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Von Mangoldt function

The von Mangoldt function, denoted by Λ(n), is an defined on the positive s n such that Λ(n) = \log p if n = p^k for a prime p and positive integer k ≥ 1, and Λ(n) = 0 otherwise. Introduced by the German mathematician Hans Carl Friedrich von Mangoldt in 1895, it serves as a key tool in for encoding information about prime powers. The function's significance stems from its connection to the distribution of prime numbers, particularly through the Chebyshev function ψ(x) = \sum_{n \leq x} Λ(n), which sums the values of Λ(n) up to x and asymptotically equals x as x → ∞, reflecting the prime number theorem. Von Mangoldt's original work established an explicit formula linking ψ(x) to the non-trivial zeros of the Riemann zeta function ζ(s), providing a precise oscillatory description of prime distribution that depends on the locations of these zeros. This formula, later refined, underscores the deep interplay between Λ(n) and the Riemann hypothesis, which posits that all non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2; under this assumption, error terms in estimates for ψ(x) - x can be bounded by O(\sqrt{x} \log^2 x). Beyond primes, generalizations of the von Mangoldt function, such as the k-th power Λ_k(n) = \sum_{d \mid n} \mu(d) \log^k (n/d) where μ is the , extend its applications to higher moments and L-functions, aiding in advanced studies of arithmetic progressions and sieve methods.

Definition and Basic Properties

Definition

The von Mangoldt function, denoted \Lambda(n), is an arithmetic function defined on the positive integers n by \Lambda(n) = \begin{cases} \log p & \text{if } n = p^k \text{ for some prime } p \text{ and integer } k \geq 1, \\ 0 & \text{otherwise}. \end{cases} This definition assigns the natural logarithm of the prime to every power of that prime, thereby encoding the logarithmic contribution of prime factors while vanishing on numbers that are either 1 or products of distinct primes raised to powers greater than 1 in a way that does not fit the pure prime power form. The function plays a key role in by distinguishing prime powers from composite numbers lacking such structure, facilitating the weighted summation over primes in asymptotic estimates. It weights primes logarithmically, providing a natural connection to the \pi(x). Introduced by Hans von Mangoldt in 1895 in the context of advancing the , the function arose in his analysis of the distribution of primes via the . For example, \Lambda(1) = 0, \Lambda(p) = \log p for any prime p, \Lambda(p^2) = \log p, and \Lambda(6) = 0 since 6 is not a prime power.

Arithmetic Properties

The von Mangoldt function interacts intimately with the natural logarithm through the framework of Dirichlet convolution and Möbius inversion. Specifically, the Dirichlet convolution of Λ with the constant function 1(n) = 1 for all positive integers n yields the logarithm: (\Lambda * 1)(n) = \sum_{d \mid n} \Lambda(d) = \log n. By the Möbius inversion theorem applied to this relation, one obtains the explicit formula \Lambda(n) = \sum_{d \mid n} \mu(d) \log(n/d), where μ is the Möbius function. This inversion highlights the arithmetic structure of Λ as a "deconvolution" of the logarithm over the divisors. A key combinatorial application arises in expressing the logarithm of the . For a positive x, one has the exact identity \log(x!) = \sum_{n \leq x} \Lambda(n) \left\lfloor \frac{x}{n} \right\rfloor. This follows by substituting \log m = \sum_{d \mid m} \Lambda(d) into \log(x!) = \sum_{m=1}^x \log m, then interchanging the sums to count the multiplicity \lfloor x/n \rfloor for each contribution of Λ(n). In the context of , \log(x!) \sim x \log x - x + \frac{1}{2} \log(2\pi x), this identity provides a bridge to asymptotic estimates involving weighted s of Λ(n). By definition, Λ(n) = \log p if n = p^k for a prime p and integer k ≥ 1, and Λ(n) = 0 otherwise. Thus, Λ vanishes on all integers that are not , including every with two or more distinct prime factors. This support property underscores the function's focus on prime power contributions in arithmetic identities.

Analytic Representations

Dirichlet Series

The Dirichlet series generating function for the von Mangoldt function \Lambda(n) is \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s} = -\frac{\zeta'(s)}{\zeta(s)} for \operatorname{Re}(s) > 1, where \zeta(s) denotes the . This relation follows from the Euler product formula \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 yields \log \zeta(s) = -\sum_p \log(1 - p^{-s}) = \sum_p \sum_{k=1}^\infty \frac{p^{-ks}}{k}. Differentiating both sides with respect to s gives \frac{\zeta'(s)}{\zeta(s)} = \sum_p \sum_{k=1}^\infty -\log p \cdot p^{-ks}, and thus -\frac{\zeta'(s)}{\zeta(s)} = \sum_p \sum_{k=1}^\infty \log p \cdot p^{-ks} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}, since the terms \log p \cdot p^{-ks} collect precisely as \Lambda(n) n^{-s} over prime powers n = p^k. The -\zeta'(s)/\zeta(s) enables of the to the critical strip $0 < \operatorname{Re}(s) < 1, where it exhibits a simple pole at s=1 and simple poles at the non-trivial zeros of \zeta(s), thereby mirroring the zero and pole structure of \zeta(s) itself. The zeros of -\zeta'(s)/\zeta(s) in this region occur where \zeta'(s) = 0 but \zeta(s) \neq 0, corresponding to the critical points of \zeta(s). Near s=1, the series diverges asymptotically as $1/(s-1), reflecting the simple pole of \zeta(s) at this point and connecting to the growth of the prime harmonic series \sum_{p \leq x} 1/p \sim \log \log x. This pole at s=1 is central to the proof of the .

Exponential Representation

A formal representation of the von Mangoldt function \Lambda(n) can be derived by applying the Mellin inversion theorem to its Dirichlet series. For \operatorname{Re}(s) > 1, -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}. Formally inverting this series yields the contour integral \Lambda(n) = \frac{1}{2\pi i} \int_{c - i\infty}^{c + i\infty} -\frac{\zeta'(s)}{\zeta(s)} \, n^{s} \, ds, where c > 1 is real. This expression involves the logarithmic derivative of the Riemann zeta function, with the term n^{s} = e^{s \log n} highlighting the exponential nature tied to the complex variable s. However, unlike absolutely convergent Dirichlet series, this integral does not provide a pointwise value for \Lambda(n) upon shifting the contour to the left, as the contributions from the shifted line do not vanish, and the resulting sum over residues (at s=1 and the zeros \rho of \zeta(s), plus trivial zeros) does not converge to \Lambda(n). Rigorous explicit formulas, involving sums over the non-trivial zeros, are instead available for the summatory function \psi(x) = \sum_{n \leq x} \Lambda(n), obtained via Perron's formula (which includes an extra x^s / s factor); see the "Explicit Formulas" section for details. These formal representations underscore the oscillatory behavior of \Lambda(n), linked to the non-trivial zeros \rho = \beta + i \gamma of \zeta(s), where terms like n^{\rho} = n^{\beta} e^{i \gamma \log n} produce waves modulated by the imaginary parts \gamma. In the critical strip $0 < \operatorname{Re}(s) < 1, where the non-trivial zeros lie on or near the critical line \operatorname{Re}(s) = 1/2, shifting the contour requires care to avoid zeros. Under the Riemann hypothesis, all non-trivial zeros lie on \operatorname{Re}(s) = 1/2, aiding error estimates in approximations. Such forms are useful in additive number theory for approximating exponential sums \sum_n \Lambda(n) e(\alpha n) via the circle method, where zeta zeros contribute to major and minor arc estimates.

Summatory Functions

Chebyshev Function

The Chebyshev function, denoted \psi(x), is defined as the summatory function of the \Lambda(n): \psi(x) = \sum_{n \leq x} \Lambda(n). This sum can be equivalently expressed as a weighted sum over prime powers, since \Lambda(n) = \log p if n = p^k for a prime p and integer k \geq 1, and zero otherwise: \psi(x) = \sum_{p^k \leq x} \log p, where the inner sum runs over all primes p and positive integers k such that p^k \leq x. This representation highlights its role in capturing the logarithmic contributions from primes and their powers up to x. In 1850, Chebyshev established the first explicit bounds for \psi(x), proving that there exist positive constants a and A such that a x < \psi(x) < A x for sufficiently large x. These bounds provided early evidence for the density of primes and were instrumental in supporting . Specifically, Chebyshev showed $0.92129 \frac{x}{\log x} < \pi(x) < 1.10555 \frac{x}{\log x}, which implies the corresponding inequalities for \psi(x) via integration or summation techniques. The prime number theorem asserts that \psi(x) \sim x as x \to \infty, meaning \lim_{x \to \infty} \psi(x)/x = 1. This equivalence to the prime number theorem was proved independently by Hadamard and de la Vallée Poussin in 1896. The classical error term in this asymptotic is \psi(x) = x + O\left(x \exp\left(-c \sqrt{\log x}\right)\right) for some constant c > 0, established by de la Vallée Poussin in his analysis of the Riemann zeta function's zero-free region. The function \psi(x) is closely related to the prime-counting function . Indeed, \psi(x) = \sum_{k=1}^\infty \sum_{p \leq x^{1/k}} \log p = \sum_{p \leq x} \log p + \sum_{p \leq \sqrt{x}} \log p + \sum_{p \leq x^{1/3}} \log p + \cdots, with higher-order terms becoming negligible. By partial summation, the for \pi(x) \sim x / \log x follows from \psi(x) \sim x, as the dominant contribution is \sum_{p \leq x} \log p \sim x.

Riesz Mean

The Riesz mean provides a smoothed version of the \psi(x) = \sum_{n \le x} \Lambda(n), where \Lambda is the , by applying fractional techniques to enhance analytic properties and in Tauberian arguments. It is defined as R_\alpha(x; \Lambda) = \frac{1}{\Gamma(\alpha+1)} \int_0^x (x-t)^\alpha \, d\psi(t) for \alpha > -1. This form corresponds to the discrete sum \frac{1}{\Gamma(\alpha+1)} \sum_{n \le x} (x-n)^\alpha \Lambda(n), leveraging the step-function nature of \psi(t). The asymptotic behavior is R_\alpha(x; \Lambda) \sim \frac{x^{\alpha+1}}{\alpha+1}, reflecting its role as a fractional of the leading term \psi(t) \sim t. This connection to fractional facilitates the analysis of higher-order smoothing effects on the distribution of prime powers encoded in \Lambda(n). For \alpha = 0, the Riesz mean reduces to R_0(x; \Lambda) = \psi(x), as \Gamma(1) = 1 and (x-t)^0 = 1. This case is foundational in Tauberian theorems, where the of the -\zeta'(s)/\zeta(s) = \sum \Lambda(n) n^{-s} near \operatorname{Re}(s) = 1 implies \psi(x) \sim x, yielding the via Ikehara's theorem or related Riesz-type results. Compared to Cesàro means, which average partial sums uniformly up to order k, Riesz means employ polynomial weights (x-t)^\alpha, providing a more flexible smoothing that better isolates contributions from prime powers in \Lambda(n) by attenuating oscillations and improving convergence in explicit formulas. The Riesz mean also aids in deriving refined error terms for \psi(x), such as O(\sqrt{x} \log^2 x) under the , by transforming the problem into one with superior summability properties.

Explicit Formulas

Von Mangoldt Formula

The von Mangoldt explicit formula provides an exact expression for the \psi(x), linking it directly to the zeros of the and other arithmetic terms. For x > 1, it states that \psi(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \log(2\pi) - \frac{1}{2} \log(1 - x^{-2}), where the sum is over the non-trivial zeros \rho of the zeta function \zeta(s), counted with multiplicity. This formula encapsulates the oscillatory behavior of \psi(x) arising from the non-trivial zeros, while the remaining terms account for the main growth and contributions from trivial . The formula was rigorously established in 1895 by Hans von Mangoldt, who provided the first complete proof building on Bernhard Riemann's 1859 in his seminal on prime . Riemann had outlined the connection between primes and zeros but left key analytic details unresolved; von Mangoldt's work filled these gaps using advanced techniques available by the late . The derivation proceeds via in the . Specifically, consider the integral \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} -\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} \, ds for c > 1, which equals \psi(x) by Perron's formula and the for -\zeta'/ \zeta(s). Shifting the contour to the left encloses the of the integrand: the at s=1 yields the term x; residues at the non-trivial zeros \rho give the sum -\sum_{\rho} x^{\rho}/\rho; the at s=0 contributes -\log(2\pi); and residues at the trivial zeros (negative even integers) sum to -\frac{1}{2} \log(1 - x^{-2}). The term -\log(2\pi) specifically arises from the simple pole of \zeta(s) at s=0, reflecting the residue computation at that point in the shifted . The non-trivial zeros introduce the primary oscillations in \psi(x), influencing the error in approximations.

Approximation via Zeta Zeros

One key application of the explicit formula involves approximating the Chebyshev function \psi(x) = \sum_{n \leq x} \Lambda(n) using a partial sum over the non-trivial zeros \rho of the . Specifically, for a height parameter T > 0, \psi(x) \approx x - \sum_{|\operatorname{Im} \rho| < T} \frac{x^\rho}{\rho}, with the error bounded by O\left( \frac{x \log^2 (x T)}{T} \right) when T is chosen sufficiently large relative to x. This truncation captures the primary oscillatory contributions from the zeros, allowing numerical computation of \psi(x) by evaluating known low-lying zeros, while the remainder term decreases as T increases. Under the Riemann Hypothesis, which posits that all non-trivial zeros lie on the critical line \operatorname{Re} \rho = 1/2, the full error \psi(x) - x simplifies dramatically to O(\sqrt{x} \log^2 x), reflecting the absence of zeros in zero-free regions to the right of this line. This bound ties directly to the distribution of the imaginary parts of the zeros, providing a quantitative measure of how closely \psi(x) follows its main term x, and has implications for the oscillation amplitude in prime distribution. While pointwise approximations for individual values \Lambda(n) can be expressed via similar sums over zeros, such as oscillatory terms of the form \sum_\rho n^{i \operatorname{Im} \rho} (normalized appropriately under RH), the primary utility lies in the summatory context for \psi(x), where these terms aggregate to reveal global patterns in prime spacing. Recent advancements, particularly post-2020, have connected these zeta zero sums to Gowers uniformity norms U^k[\Lambda] for the von Mangoldt function, establishing that the U^k norm of \Lambda (or an adjusted version) on intervals [N, 2N] is O((\log \log N)^{-c_k}) for some c_k > 0, thereby linking zero distributions to higher-order correlations in prime gaps and arithmetic uniformity.

Generalizations and Extensions

Generalized Von Mangoldt Function

The generalized von Mangoldt function \Lambda_k(n) for positive integers k \geq 1 is defined as \Lambda_k(n) = \sum_{d \mid n} \mu(d) \log^k \left( \frac{n}{d} \right), where \mu is the . This definition recovers the standard von Mangoldt function \Lambda(n) upon setting k=1. A key property is the summation formula \sum_{d \mid n} \Lambda_k(d) = \log^k n, which follows from inversion and serves as a higher-rank analog of the relation \sum_{d \mid n} \Lambda(d) = \log n. The associated is given by \sum_{n=1}^\infty \frac{\Lambda_k(n)}{n^s} = \left( -\frac{\zeta'(s)}{\zeta(s)} \right)^k for \Re(s) > 1, where \zeta(s) is the ; this extends the series -\zeta'(s)/\zeta(s) = \sum_{n=1}^\infty \Lambda(n)/n^s for the classical case. These functions find application in the analysis of k-free numbers, whose characteristic function involves the Möbius function over k-th powers, with \Lambda_k facilitating estimates for related divisor sums and asymptotic distributions. A further generalization arises in Beurling's framework of abstract semigroups with generalized primes and integers, where the von Mangoldt function is defined analogously as \Lambda(g) = \log |p| for elements g = p^k with p a generalized prime power k \geq 1, enabling prime number theorems in non-standard arithmetic settings.

Applications in Analytic Number Theory

The von Mangoldt function \Lambda(n) is fundamental to the proof of the , which asserts that \pi(x) \sim x / \log x as x \to \infty, where \pi(x) counts the primes up to x. This theorem is equivalent to the asymptotic \psi(x) := \sum_{n \leq x} \Lambda(n) \sim x, reflecting the density of primes weighted by their logarithms. The classical analytic proof, due to Hadamard and de la Vallée Poussin, establishes that -\zeta'(s)/\zeta(s) = \sum_{n=1}^\infty \Lambda(n) n^{-s} has no zeros in the half-plane \operatorname{Re}(s) > 1, ensuring the non-vanishing of the \zeta(s) there and yielding the desired asymptotic via Perron's formula or Tauberian theorems. In , the von Mangoldt function serves as a key weight in linear constructions, particularly for bounding prime gaps and detecting primes in structured sets. For example, in the GPY method extended to multidimensional settings, weights involving \Lambda(n) are applied to admissible tuples of linear forms, enabling the proof of infinitely many bounded gaps between primes by optimizing the sieve dimension and level of distribution. These techniques, building on earlier combinatorial sieves, leverage the pseudorandom behavior of \Lambda(n) to isolate prime contributions while suppressing composites. In the 2010s, studies focused on correlations between \Lambda(n) and the higher d_k(n), which counts the number of ways to write n as a product of k positive integers. Asymptotic formulas for sums such as \sum_{X < n \leq 2X} \Lambda(n) d_k(n + h) were derived for shifts h up to X^{1/2 - \epsilon} (and larger under the Elliott-Halberstam conjecture), revealing how primes interact multiplicatively with divisor structures at short distances. These results, employing the circle method, spectral analysis of the , and large sieve inequalities, extend classical shifted convolution problems and inform ongoing work on the distribution of primes in divisor-rich sequences. More recent work (as of 2024–2025) has explored the higher uniformity of the von Mangoldt function on short intervals and its decompositions in bilinear forms with trace functions. In the theory of Dirichlet L-functions L(s, \chi) associated with non-principal characters \chi modulo q, the von Mangoldt function generalizes via twisted sums \sum \Lambda(n) \chi(n) n^{-s} = -L'(s, \chi)/L(s, \chi), leading to explicit formulas that connect the partial sums \sum_{n \leq x} \Lambda(n) \chi(n) to the non-trivial zeros of L(s, \chi). These formulas underpin the for arithmetic progressions, quantifying the equidistribution of primes among residue classes coprime to q. Such generalizations facilitate applications to sieve problems over L-functions and estimates for character sums.