Spectral Convergence: Linking Rogers-Ramanujan Identities to the Riemann Hypothesis

The Convergence of Prime Sums and Continued Fractions

A central identity explored in arXiv:1003.4015 relates the sum of a function over prime numbers to the logarithmic integral and the prime-counting error term. This relationship provides a mathematical bridge between the discrete distribution of primes and the analytic properties of the Riemann zeta function. By expressing prime sums in terms of the error term pi(x) - li(x), researchers can localize the influence of zeta zeros on arithmetic sequences.

The Prime Sum Identity

The core formulation established in the research is: the sum over p less than or equal to n of f(p) equals the integral from 2 to n of f(x) divided by log(x), plus f(2)li(2), plus f(n) multiplied by the difference (pi(n) - li(n)), minus the integral of (pi(x) - li(x)) times the derivative f'(x). This identity allows the prime-counting discrepancy to be treated as a test functional, where the size and oscillation of the sum are sensitive to the location of zeros on the critical line.

Irrationality Measures and the Euler-Mascheroni Constant

The research investigates the irrationality measure (mu) of constants derived from spectral data, denoted as u_d. A striking observation is that for certain arithmetic structures, these measures oscillate around a value determined by the Euler-Mascheroni constant, specifically 1 + 2^(e^-gamma). This constant is deeply embedded in the Mertens' theorems, which are elementary equivalents to the Prime Number Theorem.

• Optimal Approximation: If the irrationality measure mu equals 1, the fluctuations of the prime-counting error are constrained, providing a new metric for bounding the error term.
• Transcendence Criteria: Continued fractions with coefficients forming rational arithmetic progressions, such as the Rogers-Ramanujan type, are shown to be transcendental. This suggests that the transcendence of these constants is inextricably linked to the non-vanishing of the zeta function on the critical line.
• Growth Invariants: The product of specific prime sequences (2, 5, 17, ... q_n) grows faster than (n!)^2, a property that can be used to construct Liouville-type numbers that probe the gaps between zeta zeros.

The r-Lambert Function in Prime Counting

To further refine the analysis of prime distribution, the research introduces the r-Lambert function W_r(y), defined as the inverse of x*e^x + r*x. This function provides a more precise asymptotic solution for the prime-counting function under the assumption of the Riemann Hypothesis, particularly when modeling the bias between different residue classes in the "prime race."

Computational Verification with Wolfram Language

The following implementation allows for the visualization of the Rogers-Ramanujan continued fraction and the verification of the prime sum identity using spectral data.

(* Define the Rogers-Ramanujan Continued Fraction function *)
RR[q_, n_] := Module[{frac = 0},
  Do[
    frac = q^(n - i + 1)/(1 + frac),
    {i, 1, n}
  ];
  q^(1/5)/(1 + frac)
];

(* Visualize the prime counting error: pi(x) - li(x) *)
piError[x_?NumericQ] := PrimePi[x] - LogIntegral[x];

(* Plotting the error term to observe zeta-driven oscillations *)
Plot[piError[x], {x, 2, 2000},
  PlotRange -> All, 
  PlotLabel -> "Prime Counting Error Term: pi(x) - li(x)",
  AxesLabel -> {"x", "Error"},
  PlotStyle -> Thick]

Future Directions for PhD Researchers

The synthesis of these structures reveals that prime distribution is not merely an analytic phenomenon but is rooted in the Diophantine properties of modular forms. Key next steps include establishing a formal proof that the irrationality measure of gamma is bounded by the Rogers-Ramanujan center and extending the computation of u_d values to observe if their oscillation frequency matches the density of zeta zeros at higher ranges.