A new expansion of the Riemann zeta function

Abstract: After a brief introduction to Ramanujan's method of summation, we give an expansion of the Riemann Zeta function in the critical strip as a convergent series $\sum_{m\geq 0}x_m P_m(s) $ where the functions $P_m$ are polynomials with their roots on the line ${\Re(s)=1/2}$, the coefficients $x_m$ being finite linear combinations of the Euler constant $γ$ and the values $ζ(2),ζ(3),\dots,ζ(m+1).$

• The paper introduces a convergent expansion for ζ(s) in the critical strip using Laguerre polynomial structures and Ramanujan summation.
• It derives explicit coefficients as finite combinations of the Euler-Mascheroni constant and ζ-values, ensuring all polynomial roots lie on the critical line.
• The expansion enables precise numerical evaluation and provides a spectral framework that may advance studies on the Riemann Hypothesis.

Analytic Expansion of the Riemann Zeta Function in the Critical Strip via Laguerre Polynomial Structures

Introduction

The paper "A new expansion of the Riemann zeta function" (2512.11405) introduces a convergent expansion for $\zeta(s)$ in the critical strip $0 < \Re(s) < 1$, using a novel polynomial basis constructed from the Mellin transforms of Laguerre polynomials. The expansion coefficients are explicit finite linear combinations of the Euler-Mascheroni constant $\gamma$ and zeta values $\zeta(k)$ at integer arguments, augmented by rational terms. The work thoroughly integrates Ramanujan's summation methods for divergent series and provides structural connections to hypergeometric and orthogonal polynomials, with precise location of the polynomial roots on the line $\Re(s) = 1/2$.

Background and Preliminary Expansions

The paper first surveys classical expansions of $\zeta(s)$, particularly those using Newton interpolation, Stieltjes constants, and expansions linked to orthogonal polynomials such as the Hermite and Meixner-Pollaczek families. Previous expansions, e.g., $\zeta(s) - 1/(s-1)$ via the Stieltjes constants, are connected to integer-shifted binomial or hypergeometric polynomial bases, but the present approach leverages analytic continuation in the critical strip with a direct polynomial structure.

Central to the construction is the use of functions $(f_n)$ related to Laguerre polynomials $L_m(x)$ and their generating functions, which, under Mellin transform, provide natural polynomial systems for expansion in $s$. This enables a systematic association: for $\Re(z) > 0$,

$$e^{-xz} = \sum_{m= 0}^{\infty} 2e^{-x} L_m(2x) \psi_m(z)$$

with $\psi_m(z)$ explicit and the Mellin transforms yielding polynomials $P_m(s)$.

Ramanujan Summation and Series Regularization

A detailed exposition is provided of Ramanujan's summation ($\mathcal{R}$-summation) for divergent series, grounded in extensions of the Euler-Maclaurin and Abel-Plana summation formulas. The construction is fully analytic on the half-plane and ensures uniqueness via Carlson's theorem under mild exponential-type growth conditions.

Key identities include the analytic continuation:

$$\zeta(s) = \sum_{n \geq 1}^{\mathcal{R}} \frac{1}{n^s} + \frac{1}{s-1},$$

valid for $s \in \mathbb{C} \setminus \{1\}$, with the constant term corresponding to the regularization of divergent terms in the series and matching the pole structure of $\zeta(s)$ at $s=1$.

Expansion of $\zeta(s)$ in the Critical Strip

Derivation of the Polynomial Expansion

The central analytic observation is the representation of $1/n^s$ in terms of a new polynomial basis:

$$\frac{1}{n^s} = \frac{1}{n+1} G_s\left(\frac{n-1}{n+1}\right)$$

where $G_s(z)$ admits a Taylor expansion in $z$ with coefficients $P_m(s)$, polynomials in $s$, provided as

$$P_m(s) = \sum_{k= 0}^m \binom{m}{k}(-1)^{k} 2^{k+1} \frac{(s)_k}{k!}.$$

The analytic structure ensures $P_m(s)$ admits all roots on $\Re(s) = 1/2$.

Ramanujan's summation is then applied to both sides, yielding

$\zeta(s) = \sum_{m=0}^\infty x_m P_m(s)$

with coefficients

$$x_m = \gamma + \sum_{k=1}^m \binom{m}{k}(-2)^k \zeta(k+1) + r_m,$$

and $r_m$ explicit rational correction terms.

Orthogonality and Basis Properties

A substantial feature is that the polynomials $Q_m(t) = \frac{1}{2}P_m(\frac12 + it)$ define an orthonormal basis for $L^2(\mathbb{R}, dt/\cosh(\pi t))$. This links the expansion to spectral decompositions in analytic number theory, and, by classical theory, ensures that all zeros of $Q_m$ (and thus $P_m$) lie on the critical line, preserving the expected functional equation symmetries.

Explicit Evaluation of Coefficients

The paper provides exact and algorithmically tractable formulae for $x_m$, involving finite sums of zeta values and binomial coefficients, with supplementary analytic properties (integral representations) derived from the relation to Laguerre polynomials.

For $\zeta(\frac12 + it)$,

$$\zeta(\tfrac{1}{2} + it) = 2 \sum_{m=0}^{\infty} x_m Q_m(t)$$

is given as a convergent, orthogonal expansion, and Parseval-type identities for the Fourier weight $\cosh(\pi t)^{-1}$ are derived.

Theoretical and Practical Implications

This expansion strengthens connections between special function theory, regularized summations of divergent series, and analytic number theory. By expressing the zeta function in the critical strip as a series in an explicit basis with computable coefficients, the result admits immediate applications to:

• Precise numerical evaluation of $\zeta(s)$ for $\Re(s)$ near $1/2$, including error bounds via convergence estimates derived in the paper.
• New perspectives on the distribution of zeros, as all roots of the polynomial basis are on the critical line, making this expansion canonically suited for spectral studies of $\zeta(s)$.
• The development of orthogonal polynomial expansions of other $L$-functions via similar regularization and Mellin transform machinery.

The work provides rigorous methods for handling divergent series using Ramanujan summation, extends classical techniques in the theory of special functions, and supplies analytic tools that may inform future progress on open questions such as the Riemann Hypothesis.

Conclusion

The polynomial expansion for the Riemann zeta function in the critical strip established in this paper merges classical techniques in analytic continuation, special functions, and Ramanujan summation. The explicit structure of the expansion, the spectral properties of the polynomial basis, and the direct connection to values of $\zeta(s)$ on and near the critical line constitute a significant technical advance with implications for both computation and theoretical number theory. The methods outlined suggest a broad spectrum of future investigations, particularly for $L$-functions and spectral structures associated with automorphic forms.