An Alternative View of the Riemann Zeta Function

This work offers a new perspective on the Riemann zeta function, based on the graphical observation of partial sums. The starting point is the discovery of a \textit{reference constant}: when plotting the partial sums of the real and imaginary parts of $\zeta(s)$, the curves oscillate around a constant value, which is precisely the analytic continuation $\zeta^p(s)$. (^p is for extended zeta). We provide a detailed explanation of genesis of the analytic continuation values $\zeta^p(s)$, a point often overlooked in the literature. By comparing the discrete sum with the integral of the corresponding continuous function, we show that the reference constant arises from the cumulative difference between the sampled signal and its integral. This difference, which accumulates mainly during the transient regime, gives rise to the value of $\zeta^p(s)$. Thus, the analytic continuation emerges naturally from the sampling process itself. We show that the partial sum $\zeta_k(s) = \sum_{n=1}^{k} n^{-s}$ decomposes into three distinct contributions: a reference constant, which is precisely the analytic continuation $\zeta^p(s)$; a divergent integral that carries the growth of the series; and a sum of corrective wavelets that tend to zero. This decomposition, established through an experimental mathematics approach, allows us to rediscover the Euler-Maclaurinformula by graphical observation. We also present three numerical methods to approximate $\zeta^p(s)$: Cesàro averaging, the sum-integral difference, and the Euler-Maclaurin formula with corrective terms. The results are validated by comparison with reference values obtained from the \texttt{mpmath} library. From this structure, we derive two remarkable invariance properties: phase-shift invariance and scaling invariance. These properties show that the non-trivial zeros of $\zeta$ manifest themselves through the vanishing of the reference constant, providing a simple graphical criterion to identify them. We thus construct a family of series sharing the same zeros as $\zeta$. We also develop a method for the visual detection of zeros, based on observing the sum of squares of the partial sums. Finally, the study of the Dirichlet $\eta$function reveals that its convergence for $\Re(s) > 0$ results from a compensation between the divergent parts of the odd and even sums, perfectly illustrating the role of the reference constant. Taken together, these results constitute an original contribution to the understanding of the Riemann zeta function and offer a visual and experimental approach complementary to classical analytical methods.