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Register a new userThere are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime
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Authors
Peter Gustav Lejeune Dirichlet
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Dirichlets proof of infinitely many primes in arithmetic progressions was published in 1837, introduced L-series for the first time, and it is said to have started rigorous analytic number theory. Dirichlet uses Eulers earlier work on the zeta function and the distribution of primes. He first proves a simpler case before going to full generality. The paper was translated from German by R. Stephan and given a reference section.
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A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2le lvert Xrvert <lvert V(G)rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.
Celebrated theorems of Roth and of Matouv{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $Theta(n^{1/4})$. We study the analogous problem in the $mathbb{Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $mathbb{Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $mathbb{Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $mathbb{Z}_n$ is $Theta(n^{1/3+r_k/(6k)})$, where $r_k in {0,1,2}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav.
We discuss the uniqueness or non-uniqueness problem of the decomposition of the gluon field into the physical and pure-gauge components, which is the basis of the recently proposed two physically inequivalent gauge-invariant decompositions of the nucleon spin. It is crucialy important to recognize the fact that the standard gauge fixing procedure is essentially a process of projecting out the physical components of the massless gauge field. A complexity of the nonabelian gauge theory as compared with the abelian case is that a closed expression for the physical component can be given only with use of the non-local Wilson line, which is generally path-dependent. It is known that, by choosing an infinitely long straight-line path in space and time, the direction of which is characterized by a constant 4-vector $n^mu$, one can cover a class of gauge called the general axial gauge, containing three popular gauges, i.e. the temporal, the light-cone, and the spatial axial gauge. Within this general axial gauge, we have calculated the 1-loop evolution matrix for the quark and gluon longitudinal spins in the nucleon. We found that the final answer is exactly the same independently of the choices of $n^mu$, which amounts to proving the gauge-independence and path-independence simultaneously, although within a restricted class of gauges and paths. By drawing on all of these findings together with well-established knowledge from gauge theories, we argue against the rapidly spreading view in the community that there are infinitely many decompositions of the nucleon spin.
We argue against the rapidly spreading idea of gauge-invariant-extension (GIE) approach in the nucleon spin decomposition problem, which implies the existence of infinitely many gauge-invariant decomposition of the nucleon spin.
This is an English translation of the Latin original De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ signum negativum (1775). E596 in the Enestrom index. Let $chi$ be the nontrivial character modulo 4. Euler wants to know what $sum_p chi(p)/p$ is, either an exact expression or an approximation. He looks for analogies to the harmonic series and the series of reciprocals of the primes. Another reason he is interested in this is that if this series has a finite value (which is does, the best approximation Euler gets is 0.3349816 in section 27) then there are infinitely many primes congruent to 1 mod 4 and infinitely many primes congruent to 3 mod 4. In section 15 Euler gives the Euler product for the L(chi,1). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $sum_p frac{chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshevs estimate for the second Chebyshev function (summing the von Mangoldt function).
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