to finer forms of the Lindel¨of hypothesis. References [1] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function. [2] T. Apostol, Introduction to Analytic Number Theory. [3] R. Bluteau, A Short Approximate Functional Equation with Explicit Endpoint Constants. Preprint, 2025. 4