Uniform Prime Number Theorem

• Uniform Prime Number Theorem is a generalization of the classical prime number theorem that provides asymptotic formulas with uniform error control across moduli and families of L-functions.
• It unifies classical results like the Siegel–Walfisz, Hoheisel, Brun–Titchmarsh, and Linnik theorems through refined zero-free regions and log-free zero-density estimates.
• The theorem employs explicit formulas and pair correlation techniques, offering robust bounds in short intervals and yielding deep corollaries on prime distributions.

The Uniform Prime Number Theorem (Uniform PNT) generalizes the classical PNT by providing asymptotic formulae for the distribution of prime numbers that remain valid uniformly across ranges of moduli, arithmetic progressions, and—more generally—across families of $L$-functions with explicit quantitative control on error terms and dependence on spectral data such as the location of zeros. Uniformity in this context pertains to the precision of error terms with respect to relevant parameters (e.g., modulus $q$ for arithmetic progressions or analytic conductor for general $L$-functions), and to the ability to deduce deep corollaries encompassing the Siegel–Walfisz theorem, Hoheisel’s short-interval result, Brun–Titchmarsh type bounds, and Linnik’s theorem on least primes. Central to uniform PNTs are refined treatments of exceptional (“Siegel”) zeros, log-free zero density estimates, robust zero-free regions, and, more globally, the pair correlation of nontrivial zeros of zeta and automorphic $L$-functions.

1. Classical and Uniform Formulations

The classical PNT asserts $\pi(x) \sim \operatorname{Li}(x)$ as $x\to\infty$, or for the logarithmic sum, $\psi(x) = x + o(x)$. The “uniform” PNT for arithmetic progressions takes the form

$$\psi(x; q, a) = \frac{\lambda(x, q, a)x}{\varphi(q)}\Bigg[1 + O\left(\exp\left\{-c\min\left(\frac{\log x}{\log q}, \frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\right\}\right)\Bigg],$$

valid for all $q\ge2$, $(a, q)=1$, and $x\ge q^{12}$, with $\lambda(x,q,a)$ adjusting for possible real zero phenomena (explicitly, $\lambda(x,q,a)=1-\chi_1(a)x^{\beta_1-1}/\beta_1$ when a Siegel zero $\beta_1$ exists), and $\varphi(q)$ the Euler totient. Partial summation yields the corresponding formula for the prime counting function $\pi(x;q,a)$, uniform in $q$ and $a$ (Thorner et al., 2021). In the “highly uniform” regime for $L$-functions, results apply to Dirichlet-series $L(s,\pi)$ in a wide family (e.g., automorphic forms) with

$$\sum_{n\le x} a_\pi(n)\Lambda(n) = r_\pi x - \frac{x^{\beta_0}}{\beta_0} + O\left(\left(x - \frac{x^{\beta_0}}{\beta_0}\right) [ m^5 x^{-c/m^4} + m^{c'm^3}A^2 e^{-(1-1/A)\eta_\pi(x)} ]\right),$$

for $x\ge C(\pi)^{cA^2m^5}$ with analytic conductor $C(\pi)$, and $\beta_0$ any real zero as above, featuring explicit dependence on the analytic and arithmetic data of $\pi$ (Kaneko et al., 2022).

2. Zero-Free Regions, Zero-Density, and Exceptional Zeros

Uniform PNTs hinge upon effective zero-free regions and log-free zero-density estimates for Dirichlet $L$-functions and their generalizations. For all Dirichlet characters $\chi\pmod q$ and all $t\ge e$, the Vinogradov–Korobov region guarantees $L(s,\chi)\neq0$ in

$$\operatorname{Re} s \ge 1 - \frac{C_{VK}}{\log q + (\log|t|)^{2/3}(\log\log|t|)^{1/3}},$$

with an at-most-single exceptional real zero $\beta_1$ (Siegel zero). Log-free zero-density estimates of Thorner–Zaman provide, uniformly,

$$N_q(\sigma,T) \ll_\varepsilon (qT)^{(12/5+\varepsilon)(1-\sigma)}$$

and, omitting a Siegel zero (if one exists),

$$N_q^*(\sigma,T) \ll_\varepsilon \nu(qT) (qT)^{(37/5+\varepsilon)(1-\sigma)},$$

where $\nu(U) = \min\{1, (1-\beta_1)\log U\}$ (Thorner et al., 2021). Deuring–Heilbronn zero repulsion ensures that, in the presence of a Siegel zero, other zeros are effectively repelled from $\operatorname{Re}s=1$ (i.e., pushed further leftward), yielding finer error terms and enabling ultimate uniformity even in the “deep” exceptional zero case.

In the automorphic context, analogs require robust nonvanishing conditions (zero-free for $\Re s=1$ up to $\delta_\pi(T)$), explicit “zero-repulsion” for any exceptional real zero, and polynomial control in analytic conductor $C(\pi)$. This allows polynomial $x$-ranges (e.g., $x\ge C(\pi)^{O(m^5)}$) as opposed to exponential (Kaneko et al., 2022).

3. Main Corollaries and Applications

The following table summarizes principal corollaries deduced from uniform PNTs (modulo technical hypotheses such as modulus size and interval length) (Thorner et al., 2021):

Result Class	Quantitative Statement (arithmetic progressions)	Remarks
Siegel–Walfisz	$$\pi(x;q,a) = \frac{\operatorname{Li}(x)}{\varphi(q)} \left[1+O\left(\exp\{-c_A(\log x)^{3/5}(\log\log x)^{-1/5}\}\right)\right]$$	$q\le(\log x)^A$; optimal uniform error
Hoheisel (short intervals)	$$\pi(x+h;q,a) - \pi(x;q,a) = \frac{h}{\varphi(q)\log x} [1+O_{A,\delta}(\exp\{-c_{A,\delta}(\log x)^{1/3}(\log\log x)^{1/3}\})]$$	$h\ge x^{7/12+\delta}(\log x)^A$
Brun–Titchmarsh bound	$\pi(x;q,a) \ll \frac{2x}{\varphi(q)\log(x/q)}$ (when $\beta_1$ exists)	$q < x$
Linnik’s theorem (least prime)	$p(a;q) \ll q^{C}$	$C$ absolute; no Siegel zero required

These results interpolate smoothly between long and short intervals, small and large modulus, and employ explicit “main-term plus secondary-term” expansions to absorb any exceptional zero.

4. Pair Correlation and Uniformity in Error Terms

Recent results (Goldston et al., 2022) demonstrate that uniform versions of the PNT also admit further quantitative refinement through hypotheses on the pair correlation of zeros of the Riemann zeta-function. Assuming the Riemann Hypothesis (RH) and a suitable pair-correlation conjecture (i.e., uniform Montgomery-type bounds on

$$F_\beta(x,T) = \sum_{0<\gamma,\gamma'\le T} x^{i(\gamma-\gamma')} w_\beta(\gamma-\gamma'),$$

where $w_\beta$ is a smooth weight and $\gamma$ enumerate zero ordinates), one obtains error bounds

$$\psi(x+H)-\psi(x)=H + O\left(x^{1/2}(\mathcal{M}(x)+(\log W(x))^2)\right),$$

with the “main term” and error term holding uniformly for $1\le H\le x$ and across long ranges in $x$. Under such hypotheses, errors can be improved to $O(x^{1/2}(\log\log x)^2)$ or $O(x^{1/2}(\log x)^a)$, enhancing the classical von Koch bound of $O(x^{1/2}\log^2 x)$. This uniformity pervades both short interval and global prime counting, with the proof leveraging truncated explicit formulae, dyadic block decompositions of zeros, and precise control over zero pair statistics (Goldston et al., 2022).

5. Techniques and Proof Outlines

Uniform PNTs are proven by combining explicit formulas (e.g., for $\psi(x;q,a)$ or for weighted prime sums for general $L$-functions) with the following analytic apparatus:

• Explicit formula: Expresses sum of prime counts in arithmetic progressions as main term plus sum over nontrivial zeros, secondary main term for possible exceptional zero, and error.
• Zero-density estimates: Legally split zero sums in dyadic intervals and rigorously bound contributions via log-free density bounds.
• Zero-free regions: Insert explicit regions into any factor $x^{\sigma-1}$ involved in error terms to achieve exponential decay as a function of $\log x$ and logarithmic quantities in $q$ or analytic conductor.
• Zero repulsion/Deuring–Heilbronn phenomenon: When a Siegel zero exists, invoke zero repulsion to bound the proximity of any additional zeros to $1$, ensuring negligible error proliferation.
• Optimization in $L$-functions: Apparent in general $L(s,\pi)$ cases, where explicit smooth weights and Laplace transforms are used, with optimization over integration heights tied to optimizing the zero-free region (Thorner et al., 2021, Kaneko et al., 2022).

This analytical framework seamlessly subsumes and strengthens classical results such as Siegel–Walfisz, Hoheisel, Brun–Titchmarsh, and Linnik’s theorem, and supports extension to higher rank automorphic forms.

6. Extensions to Automorphic and Rankin–Selberg $L$-Functions

The highly uniform PNT of Thorner and Zaman (Kaneko et al., 2022) provides for the first time general uniform PNTs (with explicit analytic conductor uniformity) for broad classes of $L$-functions, including Rankin–Selberg convolutions $L(s, \pi\times\pi')$:

$$\sum_{n\le x} a_{\pi\times\pi'}(n)\Lambda(n) = r_{\pi\times\pi'}x - \frac{x^{\beta_1}}{\beta_1} + O\left((x-x^{\beta_1}/\beta_1)\exp\left(-c\frac{\log x}{(m+m')\log(C(\pi)C(\pi'))+\sqrt{m(m+m')\log x}}\right)\right)$$

for $x\ge (C(\pi)C(\pi'))^{O((mm')^4)}$ and $r_{\pi\times\pi'}$ as the pole order. Improvements include polynomial $x$-ranges in terms of analytic conductor, error terms robust as the possible exceptional zero $\beta_1\to1$, and no reliance on $\ell^2$-type coefficient bounds—only $\ell^1$ and Ramanujan bounds are needed.

This generality yields the first uniform PNTs in complete generality for automorphic forms of high degree, with consequences for the arithmetic of associated fields and equidistribution in automorphic spectra (Kaneko et al., 2022).

7. Conceptual Significance and Future Directions

The uniform prime number theorem provides a unifying framework that transcends classical limitations regarding modulus or arithmetic complexity, directly links the quality of zero-free regions and zero-density estimates to the precision of prime distribution, and systematically organizes corollaries such as least prime bounds, short interval distribution, and mean-square error phenomena. The recent interplay between explicit correlation estimates of zeros and uniform PNTs suggests further refinements may be possible should deeper conjectures (e.g., generalized pair correlation for families of $L$-functions) be established. This suggests a promising direction involving even tighter uniform error terms, improved bounds in short intervals, and new uniform results for general families of automorphic and motivic $L$-functions (Kaneko et al., 2022, Goldston et al., 2022).