Explicit Bounds Under the Riemann Hypothesis

• The paper establishes an explicit bound on the analytic error term for the divisor function, breaking it into arithmetic and analytic components under RH.
• It employs Mellin transforms and contour integration to separate main terms from error contributions while using sharp estimates on zeta functions.
• The results confirm the optimal double-logarithmic error factor and extend the framework to similar bounds in prime-counting and L-function analysis.

An explicit bound under the Riemann Hypothesis refers to a numerically effective, fully explicit inequality—valid assuming the Riemann Hypothesis (RH)—for important number-theoretic objects. Typical targets include error terms in analytic summatory functions, prime-counting functions in both “global” and “short interval” settings, the divisor and Euler totient functions, central values and derivatives of L-functions, and oscillatory terms like the argument function of the zeta function. Such results generally distill the full force of the Riemann Hypothesis (or its generalizations) into explicit estimates with all implicit constants described, enabling quantitative validation and application across numerous problems in arithmetic.

1. Overview and Main Statements

A landmark example is the explicit upper bound on the analytic part of the divisor function error term, $E_{\sigma_1}^{\mathrm{AN}}(x)$, as established in "On a relation to the Riemann Hypothesis and an analytic part for the divisor function" (Iwata, 16 Jan 2026). Writing

$$E_{\sigma_1}(x) = \sum_{n\le x} \sigma_1(n) - \frac{\pi^2}{12} x^2,$$

with

$$E_{\sigma_1}(x) = E_{\sigma_1}^{\mathrm{AR}}(x) + E_{\sigma_1}^{\mathrm{AN}}(x) + O(x^{1/2}),$$

where the arithmetic and analytic parts decompose as \begin{align*} E_{\sigma_1}^{\mathrm{AR}}(x) & = x f_2(x), \qquad f_2(x) = -\sum_{n\ge1} n^{-1} {x/n}, \ E_{\sigma_1}^{\mathrm{AN}}(x) & = \frac{1}{2} g_2(x) + \frac{1}{2} x(\log x + 2\gamma - 1), \qquad g_2(x) = \sum_{n\ge1} {x/n}^2. \end{align*} The principal explicit bound under RH is

$$E_{\sigma_1}^{\mathrm{AN}}(x) \ll x^{\delta'} \exp\left(\frac{\log x}{\log\log x}\right),$$

for any $\delta\in(0,1)$ and $\delta' = \max\{\frac{1}{2},\delta\}$, for all $x \ge e^e$ [(Iwata, 16 Jan 2026), Thm 1.2.1]. An $\varepsilon$-refined bound for any $\varepsilon>0$ asserts

$$E_{\sigma_1}^{\mathrm{AN}}(x) \ll_\varepsilon x^{\delta'+\varepsilon} \qquad (x\ge1).$$

The optimal exponent in the double-logarithmic exponential factor is known not to be improvable without going beyond RH, as it matches the subconvexity in bounds for $\zeta(1+it)$.

2. Analytic Methodology: Mellin Transforms and Contour Integration

The derivation employs the contour-integral approach associated with explicit formula techniques. The Mellin transform of $E_{\sigma_1}^{\mathrm{AN}}(x)$ for $\Re s>2$ is computed explicitly: $$\int_1^\infty E_{\sigma_1}^{\mathrm{AN}}(x) x^{-s-1} dx = \frac{\pi^2}{12}\frac{1}{s-2} + \frac{\zeta(s)\zeta(s-1)}{s(1-s)} + O(1).$$ The inverse Mellin representation reads

$$E_{\sigma_1}^{\mathrm{AN}}(x) = \frac{1}{2\pi i} \int_{\Re s=3+\delta} \frac{\zeta(s)\zeta(s-1) x^s}{s(1-s)} ds + O(x^{3+\delta}) \qquad (0<\delta<1).$$

A crucial step is shifting the contour to $\sigma = \frac{1}{2} + \eta$, where $\eta = 1/\log\log x^2$, invoking Cauchy's theorem. The main residue at $s=2$ produces the leading $x^2$ term, which cancels in the analytic decomposition, leaving the estimation of five contour integrals.

3. Key Estimates and Exploiting the Riemann Hypothesis

The critical estimates rest on sharp upper bounds for $\zeta(s)\zeta(s-1)$ on the contour, under RH: $$\log \zeta(s) = O\left( \frac{(\log t)^{2-2\sigma}}{\log\log t} \right), \quad \left(\frac{1}{2} < \sigma < 1 \right),$$ and, generally, for $\frac{1}{2}+\eta \le \sigma \le 2+\delta$, $|t| \le x^2$,

$$\zeta(s)\zeta(s-1) \ll |s| \exp\left( \frac{\log|s|}{\log\log|s|} \right).$$

The “critical” vertical contour ($\sigma = \frac{1}{2} + \eta$, $|t| \le x^2$) dominates the error, yielding

$$\int_{-x^2}^{x^2} \left| \frac{x^{1/2+\eta} \exp((\log x)/(\log\log x))}{t^2} \right| dt = O\left( x^{1/2+o(1)} \exp\left( \frac{\log x}{\log\log x} \right) \log x \right).$$

Horizontal and other vertical segments contribute $O(x^{\delta}\exp((\log x)/(\log\log x)))$ or are negligible.

4. Comparative Framework and Relationship to Preceding Results

The methodology generalizes smoothing and contour techniques developed for related arithmetic error terms. For the totient error term, Kaczorowski–Wiertelak [as summarized in (Iwata, 16 Jan 2026)] obtained, under RH,

$$E_\varphi^{\mathrm{AN}}(x) \ll x^{1/2} \exp\left(A \frac{\log x}{\log\log x}\right),$$

where the Mellin integral features $\zeta(s-1)/\zeta(s)$ rather than $\zeta(s)\zeta(s-1)$. The analytic structure—absence of a reciprocal $\zeta(s)$ factor for $\sigma_1(n)$—precludes any equivalence statement to RH for this divisor function bound.

Explicit bounds for prime-counting error terms such as those for $\psi(x)$ (Chebyshev's function) (Büthe, 2014, Cully-Hugill et al., 2019) have analogous methodology: formulate a smoothed explicit formula, express as a sum over zeros plus main terms, and execute precise error control after contour movement, with explicit constants. However, for the divisor problem, the prominent exponential arising from the subconvexity estimate for $\zeta(s)$ on the critical line is provably necessary, which “saturates” the potential improvements under RH.

5. Broader Landscape: Explicit RH Bounds Across Number Theory

Explicit RH-based bounds for summatory, oscillatory, and value-distribution problems now exist across a wide spectrum:

• For the Chebyshev function in short intervals, (Cully-Hugill et al., 2019) provides,

  $$|\psi(x+h)-\psi(x)-h| < \frac{1}{\pi}\sqrt{x} \ln x \ln\left(\frac{h}{\sqrt{x} \ln x}\right) + 2 \sqrt{x} \ln x,$$

  for $h \ge \sqrt{x}\ln x$.

• For the argument function $S(t) = \frac{1}{\pi}\arg \zeta(\frac{1}{2} + it)$, sharp bounds are obtained (Carneiro et al., 2013, Carneiro et al., 2017),

  $$|S(t)| \le \left(\frac{1}{4}+o(1)\right) \frac{\log t}{\log\log t}.$$

• For the number of primes in intervals and arithmetic progressions, explicit error bounds with all constants (e.g., $(\sqrt{x}\log x)/(8\pi)$) are available (Büthe, 2014, Ernvall-Hytönen et al., 2020), ready for computational validation and application.
• For $|\zeta(\sigma+it)|$ and $|\zeta'/\zeta(\sigma+it)|$ in the critical strip, (Simonič, 2021, Chirre et al., 2021, Carneiro et al., 2017, Palojärvi et al., 2024) provide fully explicit constants and dependence on distance to the critical line, as well as generalizations to the Selberg class.

A consistent structural feature is that such results rest on explicit formulae connecting the analytic object to sums over zeros and Dirichlet–Arithmetical data, then bound each analytic term with precision using the full force of (G)RH.

6. Significance, Optimality, and Limits of Improvement

The “exp(log x / log log x)” factor in the error bound for $E_{\sigma_1}^{\mathrm{AN}}(x)$ under RH is optimal in the sense that this structure arises precisely from the best possible upper bound on $|\zeta(1+it)|$ implied by RH. The only possible refinement consists of $o(1)$-sized improvements in the exponent or explicit constants. Major advances—such as replacing it with a power-saving as in the Lindelöf Hypothesis regime—would necessitate progress beyond RH.

No new zero-free regions or zero-density results are involved; the full strength of what RH gives for the zeta function on the critical line is utilized (Iwata, 16 Jan 2026). This structural optimality and technical rigidity are mirrored in analogous error-term results for other summatory and oscillatory functions in analytic number theory, for instance, explicit primes gaps (Dudek et al., 2015), explicit error terms for $\zeta'/\zeta(s)$ (Chirre et al., 2021), and the bounds for the argument function and all its iterates (Carneiro et al., 2017, Wakasa, 2012).

7. Summary of Methodological Paradigms

The paradigmatic structure for deriving explicit RH bounds—whether for divisor sums, prime-counting error terms, or L-functions—proceeds as follows:

1. Express the target error or oscillatory object via an explicit formula or Mellin inversion, isolating main terms and error terms with transparent arithmetical or analytic meaning.
2. Shift contours to regions where the error is minimized, precisely controlling pinched poles and residues.
3. Employ the sharpest uniform analytic bounds for zeta/L-functions on the relevant lines or vertical segments, utilizing RH or GRH in the critical region.
4. Aggregate the respective contributions, minimizing and balancing exponents and leading double-logarithmic factors, and optimize over auxiliary small parameters (e.g., shifting $\delta \to 0+$ as needed).
5. Check for optimality against the best known subconvexity/sub-logarithmic exponents derivable under RH.

This backbone unites explicit divisor bounds, short-interval prime theorems, Chebyshev error bounds, argument function estimates, and value-distribution control for zeta and L-functions, illustrating the deep interconnection of analytic and arithmetic structure under the Riemann Hypothesis (Iwata, 16 Jan 2026, Cully-Hugill et al., 2019, Carneiro et al., 2013, Simonič, 2021, Palojärvi et al., 2024).