In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers. Our short paper uses a simple modification of Euler's argument to obtain new results about the distribution of prime factors of sets of integers, including a weak one-sided Tschebyshev inequality. An example shows that there cannot be a prime number theorem in this situation, or even a pair of Tschebyshev inequalities, but it would be very interesting to know if a one-sided Tschebyshev inequality holds.
In this paper we establish the Eulers criterion of order l (a prime) when the ring of integers in the cyclotomic extension of Q of order l is a PID. Conditions are obtained in terms of Jacobi sums of order l.
We study the distribution of primes from a topological viewpoint. Certain conjecture is introduced, and we show that it is equivalent to the Riemann Hypothesis.
For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n\le x such that q does not divide phi(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the prime k-tuples conjecture and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.
We present an elementary inductive proof which Euler could have obtained, for the corresponding result as the title indicates, had he refined a bit his proof for Fermat's assertion on representing primes as two squares.
This note provides truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$ when $F$ is a two-variable polynomial with coefficients in $\mathbb{C}$ and satisfying some restrictive conditions.
The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in the prime distribution.
An odd prime $p$ is called irregular with respect to Euler polynomials if it divides the numerator of one of the numbers $$E_1(0),E_{3}(0),\ldots,E_{p-2}(0),$$ where $E_n(x)$ is the $n$-th Euler polynomial. As in the classical case, we link the regularity of primes to the divisibility of some class numbers. Besides, we obtain some results on the distribution of such irregular primes. Remark: This preprint has been withdrawn, because all the results in it have been included or improved in a recent preprint. (See arXiv:1809.08431).
In this paper we extend the hybrid Euler-Hadamard product model for quadratic Dirichlet $L$-functions associated to irreducible polynomials over function fields. We also establish an asymptotic formula for the first twisted moment in this family of L-functions and then we provide further evidence for the conjectural asymptotic formulas for its moments.
Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number E_q(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K. Spearman and K.S. Williams [Ark. Mat. 44 (2006), 166-181]. Furthermore we resolve, under the Generalized Riemann Hypothesis, which of two approximations to E_q(x) is asymptotically superior using recent results of Y. Ihara on the Euler-Kronecker constant of a number field.
Let $p$ be an odd prime number, and let $E$ be an elliptic curve defined over a number field $F'$ such that $E$ has semistable reduction at every prime of $F'$ above $p$ and is supersingular at at least one prime above $p$. Under appropriate hypotheses, we compute the Akashi series of the signed Selmer groups of $E$ over a $\mathbb{Z}_p^d$-extension over a finite extension $F$ of $F'$. As a by-product, we also compute the Euler characteristics of these Selmer groups.
We prove unconditionally that for each $\ell \geq 1$, the difference $ϕ(p-\ell) - ϕ(p+\ell)$ is positive for $50\%$ of odd primes $p$ and negative for $50\%$.
The Euler's totient function $ φ(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ φ(n)\mid n-1 $. We also note that there are infinitely many primes that are not Sophie Germain primes.
In this article, we investigate how Euler might have been led to conjecture the Prime Number Theorem, based on what he knew. We also speculate on why he did not do so.
In a previous paper we proved that if an $L$-function $F$ from the Selberg class has degree $2$, its conductor $q_F$ is a prime number and $F$ is weakly twist-regular at all primes $p\neq q_F$, then $F$ has a polynomial Euler product. In this paper we extend this result to $L$-functions of degree 2 with square-free conductor $q_F$, which are weakly twist-regular at all primes $p\nmid q_F$
Let $p>3$ be a prime. Euler numbers $E_{p-3}$ first appeared in H. S. Vandiver's work (1940) in connection with the first case of Fermat Last Theorem. Vandiver proved that $x^p+y^p=z^p$ has no solution for integers $x,y,z$ with $\gcd(xyz,p)=1$ if $E_{p-3}\equiv 0 (\bmod p)$. Numerous combinatorial congruences recently obtained by Z.-W. Sun and by Z.-H. Sun involve the Euler numbers $E_{p-3}$. This gives a new significance to the primes $p$ for which $E_{p-3}\equiv 0 (\bmod p)$. For the computation of residues of Euler numbers $E_{p-3}$ modulo a prime $p$, we use the congruence which runs signi…
Let $p$ be an odd prime number, and $E$ an elliptic curve defined over a number field with good reduction at every prime of $F$ above $p$. In this short note, we compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\Zp$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and that we allow mixed signs in the definition of the signed Selmer groups.
We examine the sums $S(k,\,n)$ of the $k-$th powers of the $ϕ(n)$ integers $α_1<α_2<\cdots<α_{ϕ(n)}$ less than and prime to $n$ (Euler set) and prove a formula (new) for $S(3,\,n)$. If $n$ equals a prime $p$, we prove a theorem showing a connection of $S(3,\,p)$ with Artin's conjectural constant for primitive roots and with other functions involving primes.
Let $X$ be a set of positive integers, and let $\mathbb Z_K$ be the ring of integers of a number field $K$ of degree $n$. Denote by $N(I)$ the absolute norm of an ideal $I$ of $\mathbb Z_K$, and by $\mathcal A$ the set of principal ideals $a\mathbb Z_K$ such that $a$ is an atom of $\mathbb Z_K$ and $a$ divides $m$ for some $m \in X$. Building upon the ideas of Clarkson from [Proc. Amer. Math. Soc. 17 (1966), 541], we show that, if the series $\sum_{m \in X} 1/m$ diverges, then so does the series $\sum_{\mathfrak a \in \mathcal A} |N(\mathfrak a)|^{-1/n}$. Most notably, this generalizes a class…
We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the $k$-tuple conjecture and more general problems of polynomial representation of primes. We show that the singular series for the $k$-tuple conjecture have a limiting distribution when taken over $k$-tuples with (distinct) entries of growing size, and observe that its moments have a curious symmetry property. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in…
We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler-Kronecker constants $\mathfrak{G}_q$ for the prime cyclotomic fields $\mathbb{Q}(ζ_q)$, where $q$ is an odd prime and $ζ_q$ is a primitive $q$-root of unity. With such a new algorithm we evaluated $\mathfrak{G}_q$ and $\mathfrak{G}_q^+$, where $\mathfrak{G}_q^+$ is the Euler-Kronecker constant of the maximal real subfield of $\mathbb{Q}(ζ_q)$, for some very large primes $q$ thus obtaining two new negative values of $\mathfrak{G}_q$: $\mathfrak{G}_{9109334831}…
We define a family {$γ(P)$} of generalized Euler constants indexed by finite sets of primes $P$ and study their distribution. These arise from partial sums of reciprocals of integers not divisible by any prime in $P$. An apparent monotonicity is investigated. We also prove that a certain property of these numbers is equivalent to the Riemann Hypothesis.
An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. Similarly, an even perfect number $M$ is said to be given in Euclidean form if $M = (2^p - 1)\cdot{2^{p - 1}}$ where $p$ and $2^p - 1$ are primes. In this article, we show how simple considerations surrounding the differences between the underlying properties of the Eulerian and Euclidean forms of perfect numbers give rise to what we will call the Euclid-Euler heuristics for perfect numbers.
In this paper, we calculate the absolute tensor square of the Dirichlet $L$-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet $ L $-functions. This study is a generalization of Akatsuka's theorem on the Riemann zeta function, and gives a proof of Kurokawa's prediction proposed in 1992.
A prime $p$ is a Wolstenholme prime if $\binom{2p}{p}\equiv2$ mod $p^4$, or, equivalently, if $p$ divides the numerator of the Bernoulli number $B_{p-3}$; a Vandiver prime $p$ is one that divides the Euler number $E_{p-3}$. Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of $10$, and show that no additional Wolstenholme primes exist up to $10^{11}$, and in the second case by a factor of $20$, proving that no additional Vandiver primes occur up to this same bound. To facilitate this, we develop a number of new congruen…
it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.
On the second partial Global Euler-Poincare characteristics for Galois cohomology
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学复杂系统研究所）
In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.
We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $φ(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12, 14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $\mathbb{Q}(e^{2πi/q})$ is true if and only if for all integers $k\geq 1$ we have \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,1)}.\] Here $\bar{N}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equ…
The notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the cohomology groups are not finite. Let $p$ be an odd prime, $E_1$ and $E_2$ be elliptic curves over a number field $F$ with semistable reduction at all primes $v|p$ such that the $\operatorname{Gal}(\bar{F}/F)$-modules $E_1[p]$ and $E_2[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_1$ and $E_2$. Leveraging these results, congruence relations for the truncated Euler charac…
We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.
In 1874, Mertens proved the approximate formula for partial Euler product for Riemann zeta function at $s=1$, which is called Mertens' theorem. In this paper, we generalize Mertens' theorem for Selberg class and show the prime number theorem for Selberg class.
We give some corrections of our paper "Primes in arithmetic progressions to large moduli"[BFI]. The corrections do not affect the statements of any of the theorems in the paper. The contents of our two sequel papers [BFI2, BFI3] also remain unchanged.
Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab + cd with min(a, b) > max(c, d) of two ordered products. This gives a new proof Fermat's Theorem expressing primes of the form 1 + 4N as sums of two squares 1 .
Consider the product of (1-p^(-s))^(-4) over all primes p=1 mod 5. We evaluate its residue at s=1 and compare with the corresponding Mertens constant of Languasco & Zaccagnini. We also count primitive quintic Dirichlet characters mod n and determine their average number as n->infty.
In the present paper, our goal is to introduce a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order dedekind-type sums with weight alpha related to Extended q-Euler polynomials by using p-adic q-integral in the p-adic integer ring.
We give an outline of a generalization of the Gelfond-Schnirelmann method in elementary number theory. It is related to an integral of Selberg (1944) generalizing the Euler beta integral. The result we explain was obtained by Nair and Chudnovsky independently in the early eighties.
We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-γ})^{-1} = 2.280...$ and $γ$ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.
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 $χ$ be the nontrivial character modulo 4. Euler wants to know what $\sum_p χ(p)/p$ 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 …
Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(ζ_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(ζ_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(ζ_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $ζ_{\mathbb Q(ζ_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a…
Let $ξ$ and $m$ be integers satisfying $ξ\ne 0$ and $m\ge 3$. We show that for any given integers $a$ and $b$, $b \neq 0$, there are $\frac{φ(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$ such that $a-bF_p(ξ) \ne 0$, where $F_p(ξ)=\sum_{n=0}^\infty n!ξ^n$ is the $p$-adic evaluation of Euler's factorial series at the point $ξ$.
We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer n and prime number p>2n+1 we have $$\sum_{k=0}^{p-1+2n}E_kE_{p-1+2n-k}=s(n) (mod p)$$ where s(n) is an integer only depending on n.
Jacobsthal's function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to P_k, the product of the first k primes. The best known bound on h(k) is h(k) < C (k ln k)^2 for some unknown constant C, due to Iwaniec. We use a generalisation of Euler's totient function to give a stronger bound on h(k).
We give a new proof of the norm relations for the Asai-Flach Euler system built by Lei-Loeffler-Zerbes. More precisely, we redefine Asai-Flach classes in the language used by Loeffler-Skinner-Zerbes for Lemma-Eisenstein classes and prove both the vertical and the tame norm relations using local zeta integrals. These Euler system norm relations for the Asai representation attached to a Hilbert modular form over a quadratic real field $F$ have been already proved by Lei-Loeffler-Zerbes for primes which are inert in $F$ and for split primes satisfying some assumption; with this technique we are a…
We define the $k$-dimensional generalized Euler function $φ_k(n)$ as the number of ordered $k$-tuples $(a_1,\ldots,a_k)\in {\Bbb N}^k$ such that $1\le a_1,\ldots,a_k\le n$ and both the product $a_1\cdots a_k$ and the sum $a_1+\cdots +a_k$ are prime to $n$. We investigate some of properties of the function $φ_k(n)$, and obtain a corresponding Menon-type identity.
