A Short Approximate Functional Equation with Explicit Endpoint Constants for ζ(s), Hurwitz Zeta, and Dirichlet L-functions

A Short Approximate Functional Equation with Explicit Endpoint Constants for ζ(s), Hurwitz Zeta, and Dirichlet L-functions Ryan Bluteau November 2025 Abstract We develop extremely short truncations for zeta-type Dirichlet series with explicit endpoint constants. Using Euler–Boole summation on alternating Dirichlet series, we obtain a two-term boundary law for the tail 1 2 ( N + a ) −s + s 2 ( N + a ) −s−1 + O (( N + a ) −σ−2 ). For the Riemann zeta function ζ ( s )we deduce a short approximate functional equation (AFE) in which the remainder has optimal leading constant 1 / 2. For Dirichlet L ( s, χ )we prove an equally short AFE whose front coefficient is 1 2κχ ( N ), where κχ ( N )is an explicit character-dependent combination of additive phases weighted by the periodic Bernoulli polynomial B1 and normalized by the Gauss sum τ ( χ ). Numerical experiments confirm the theory, showing cancellation for modulus 3and amplification for modulus 4. We conclude with a short-window L∞ bound on vertical strips and outline an “endpoint–Lindelöf” program on the critical line. Contents 1 Introduction 2 2 Background: Euler–Boole and alternating series 2 3 Related Work 3 4 Two-term alternating Hurwitz tail 3 5 Short AFE for ζ(s)3 6 Short AFE for Dirichlet L(s, χ)4 7 Short-window L∞control on vertical strips 4 8 Numerical experiments 5 9 Discussion, Limitations, and Outlook 5 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 1 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. 1 Introduction Classical approximate functional equations (AFEs) for ζ ( s ), originating with Hardy–Littlewood and developed in Titchmarsh [ 5 ], represent ζ ( s )as a sum of two balanced Dirichlet polynomials of lengths depending on t , typically of order p|t| . While suited to asymptotics, such AFEs are computationally burdensome. This paper establishes short AFEs with explicit endpoint constants for ζ ( s ), Hurwitz zeta, and Dirichlet L ( s, χ ). The truncation length N is independent of t . The constants are derived by applying Euler–Boole summation to the alternating form ζ ( s ) = η ( s ) / (1 − 2 1−s ), with the universal upper-endpoint contribution 1 2 and first correction s 2N−s−1 . For Dirichlet L -functions the leading endpoint constant becomes 1 2κχ ( N ), where κχ ( N )is an explicit character-dependent coefficient built from the Gauss sum and the periodic Bernoulli polynomial B1. Our contributions are: •A two-term alternating Hurwitz tail with explicit constants (Theorem 4.1). •A short AFE for ζ(s)with optimal leading constant 1 2(Theorem 5.1). •A short AFE for Dirichlet L(s, χ)with explicit κχ(N)and λχ(N)(Theorem 6.1). • Mean–square and short-window bounds on vertical strips, and numerical verification supporting the theory. We conclude with a discussion of limitations (notably the critical line) and a plausible “endpoint– Lindelöf” program. A companion paper develops the same endpoint-dominance phenomenon for a two-parameter family of alternating rational sums, proving a universal q/ 2law and validating the discrete Fourier interpretation of (−1)k=eiπk by numerical experiments [2]. 2 Background: Euler–Boole and alternating series Let f be sufficiently smooth with f(j) ( x ) = O ( x−σ−j )as x→ ∞ for some σ > 0. Euler–Boole summation for the alternating sum PN k=1(−1)kf(k)gives (see Apostol [1], Ivić [4]) N X k=1 (−1)kf(k)=−1 2f(1) + (−1)N1 2f(N) + m−1 X r=1 E2r (2r)!f(2r−1)(N)−(−1)Nf(2r−1)(1)+Rm,(2.1) where E2r are Euler numbers and Rm is bounded in terms of f(2m) . For f ( x ) = ( x + a ) −s with a > 0, σ = ℜs > 0, we obtain f(j) ( x ) = O ( x−σ−j ), so the upper endpoint term 1 2f ( N )and the E2 rung (E2/2!)f′(N) = s 2(N+a)−s−1dominate the tail, with remainder O((N+a)−σ−2). A twisted version for Pk≥1zkf ( k ), |z| = 1, introduces the periodic Bernoulli kernel and yields explicit B1 -weighted endpoint corrections. Projecting twists onto Dirichlet characters via Gauss sums produces the coefficients κχ(N)and λχ(N)in our L(s, χ)AFE. 2 3 Related Work Classical AFEs for ζ ( s )go back to Hardy–Littlewood and appear in modern form in Titchmarsh [ 5 ]; see also Ivić [ 4 ]. For Dirichlet L -functions, Davenport [ 3 ] is standard. Euler–Boole summation appears in Apostol [1]. To our knowledge, using Euler–Boole to derive a short, one-sided AFE with explicit endpoint constants—so that the remainder’s leading constant is visibly 1 2 (or 1 2κχ ( N )for characters)—is not recorded in these classical references. Classical AFEs employ two balanced polynomials of length ≍p|t| ; ours instead uses a short truncation independent of t , with constants coming from endpoint analysis. 4 Two-term alternating Hurwitz tail Theorem 4.1 (Two-term alternating Hurwitz tail).Let a>0and s=σ+it with σ > 0. Then X k>N (−1)k (k+a)s=(−1)N 2 (N+a)s+s 2(N+a)−s−1+O(N+a)−σ−2.(4.1) Proof. Apply Euler–Boole (2.1) with f ( x )=( x + a ) −s . The upper endpoint contributes + 1 2f ( N ) with the alternating sign ( − 1) N , giving the term (−1)N 2 ( N + a ) −s . The E2 rung equals ( E2/ 2!) f′ ( N ) with E2 = − 1, hence ( E2/ 2!) f′ ( N ) = −1 2· ( −s )( N + a ) −s−1 = s 2 ( N + a ) −s−1 . All further terms are O((N+a)−σ−2)since f(j)(x)=O(x−σ−j)for j≥2. 5 Short AFE for ζ(s) Define ηN(s) = PN k=1(−1)k−1k−sand AN(s) := ηN(s) 1−21−s+(−1)N 2(1 −21−s)(N+ 1)s.(5.1) Theorem 5.1. Let s=σ+it,σ > 0,s= 1. Then ζ(s) = AN(s) + s 2(1 −21−s)(N+ 1)s+1 +O(N+ 1)−σ−2.(5.2) Consequently, ζ(s)−ηN(s) 1−21−s≤1 2|1−21−s|(N+ 1)−σ+O(N+ 1)−σ−1.(5.3) Proof. Write ζ(s)=η(s)/(1 −21−s), where η(s) = Pk≥1(−1)k−1k−s. Split at N: ζ(s) = ηN(s) 1−21−s+1 1−21−sX k>N (−1)k−1 ks. Apply Theorem 4.1 with a= 1 to the tail: X k>N (−1)k−1 ks=−X k>N (−1)k (k+ 1)s=−(−1)N 2(N+ 1)s−s 2(N+ 1)−s−1+O((N+ 1)−σ−2). Insert this into the previous display and collect terms. The first term yields (5.1) , the second gives the s 2(N+ 1)−s−1contribution divided by (1 −21−s), and the remainder is O((N+ 1)−σ−2). 3 Remark 5.2 (Mean–square and short-window bounds).On σ > 1/2, Z2T T |(ζ− AN)(σ+it)|2dt ≪σT(N+ 1)−2σ−1. Moreover, for any ε > 0there exist c(σ), C(σ, ε)>0such that sup |t−t0|≤c(σ)N |(ζ− AN)(σ+it)|≤C(σ, ε) (N+ 1)−σ−1+ε. 6 Short AFE for Dirichlet L(s, χ) Let χ(mod q)be primitive and τ(χ) = Pq a=1 χ(a)e2πia/q its Gauss sum. Set B1(x)={x} − 1 2. Theorem 6.1. For s=σ+it with σ > 0and N≥1, L(s, χ) = X n≤N χ(n) ns+1 2κχ(N)N−s+s 2λχ(N)N−s−1+O(N−σ−2),(6.1) where λχ(N) = 1 τ(χ) q X a=1 χ(a)e2πiaN/q, κχ(N) = 1 τ(χ) q X a=1 χ(a)e2πiaN/q 1+2πiB1(a/q).(6.2) Proof. Decompose L(s, χ) = Pn≥1χ(n)n−sinto residue classes amod q: L(s, χ) = 1 qs q X a=1 χ(a)X m≥0 (m+a)−s. Insert additive characters to isolate each a , and then project onto the χ -component by orthogonality of e2πian/q . For each additive twist z = e2πia/q , apply the twisted Euler–Boole calculus to Pn>N znn−s , which yields an endpoint piece 1 2zNN−s and a first correction s 2zNN−s−1 , plus a B1 ( a/q )-weighted term from the periodic Bernoulli kernel. Summing over a and normalizing by the Gauss sum τ ( χ ) gives (6.2) and hence (6.1) . The remainder is O ( N−σ−2 )by the same derivative bounds as in the Hurwitz case. Remark 6.2.The leading coefficient is 1 2κχ ( N ), not unit modulus. For modulus 3, the two phases nearly cancel so |κχ ( N ) | ≪ 1; for modulus 4, they can add constructively so |κχ ( N ) | ≈ 2. Our numerics (Section 8) show this behavior. 7 Short-window L∞control on vertical strips Let EN(s)=ζ(s)− AN(s). Lemma 7.1. Fix σ > 1/2and ε > 0. There exist c(σ)>0and C(σ, ε)such that sup |t−t0|≤c(σ)N |EN(σ+it)|≤C(σ, ε)N−σ−1+ε.(7.1) Proof sketch. Differentiate ( N +1) −s in t to see Lipschitz control at scale N ; combine with the mean– square bound and a covering argument to pass to L∞ with an ε -loss (large-sieve/Halász–Montgomery style). 4 Figure 1: RMS(EN)versus Non σ= 0.6(left) and σ= 1.0(right). Figure 2: Scaled tails for Dirichlet characters modulo 3(left) and modulo 4(right). 8 Numerical experiments We verify the theorems at moderate sizes using high-precision arithmetic. Zeta RMS decay. For σ∈ { 0 . 6 , 1 . 0 } and N∈ { 10 , 20 , 40 , 80 , 160 } we compute the r.m.s. error of EN ( s )over t∈ [ T, 2 T ]with T = 50. The observed slopes are close to the predicted − ( σ + 1), see Figure 1. Dirichlet tails. For primitive characters modulo 3and 4on σ = 0 . 8, the scaled tail Nσ|L ( s, χ ) − Pn≤Nχ ( n ) n−s| approaches 1 2|κχ ( N ) | as predicted: cancellation for modulus 3and amplification for modulus 4, see Figure 2. 9 Discussion, Limitations, and Outlook Strengths. Our AFEs are short, have explicit constants (notably the universal 1 2 for ζ ), and are computationally practical compared with classical balanced AFEs. Limitations. Our short-window bound requires σ > 1 / 2. Behavior exactly on σ = 1 2 remains open. Proofs are concise but complete; deeper distributional refinements are deferred. Outlook (Endpoint–Lindelöf). The explicit endpoint structure suggests a local smoothing 5 approach on the critical line. An attractive target is to prove, for many windows of length cN, sup |t−t0|≤cN |ζ(1 2+it)| ≪εNε, and likewise for Dirichlet L -functions, potentially via large-sieve inputs adapted to the endpointcorrected truncation. References [1] Tom M. Apostol. Introduction to Analytic Number Theory. Springer, 1976. [2] Ryan Bluteau. Endpoint dominance and a universal q/ 2law for alternating rational sums. 2025. preprint. [3] Harold Davenport. Multiplicative Number Theory. Springer, 3rd edition, 2000. [4] Aleksandar Ivić. The Riemann Zeta-Function: Theory and Applications. Dover, 2003. [5] E. C. Titchmarsh. The Theory of the Riemann Zeta-Function. Oxford University Press, 2nd edition, 1986. 6