An approach to the Lindelöf Hypothesis for Dirichlet $L$-functions

The suggested approach is based on a known representation of Dirichlet $L$-functions via the incomplete gamma functions. Some properties of the Taylor coefficients of the lower incomplete gamma function at infinity seem to be new. Specifically, these coefficients can be expressed in terms of Touchard polynomials. Furthermore, these same coefficients can be used to reformulate the functional equation for Dirichlet $L$-functions. This relationship “explains”’ why $\vert L_χ(1/2+i t)\vert $ should be small. To present the new ideas in a nutshell, we start by giving (in Section 1) a “formula proof” of the Lindelöf hypothesis. This is not a genuine proof, as we are not concerned with the convergence of our series nor do we justify changing the order of summation. In Section 2, we suggest some hypothetical ways of transforming the “proof” from Section 1 into a rigorous mathematical proof. Sections 3-5 contain some technical details and bibliographical references.

💡 Research Summary

The manuscript proposes a novel‑looking “formula proof” of the Lindelöf hypothesis for Dirichlet L‑functions by exploiting a representation of these functions through lower incomplete gamma functions and by expressing the Taylor coefficients of the incomplete gamma at infinity in terms of Touchard polynomials. The author focuses on the non‑principal character modulo 3, denoted χ₃, and defines the associated L‑function L₃(s)=∑_{n≥1}χ₃(n)n^{-s}. After completing the function with the factor g₃(s)=π^{-s+½}Γ(s+½) to obtain ξ₃(s)=g₃(s)L₃(s), the standard functional equation ξ₃(s)=ξ₃(1−s) is recalled.

Using Kummer’s series for the lower incomplete gamma, γ(w,m)=m^{w}∑_{k≥0}(-1)^{k}k!^{-1}(w+k)m^{k}, the author substitutes w=a+ib and expands 1/(a+ib)^{l} into a binomial‑type series. This manipulation yields the compact identity

γ(a+ib,m)=m^{a+ib}e^{-m}∑{l≥1}(-1)^{l-1}T{l-1}(-m)(a+ib)^{l},

where T_n(x)=∑_{j=0}^{n}S(n,j)x^{j} are Touchard (or Bell) polynomials. The author claims this representation is new, although similar connections between incomplete gamma expansions and Bell numbers are known in the literature.

Inserting this expansion into the incomplete‑gamma representation of ξ₃(½+it) leads to a double series of the form

ξ₃(½+it)=∑{k≥1}2^{-2k-1}t^{-2k}∑{n≥1}χ₃(n)n U_{2k}(¾,πn²/3) e^{-πn²/3},

where the coefficients U_k(a,m) are explicit linear combinations of Touchard polynomials. The crucial “Key Discovery” is the claim that for every k

∑{n≥1}χ₃(n)n U{2k}(¾,πn²/3) e^{-πn²/3}=0,

which the author shows is essentially equivalent to the functional equation. From this, the author formally deduces the absurd identity ξ₃(½+it)=0, calling it a “paradoxical identity” that nevertheless “explains” why |L₃(½+it)| should be small. The paper openly acknowledges that the step is incorrect, attributing the failure to an unjustified interchange of infinite sums and to the misuse of the expansion (13), which is valid only when k<|w|.

Section 2 attempts to turn the informal argument into a rigorous one. Two strategies are outlined: (i) truncate the outer and inner sums at finite N and K, estimate the truncation errors using the known asymptotics of the incomplete gamma (equations (33)–(34)), and then let N,K→∞; (ii) apply linear series‑acceleration by introducing weights ν_n(N) and κ_k(K) to approximate the infinite sums with finite weighted sums. The author suggests that suitable choices of these weights could keep N and K modest while still delivering the desired O(t^{ε}) bound, but provides no concrete construction, nor any quantitative error analysis.

Sections 3–5 place the work in context. Section 3 recalls earlier representations of Dirichlet L‑functions via incomplete gamma functions due to Riemann, La­vrik, and others, and shows how the present formula (11) is a special case of a more general expression (61). Section 4 derives a family of polynomial identities (G_m(n), E_m(l), F_{d,m}(l)) from the functional equation for the theta‑type series θ_χ(τ) and demonstrates that the vanishing sums (31) are equivalent to these identities. Section 5 generalises the whole framework to an arbitrary primitive character χ modulo q, introducing the parameters δ (parity) and ω (root number), and obtains analogous “paradoxical” equalities (86) for any χ.

Overall, the paper contributes two observations of interest: (1) an explicit link between the lower incomplete gamma’s Taylor coefficients at infinity and Touchard polynomials, and (2) a reformulation of the functional equation for Dirichlet L‑functions as an infinite family of linear relations among weighted exponential sums. However, the central claim – that these observations yield a proof of the Lindelöf hypothesis – remains unsubstantiated. The main mathematical gaps are: lack of convergence proofs for the double series, unjustified rearrangement of infinite sums, misuse of the expansion valid only in a restricted region, and absence of any quantitative bound on the truncated or weighted sums. Consequently, the manuscript should be regarded as a speculative exploration of a potentially fruitful representation rather than a rigorous proof. Future work would need to (i) establish precise domains of validity for the incomplete‑gamma expansions, (ii) develop explicit weight constructions that guarantee error terms smaller than t^{ε}, and (iii) connect the Touchard‑polynomial coefficients to known subconvexity estimates, possibly opening a new pathway toward the Lindelöf hypothesis.