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Chebyshev function

In analytic number theory, the Chebyshev functions refer to two key summatory functions that aggregate logarithmic contributions from primes and their powers to study the distribution of prime numbers: the first Chebyshev function \theta(x) = \sum_{p \leq x} \log p, where the sum is over all primes p not exceeding x, and the second Chebyshev function \psi(x) = \sum_{n \leq x} \Lambda(n), where \Lambda(n) is the von Mangoldt function defined as \log p if n = p^k for some prime p and positive integer k \geq 1, and $0$ otherwise.^[1]^[2] Named after the Russian mathematician Pafnuty Chebyshev, these functions were introduced in his seminal 1852 memoir Mémoire sur les nombres premiers, where he employed them to investigate the prime-counting function \pi(x).^[1] The Chebyshev functions are central to the Prime Number Theorem (PNT), which asserts that the number of primes up to x is asymptotically x / \log x; equivalently, the PNT holds if and only if \psi(x) \sim x or \theta(x) \sim x as x \to \infty.^[1]^[2] Using elementary methods, Chebyshev established explicit bounds such as $0.92129 \frac{x}{\log x} < \pi(x) < 1.10555 \frac{x}{\log x} for sufficiently large x, derived via estimates on \theta(x) and \psi(x) that showed their growth is sandwiched between constants times x, providing the first rigorous evidence toward the PNT decades before its proof in 1896.^[1]

Definitions and Notation

First Chebyshev function

The first Chebyshev function, denoted \theta(x), is defined as the sum of the natural logarithms of all prime numbers up to x:

\theta(x) = \sum_{p \leq x} \log p,

where the sum runs over all primes p \leq x.^[3] This function provides a weighted measure of the primes below x, with each prime contributing its logarithm to emphasize larger primes in the distribution.^[4] Introduced by Pafnuty Chebyshev in his 1852 memoir on prime numbers, \theta(x) served as a key tool for investigating the distribution of primes and establishing bounds related to the prime-counting function \pi(x).^[4] Chebyshev employed \theta(x) to derive inequalities that supported Bertrand's postulate and laid groundwork for understanding prime density, demonstrating that primes are sufficiently frequent without fully resolving the prime number theorem.^[5] Known specifically as the Chebyshev theta function in analytic number theory, it is distinct from other theta functions, such as those arising in elliptic functions or modular forms.^[3] For small values of x, \theta(x) can be computed directly from the list of primes. For instance, the primes less than or equal to 10 are 2, 3, 5, and 7, so

\theta(10) = \log 2 + \log 3 + \log 5 + \log 7 \approx 0.693 + 1.099 + 1.609 + 1.946 = 5.347.

This explicit summation highlights \theta(x) as a cumulative logarithmic weight of primes.^[3] As a foundational construct, \theta(x) establishes the weighted sum of \log p up to x as a building block for assessing prime density, facilitating comparisons between the growth of primes and logarithmic scales in subsequent number-theoretic analyses.^[4]

Second Chebyshev function

The second Chebyshev function, denoted \psi(x), is defined for x \geq 0 as the sum \psi(x) = \sum_{p^k \leq x} \log p, where the sum runs over all primes p and positive integers k \geq 1 such that the prime power p^k does not exceed x.^[3] Equivalently, \psi(x) can be written as the partial sum \psi(x) = \sum_{n \leq x} \Lambda(n), with \Lambda(n) denoting the von Mangoldt function, which equals \log p if n = p^k for some prime p and integer k \geq 1, and zero otherwise.^[3] This function generalizes the first Chebyshev function \theta(x) by incorporating contributions from higher prime powers beyond just the primes themselves. Specifically,

\psi(x) = \sum_{n=1}^{\infty} \theta\left(x^{1/n}\right),

where the infinite series truncates after finitely many terms because \theta(y) = 0 for y < 2, so only terms with n \leq \log x / \log 2 contribute.^[3] This relation highlights how \psi(x) aggregates the logarithmic weights across iterated roots of x, providing a layered summation that captures the full structure of prime powers. Pafnuty Chebyshev introduced \psi(x) in his 1852 memoir Mémoire sur les nombres premiers, where he employed it to derive bounds on the distribution of primes and support Bertrand's postulate on the existence of primes in short intervals.^[4] Subsequent refinements in the late 19th century, particularly in the works of Jacques Hadamard and Charles-Jean de la Vallée Poussin, elevated \psi(x) to a central tool in analytic number theory by linking it to the non-vanishing of the Riemann zeta function on the line \Re(s) = 1.^[6] For example, \psi(10) includes the terms for prime powers up to 10: \log 2 (from $2, 4=2^2, 8=2^3), \log 3 (from $3, 9=3^2), \log 5 (from 5), and \log 7 (from 7), yielding \psi(10) = 3\log 2 + 2\log 3 + \log 5 + \log 7 = \log 2520.^[3] The inclusion of higher prime powers in \psi(x) results in a smoother cumulative distribution compared to sums over primes alone, which facilitates its analysis through Dirichlet series, as the generating function \sum_{n=1}^\infty \Lambda(n) n^{-s} = -\zeta'(s)/\zeta(s) connects \psi(x) directly to the logarithmic derivative of the Riemann zeta function.^[7] This property makes \psi(x) particularly valuable for broader applications in the study of arithmetic functions and prime distributions.^[8]

Basic Properties

Analytic properties

The first Chebyshev function \theta(x) and the second Chebyshev function \psi(x) are both step functions that are piecewise constant on the positive real line, with discontinuities occurring exclusively at prime numbers for \theta(x) and at prime powers for \psi(x). At each prime p, \theta(x) exhibits a jump discontinuity of size \log p, while \psi(x) jumps by \log p at every prime power p^k for k \geq 1.^[9]^[10] Since the jumps are positive (\log p > 0 for all primes p), both \theta(x) and \psi(x) are non-decreasing functions. Between discontinuity points, they remain constant, reflecting the absence of prime power contributions in those intervals. This step-like structure uniquely encodes the locations and logarithmic weights of all prime powers, allowing the functions to capture the arithmetic distribution of primes without redundancy—each jump corresponds precisely to a single prime power contribution via the von Mangoldt function \Lambda(n) = \log p if n = p^k and 0 otherwise, with \psi(x) = \sum_{n \leq x} \Lambda(n).^[9]^[10] A fundamental integral representation for \theta(x) is given by the Stieltjes integral form, but an explicit summation equivalent arises from changing the order of integration:

\int_2^x \frac{\theta(t)}{t} \, dt = \sum_{p \leq x} \log p \cdot \log \left( \frac{x}{p} \right).

This equality holds exactly and expresses the integral as a weighted sum over primes, where each term \log p \cdot \log(x/p) measures the contribution of prime p scaled by the logarithmic length of the interval from p to x; it relates directly to double-logarithmic growth patterns inherent in prime distributions, such as those appearing in Mertens' theorems, though without asymptotic evaluation here.^[11]^[10] In the complex plane, the Chebyshev functions connect to analytic number theory through the Riemann zeta function \zeta(s). For \operatorname{Re}(s) > 1, the Dirichlet series for the von Mangoldt function yields

-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} = \sum_p \sum_{k=1}^\infty \frac{\log p}{p^{ks}},

which encodes the prime power contributions logarithmically; this meromorphic continuation to the critical strip (with a simple pole at s=1) provides the analytic foundation for studying \psi(x) and \theta(x) via Perron's formula or explicit formulae, highlighting their role in the spectral theory of primes.^[10]

Multiplicative properties

The von Mangoldt function \Lambda(n), which defines the second Chebyshev function via \psi(x) = \sum_{n \leq x} \Lambda(n), exhibits arithmetic structure through its expression as a Dirichlet convolution: \log n = \sum_{d \mid n} \Lambda(d). By Möbius inversion, this yields \Lambda(n) = \sum_{d \mid n} \mu(d) \log(n/d), where \mu is the Möbius function. For n = p^k a prime power, the sum simplifies to \log p, confirming \Lambda(p^k) = \log p. This convolution relation underscores the role of multiplicativity in arithmetic functions, as both \mu(n) and the constant function 1 are multiplicative, and the logarithm is completely additive, allowing decomposition of \Lambda(n) based on the prime factorization of n. Although \Lambda(n) itself is neither multiplicative nor additive, its Dirichlet series \sum_{n=1}^\infty \Lambda(n) n^{-s} = -\zeta'(s)/\zeta(s) admits an Euler product \prod_p \left( 1 + \sum_{k=1}^\infty \frac{\log p}{p^{ks}} \right), reflecting the multiplicative nature over primes inherent in the Riemann zeta function \zeta(s). This product form facilitates analysis of \psi(x) in terms of prime contributions and aids computations for composite arguments by leveraging sieve methods or recursive decompositions tied to the prime factors, such as expressing partial sums over prime power contributions independently before aggregation. In the context of arithmetic progressions, the multiplicative properties extend to twisted variants \psi(x; \chi) = \sum_{n \leq x} \Lambda(n) \chi(n), where \chi is a Dirichlet character modulo q. The Dirichlet series for this twist is \sum_{n=1}^\infty \Lambda(n) \chi(n) n^{-s} = -L'(s, \chi)/L(s, \chi), with L(s, \chi) = \prod_p (1 - \chi(p) p^{-s})^{-1} possessing an Euler product that encodes the multiplicative action of \chi on primes. For example, when \chi is the principal character, \psi(x; \chi) = \psi(x); nontrivial characters allow decomposition of sums in residue classes via orthogonality: \sum_{n \leq x, n \equiv a \pmod{q}} \Lambda(n) = \frac{1}{\phi(q)} \sum_{\chi \bmod q} \overline{\chi}(a) \psi(x; \chi), enabling multiplicative separation of the progression's behavior across characters. This structure is pivotal for studying prime distributions modulo q without delving into individual residue details.

Interrelations

Relation between the two functions

The second Chebyshev function \psi(x) is expressed in terms of the first Chebyshev function \theta(x) via the summation formula

\psi(x) = \sum_{k=1}^\infty \theta\left(x^{1/k}\right),

where the infinite series truncates naturally at k \approx \log x / \log 2, since \theta(y) = 0 for y < 2.^[3]^[9] This relation arises from the definitions: \theta(x) sums \log p over primes p \leq x, while \psi(x) extends this to all prime powers p^m \leq x with multiplicity m, grouping terms by the exponent k = m.^[12] The difference between the functions follows directly as

\psi(x) - \theta(x) = \sum_{k=2}^\infty \theta\left(x^{1/k}\right),

with the tail bounded by \psi(x) - \theta(x) = O(\sqrt{x} \log x), reflecting the rapid decay of higher powers.^[9]^[12] This bound ensures that the contribution from prime powers beyond the first is negligible compared to the main term for large x. Consequently, \psi(x) \sim \theta(x) as x \to \infty, as the difference is asymptotically smaller than either function's leading growth.^[9] For practical numerical computation, the finite number of terms O(\log x) in the summation allows efficient approximation of \psi(x) from tabulated or computed values of \theta(x^{1/k}) for small integers k, a method used in algorithmic prime counting; the inverse relation via Möbius inversion,

\theta(x) = \sum_{k=1}^\infty \mu(k) \psi\left(x^{1/k}\right),

enables similar approximations in the reverse direction.^[3]^[12] This summation relation, first established by Chebyshev in his 1850 memoir on prime numbers, played a key role in early analytic number theory by linking the behaviors of \theta(x) and \psi(x), allowing bounds on one to imply results for the other and advancing toward the prime number theorem.^[9]^[12]

Relation to the logarithmic integral

The Chebyshev functions θ(x) and ψ(x) play a central role in approximating the distribution of primes, linking directly to the logarithmic integral li(x), which serves as the primary asymptotic for the prime counting function π(x). The prime number theorem asserts that π(x) ∼ li(x) as x → ∞, where li(x) is defined as the Cauchy principal value

\li(x) = \lim_{\epsilon \to 0^+} \left( \int_0^{1-\epsilon} \frac{\mathrm{d}t}{\log t} + \int_{1+\epsilon}^x \frac{\mathrm{d}t}{\log t} \right).

This equivalence holds because θ(x) ∼ x and ψ(x) ∼ x, with ψ(x) incorporating higher prime powers but asymptotically equivalent to θ(x). Through partial summation, one obtains

\pi(x) = \frac{\theta(x)}{\log x} + \int_2^x \frac{\theta(t)}{t (\log t)^2} \, \mathrm{d}t,

and substituting θ(t) ∼ t yields π(x) ∼ li(x), as the integral approximates the tail of li(x). The ratio ψ(x)/x ≈ 1 reflects the prime density around 1/log x, since ψ(x) ≈ ∫_2^x log t , \mathrm{d}\pi(t) ≈ ∫_2^x \mathrm{d}t = x - 2 under this density.^[13] Historically, Pafnuty Chebyshev employed θ(x) in his 1852 memoir to derive explicit bounds for π(x), showing that if lim_{x→∞} π(x) log x / x exists, it equals 1, with 0.92129 < lim inf ≤ lim sup < 1.10555. These bounds involved integral expressions akin to variations of li(x), such as lower estimates exceeding ∫_2^x dt / log t minus a small error term, providing early evidence for the prime number theorem's form without complex analysis. Chebyshev's approach bridged elementary estimates to the logarithmic scale, influencing later refinements by Hadamard and de la Vallée Poussin in 1896.^[14]^[15] The von Mangoldt explicit formula further connects ψ(x) to li(x) via oscillatory terms: briefly, ψ(x) - x ≈ ∑_ρ x^ρ / ρ, where ρ are the nontrivial zeros of the Riemann zeta function, with the sum's magnitude controlling the error in both ψ(x) ∼ x and π(x) ∼ li(x). Assuming the Riemann hypothesis, sharper bounds hold, such as |ψ(x) - x| ≤ 0.83 √x log x for x ≥ 2. Unconditionally, analytic methods yield bounds like |ψ(x) - x| < x exp(-√(log x)/5.7) for x ≥ exp(10,000), verified via zero-free regions.^[16] Computations using Lagarias-Odlyzko methods confirm these relations with high precision for large x, such as up to x ≈ 10^{32} as of 2016, underscoring ψ(x)/x ≈ 1 and the minimal deviation of li(x) from π(x). These results bridge Chebyshev's foundational estimates to modern explicit error controls.^[17]^[16]

Asymptotic Behavior

Main asymptotics

The prime number theorem asserts that the Chebyshev functions satisfy \theta(x) \sim x and \psi(x) \sim x as x \to \infty. These relations are equivalent to the classical form \pi(x) \sim x / \log x for the prime-counting function \pi(x).^[9]^[18] The derivation of this asymptotic stems from the Euler product formula for the Riemann zeta function, \zeta(s) = \prod_p (1 - p^{-s})^{-1} for \operatorname{Re}(s) > 1, where the product runs over all primes p. Taking the natural logarithm gives

\log \zeta(s) = -\sum_p \log(1 - p^{-s}) = \sum_p \sum_{k=1}^\infty \frac{1}{k} p^{-ks},

which expands into a Dirichlet series whose coefficients involve the von Mangoldt function \Lambda(n), directly linking to \psi(x) = \sum_{n \le x} \Lambda(n). The limit \psi(x)/x \to 1 follows from analyzing the behavior of \log \zeta(s) near s=1 combined with analytic continuation and growth estimates.^[19]^[20] The first rigorous proof of these asymptotics was provided by Charles Jean de la Vallée Poussin in 1896, building on Riemann's ideas. He established a zero-free region for \zeta(s) to the right of the line \operatorname{Re}(s) = 1, specifically \sigma > 1 - c / \log(|t| + 2) for some constant c > 0 and |t| \ge 2, where \sigma = \operatorname{Re}(s). This region ensures that \zeta(s) does not vanish close to the critical line, allowing Tauberian theorems or contour integration to yield \psi(x) \sim x and hence the prime number theorem.^[21] The relation between \theta(x) and \pi(x) arises via partial summation: since \theta(x) = \int_2^x \log t \, d\pi(t), integration by parts gives \pi(x) = \theta(x)/\log x + \int_2^x \theta(t)/t (\log t)^2 \, dt. Under \theta(x) \sim x, the dominant term yields \pi(x) \sim x / \log x, with the integral contributing a lower-order term; equivalently, \pi(x) \sim \operatorname{li}(x), where \operatorname{li}(x) = \int_0^x dt / \log t is the logarithmic integral (principal value).^[9]^[22] Computational verifications continue to affirm these asymptotics to high precision. For instance, explicit bounds improving on the main term have been derived, confirming \psi(x)/x \to 1 with quantified errors for all x \ge 1, consistent with numerical evaluations up to large scales.^[17]

Error terms and bounds

The Chebyshev functions satisfy certain elementary inequalities established by Chebyshev in his seminal 1852 memoir. Specifically, for the first Chebyshev function, there exist positive constants A < 1 < B such that A x < \theta(x) < B x for sufficiently large x, with explicit values like $0.92129 x < \theta(x) < 1.10555 x holding for x \geq 1. These bounds demonstrate that \theta(x) \sim x in a weak sense but fall short of the full prime number theorem, as the constants do not approach 1. Building on this, the classical proof of the prime number theorem by de la Vallée Poussin in 1899 introduced a zero-free region for the Riemann zeta function \zeta(s), namely \sigma > 1 - \frac{c}{\log(|t| + 2)} for some c > 0, which yields the unconditional error bound \psi(x) = x + O\left(x \exp\left(-c \sqrt{\log x}\right)\right) for the second Chebyshev function. This bound arises from integrating the logarithmic derivative of \zeta(s) over a suitable contour avoiding the zero-free region, establishing the asymptotic \psi(x) \sim x with a subexponential error term.^[23] Similar estimates hold for \theta(x), with \theta(x) = x + O\left(x \exp\left(-c \sqrt{\log x}\right)\right), confirming the prime number theorem without the Riemann hypothesis.^[12] Significant improvements came from zero-density estimates developed independently by Korobov and Vinogradov in 1958, providing a wider zero-free region of the form \sigma > 1 - c (\log t)^{-2/3} (\log \log t)^{-1/3}. This leads to the enhanced unconditional bound \psi(x) = x + O\left(x \exp\left(-c (\log x)^{3/5} (\log \log x)^{-1/5}\right)\right), which is sharper for large x and has been made explicit in subsequent works with optimized constants. These methods rely on bounds for Dirichlet polynomials and have influenced modern analytic number theory. Despite these upper bounds, the error term exhibits persistent oscillations, as shown by Littlewood in 1914: \psi(x) - x = \Omega_\pm \left( \sqrt{x} \log \log \log x \right), meaning the error changes sign infinitely often and achieves both positive and negative values of this magnitude. This \Omega-result underscores that the error cannot be improved to o(\sqrt{x}) unconditionally, highlighting the limitations of zero-free regions alone.^[24] In the 2020s, advances in subconvexity bounds for \zeta(1/2 + it) have indirectly refined estimates for the Chebyshev functions by improving approximations in the critical strip. For instance, Hiary, Patel, and Yang (2022) established an explicit subconvex bound |\zeta(1/2 + it)| \ll t^{1/6} (\log t)^{2/9} \exp(0.012 \sqrt{\log t}) for t \geq 2, surpassing classical Weyl-type estimates and enabling tighter explicit versions of the Korobov-Vinogradov error terms for \psi(x).^[25] Building on such progress, Fiori, Kadiri, and Swidinsky (2023) derived sharper unconditional explicit bounds, including \psi(x) = x + O(x^{0.525} \log x) for all x \geq 2.^[26] These developments support ongoing efforts to optimize unconditional bounds for prime distribution.

Exact Formulas

Von Mangoldt explicit formula

The Von Mangoldt explicit formula expresses the smoothed second Chebyshev function \psi_0(x), defined as \psi_0(x) = \frac{1}{2} \left( \lim_{\epsilon \to 0^+} \psi(x + \epsilon) + \lim_{\epsilon \to 0^+} \psi(x - \epsilon) \right) for non-integer x > 0 and \psi_0(x) = \psi(x) at integers, exactly in terms of the non-trivial zeros of the Riemann zeta function \zeta(s). For x > 1,

\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \log(2\pi) - \frac{1}{2} \log\left(1 - x^{-2}\right),

where the sum runs over all non-trivial zeros \rho of \zeta(s), counted with multiplicity, and the series converges conditionally.^[27] This formula was rigorously proved by Hans von Mangoldt in 1895, providing the first exact non-asymptotic relation between the distribution of primes and the zeta zeros; von Mangoldt also derived an explicit formula for the prime-counting function \pi(x) by integrating the expression for \psi_0(x).^[28] The derivation begins with the Dirichlet series representation -\zeta'(s)/\zeta(s) = \sum_{n=1}^\infty \Lambda(n) n^{-s} for \operatorname{Re}(s) > 1, where \Lambda(n) is the von Mangoldt function. Perron's inversion formula is then applied to express \psi_0(x) as a contour integral of -\zeta'(s)/\zeta(s) \cdot x^s / s over a vertical line in the critical strip, with the integral evaluated by shifting the contour to the left and collecting residues at the zeros and poles of \zeta(s), yielding the explicit sum over zeros plus contributions from the trivial zeros (captured in the logarithmic terms). To approximate \psi_0(x) numerically, the infinite sum over zeros can be truncated to the first N zeros \rho_1, \dots, \rho_N, with the remainder error bounded by the tail \sum_{|\operatorname{Im} \rho| > T} |x^\rho / \rho|, where T \approx N \log N reflects the zero density; this error is typically O(\sqrt{x} \log^2 x) for large x, allowing practical computations when combined with known zero tabulations up to heights of order $10^{32}. The formula's significance lies in its exact linkage of prime distribution—via the summatory function \psi_0(x)—to the precise locations of zeta zeros, enabling direct study of oscillatory deviations in prime counts without asymptotic assumptions.^[28]

Smoothing variants

The smoothed variant of the Chebyshev function, denoted \psi_0(x), addresses the discontinuities inherent in \psi(x) at prime powers by averaging the left and right limits: for non-integer x > 0,

\psi_0(x) = \frac{1}{2} \left( \lim_{\epsilon \to 0^+} \psi(x + \epsilon) + \lim_{\epsilon \to 0^+} \psi(x - \epsilon) \right),

and \psi_0(x) = \psi(x) at integers x.^[13] This definition renders \psi_0(x) continuous while preserving the asymptotic behavior of \psi(x), facilitating smoother approximations in analytic expressions.^[20] Von Mangoldt's explicit formula from 1895 is originally stated for this smoothed form \psi_0(x):

\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \log(2\pi) - \frac{1}{2} \log\left(1 - x^{-2}\right), \quad x > 1,

where the sum runs over the non-trivial zeros \rho of the Riemann zeta function. This incorporates a kernel-like averaging in the oscillatory terms, mitigating the Gibbs phenomenon that would arise from abrupt jumps in the unsmoothed \psi(x). Riemann sketched a precursor to this smoothed approach in his 1859 memoir on the zeta function, envisioning the prime distribution through zero sums without full rigor. Further smoothing variants integrate \psi(t) to enhance continuity, such as \psi_1(x) = \int_2^x \psi(t) \, dt or the weighted form \int_0^\infty \frac{\psi(x + h) - \psi(x - h)}{2h} \, dh for interval approximations like \psi(x + h) - \psi(x - h) \approx 2h + \sum_{\rho} \int \frac{x^{\rho}}{\rho} k(h) \, dh, where k(h) is a smoothing kernel (e.g., a Gaussian or rectangular window). These reduce step-like errors in the explicit sum, improving convergence in numerical evaluations.^[13] Such variants enable precise numerical computations of \psi_0(x) for large x, essential for error analysis in prime gap estimates and verification of zeta zero locations. For instance, they support bounds on gaps between primes in short intervals under the Riemann hypothesis, with error terms controlled via the smoothed oscillations.^[29] Modern refinements leverage Hiary's algorithm for rapid evaluation of \zeta(1/2 + it), allowing efficient summation over millions of zeros to compute \psi_0(x) up to x \approx 10^{30} with relative errors below $10^{-10}. These techniques have advanced large-scale verifications of the Riemann hypothesis through 2025, confirming zero alignments via smoothed explicit sums at unprecedented heights.^[25]

Applications to Number Theory

Connection to prime counting

The second Chebyshev function \psi(x) provides a weighted measure of primes and their powers up to x, defined as \psi(x) = \sum_{n \leq x} \Lambda(n), where \Lambda(n) is the von Mangoldt function. This function connects directly to the prime counting function \pi(x), which enumerates the primes p \leq x, through the Stieltjes integral representation

\psi(x) = \int_{1}^{x} \log t \, d\pi(t).

This relation arises because d\pi(t) jumps by 1 at each prime, and the logarithmic weighting aligns with \Lambda(n) = \log p for prime powers n = p^k.^[9] Applying integration by parts to the Stieltjes integral yields

\psi(x) = \pi(x) \log x - \int_{2}^{x} \frac{\pi(t)}{t} \, dt,

assuming the lower limit adjustment for convenience (with \pi(t) = 0 for t < 2). Rearranging gives an expression for \pi(x):

\pi(x) = \frac{\psi(x)}{\log x} + \frac{1}{\log x} \int_{2}^{x} \frac{\pi(t)}{t} \, dt.

Although this form is implicit, it highlights how \psi(x) drives the growth of \pi(x). The asymptotic \psi(x) \sim x implies \pi(x) \sim x / \log x, but more refined analysis leads to the stronger equivalence \pi(x) \sim \mathrm{li}(x), where \mathrm{li}(x) = \int_{2}^{x} \frac{dt}{\log t} is the logarithmic integral.^[9] A detailed derivation of \pi(x) from \psi(x) (or the first Chebyshev function \theta(x) = \sum_{p \leq x} \log p) employs the partial summation formula, the discrete analogue of integration by parts. Let A(y) = \sum_{n \leq y} a_n be a cumulative sum and f a smooth function; then

\sum_{n \leq x} a_n f(n) = A(x) f(x) - \int_{1}^{x} A(t) f'(t) \, dt.

For \theta(x), set a_p = \log p for primes p and a_n = 0 otherwise, so A(x) = \theta(x) and f(t) = 1 / \log t. This yields

\sum_{p \leq x} 1 = \pi(x) = \frac{\theta(x)}{\log x} + \int_{2}^{x} \frac{\theta(t)}{t (\log t)^2} \, dt.

Since \theta(x) \sim x and \psi(x) = \theta(x) + O(\sqrt{x} \log x), substituting the asymptotic for \theta(t) into the integral approximates \pi(x) \approx \int_{2}^{x} \frac{dt}{(\log t)^2} + \frac{x}{(\log x)^2} \sim \mathrm{li}(x). The higher powers in \psi(x) contribute negligibly to this leading behavior.^[9] Numerically, \psi(x) / \log x offers a basic approximation to \pi(x). For x = 10^6, \pi(10^6) = 78498, while since \psi(10^6) \approx 10^6, the ratio \psi(10^6) / \log(10^6) \approx 10^6 / \ln(10^6) \approx 72382, yielding a relative error of about 7.8%. This demonstrates the approximation's utility even at moderate scales, improving for larger x as the asymptotic \psi(x) \sim x sharpens. The more precise \mathrm{li}(10^6) \approx 78628 reduces the error to under 0.2%.^[30]^[31] Explicit bounds on \psi(x) translate to rigorous inequalities for \pi(x). For instance, Dusart established that |\psi(x) - x| < 59.18 \, x / \ln^4 x for x \geq 2, which implies tight controls on the error in \pi(x) - \mathrm{li}(x). Using this, for x \geq 599, \pi(x) \leq (x / \ln x) (1 + 1/\ln x + 1.2762 / \ln x), providing verifiable upper bounds beyond mere asymptotics. Such results fill gaps in early estimates by enabling precise computations and verifications for prime distribution up to large x.^[31]

Connection to primorials

The primorial associated with the nth prime p_n is defined as p_n\# = \prod_{k=1}^n p_k, and its natural logarithm equals the Chebyshev function evaluated at p_n: \log(p_n\#) = \theta(p_n).^[3] This relation extends to a continuous analogue, where the product of all primes up to x satisfies \prod_{p \leq x} p = \exp(\theta(x)), providing an exponential representation of the cumulative logarithmic contribution of primes up to x.^[3] This logarithmic connection ties directly into Mertens' theorems on prime products. Specifically, the asymptotic \theta(x) \sim x from the prime number theorem implies \prod_{p \leq x} p \sim e^x, establishing the scale of primorial growth as roughly exponential in x.^[32] Mertens' third theorem, \prod_{p \leq x} \left(1 - \frac{1}{p}\right) \sim \frac{e^{-\gamma}}{\log x} (where \gamma is the Euler-Mascheroni constant), complements this by bounding the reciprocal density of primes, indirectly supporting estimates for \theta(x) through summation techniques.^[33] For small values, the relation is explicit and computable. Consider the first four primes: p_4 = 7 and $7\# = 2 \times 3 \times 5 \times 7 = 210, so \log(210) = \theta(7) = \log 2 + \log 3 + \log 5 + \log 7 \approx 5.347. Similarly, for p_{10} = 29 (primes up to 29), $29\# is the product of the first ten primes equaling 6,469,693,230, with \log(29\#) = \theta(29) \approx 22.590. These examples illustrate how \theta(x) precisely measures the logarithm of primorial-like products.^[3] In applications, primorials leverage this connection in sieve theory, where they act as moduli for sieving intervals by small primes; the size of such moduli is quantified logarithmically via \theta(x), aiding bounds on sifted sets.^[34] For factorial approximations, the divisibility p_n\# \mid n! (since all primes up to p_n \leq n divide n!) and comparable growth \log(n!) \sim n \log n \sim \theta(p_n) enable inequalities bounding factorials by primorials, useful in estimating prime factors in n!.^[35]

Implications for the Riemann hypothesis

The Riemann hypothesis (RH) asserts that all non-trivial zeros ρ of the Riemann zeta function ζ(s) satisfy Re(ρ) = 1/2. This condition implies a strong bound on the error term in the asymptotic expansion of the Chebyshev function, specifically ψ(x) = x + O(√x log x) for x ≥ 2.^[36] The proof relies on the explicit formula linking ψ(x) to the zeros of ζ(s), where the contribution from off-critical-line zeros would dominate the error otherwise. A closely related result due to Littlewood establishes an equivalence between RH and a slightly weaker bound: RH holds if and only if ψ(x) - x = O(√x log² x) for all sufficiently large x. This equivalence arises from refining the explicit formula and analyzing the oscillatory terms from the zeros, showing that the log² x factor accounts for the density of zeros on the critical line. Under RH, the explicit formula for ψ(x) also yields implications for prime gaps. In particular, RH implies that π(x + x^{1/2 + ε}) - π(x) ≫ log x for any fixed ε > 0 and sufficiently large x, meaning short intervals of length roughly √x contain asymptotically many primes. This follows from the controlled oscillations in the explicit formula, ensuring the prime distribution remains regular without large deviations. Unconditionally, Littlewood's Ω-results demonstrate the sharpness of the RH bound: ψ(x) - x = Ω(√x (log log log x)/log x), showing that the error term cannot be improved beyond √x up to logarithmic factors, even if RH holds. These lower bounds contrast with the O-bound under RH, highlighting that RH provides the optimal control on the error. As of 2021, numerical verifications of RH through computations of zeta zeros have confirmed no off-line zeros up to height ~3×10^{12}, supporting consistency with the RH error term for ψ(x) up to extremely large x.^[37]

References

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