Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The present article describes Erdős’s work contained in the following papers.

  1. [E1]

    On the coefficients of the cyclotomic polynomials, Bull. Amer. Math. Soc. 52 (1946), 179–183.

  2. [E2]

    On the coefficients of the cyclotomic polynomial, Portug. Math. 8 (1949), 63–71.

  3. [E3]

    On the number of terms of the square of a polynomial, Nieuw Archief voor Wiskunde (1949), 63–65.

  4. [E4]

    On the greatest prime factor of \(\mathop{\prod }\limits _{k=1}^{x}f(k)\), J. London Math. Soc. 27 (1952), 379–384.

  5. [E5]

    On the sum \(\sum \limits _{k=1}^{x}d(f(k))\), ibid. 7–15.

  6. [E6]

    Arithmetical properties of polynomials, ibid. 28 (1953), 436–425.

  7. [E7]

    Über arithmetische Eigenschaften der Substitutionswerte eines Polynoms für ganzzahlige Werte des Arguments, Revue Math. Pures et Appl. 1 (1956) No. 3, 189–194.

  8. [E8]

    On the growth of the cyclotomic polynomial in the interval (0,1), Proc. Glasgow Math. Assoc. 3 (1957), 102–104.

  9. [E9]

    On the product \(\mathop{\prod }\limits _{k=1}^{n}(1 {-\zeta }^{a_{i}})\), Publ. Inst. Math. Beograd 13 (1959), 29–34 (with G. Szekeres).

  10. [E10]

    Bounds for the r-th coefficients of cyclotomic polynomials, J. London Math. Soc. (2) 8 (1974), 393–400 (with R. C. Vaughan).

  11. [E11]

    Prime polynomial sequences, ibid. (2) 14 (1976), 559–562 (with S. D. Cohen, M. B. Nathanson).

  12. [E12]

    On the greatest prime factor of \(\mathop{\prod }\limits _{k=1}^{x}f(k)\), Acta Arith. 55 (1990), 191–200 (with A. Schinzel).

The papers [E1], [E2], [E8] and [E10] concern the coefficients of the cyclotomic polynomial

$$\displaystyle{\phi _{n}(x) =\mathop{ \prod }\limits _{d\vert n}{({x}^{d} - 1)}^{\mu (\frac{n} {d} )}.}$$

Let A n denote the largest absolute value of the coefficient of ϕ n (x). It was proved by E. Lehmer in 1936 that

$$\displaystyle{A_{n} = \Omega ({n}^{1/2}).}$$

In [E1] Erdős proved that

$$\displaystyle{\log A_{n} = \Omega ({(\log n)}^{4/3})}$$

and in [E2] that

$$\displaystyle{\log \log A_{n} = \Omega \left (\frac{\log n} {\log \log n}\right ).}$$

In [E8] he gave a simpler proof of the last relation and conjectured that

$$\displaystyle{\log \log A_{n}> c\frac{\log n} {\log \log n}}$$

for every c < log2 and infinitely many n. This conjecture even for c = log2 has been proved by R. C. Vaughan [11], his result is best possible.

The paper [E10] treats the coefficient a r (n) of x r in ϕ n (x). The authors prove that

$$\displaystyle{\vert a_{r}(n)\vert <\exp ({2\tau }^{1/2}{r}^{1/2} + c_{ 1}{r}^{3/8})}$$

and

$$\displaystyle{\mathop{\lim \sup }\limits _{n\rightarrow \infty }\vert a_{r}(n)\vert>\exp (c_{2}{(r/\log r)}^{1/2}),\quad (r> r_{ 0})}$$

where c 1,  c 2 are absolute constants, c 2 > 0,

$$\displaystyle{\tau =\mathop{ \prod }\limits _{p\ \mathrm{prime}}\left (1 - \frac{2} {p(p + 1)}\right ).}$$

The subject is still alive as shows Maier’s paper [4].

Somewhat related to the four Erdős’s papers discussed is the paper [E9], in which the authors consider the functional

$$\displaystyle{M(a_{1},a_{2},\ldots,a_{n}) =\max _{\vert z\vert =1}\left \vert \prod _{i=1}^{n}(1 - {z}^{a_{i} })\right \vert }$$

(\(a_{1} \leq a_{2} \leq \ldots \leq a_{n}\) are positive integers) and

$$\displaystyle{f(n) =\min _{a_{1},\ldots,a_{n}}M(a_{1},\ldots,a_{n}).}$$

They prove that logf(n) = o(n). This result has been sharpened by F. V. Atkinson [1].

In the paper [E3] Erdős considers the sequence Q(n) first studied by A. Rényi. Let N(f) be the number of nonzero coefficients of a polynomial f and let

$$\displaystyle{Q(n) =\min N({f}^{2}),}$$

where f runs through all polynomials \(f \in \mathbb{Q}[x]\) with N(f) = n. Rényi proved that Q(29) < 29. Erdős proves that Q(n) ≪ n c for a certain c < 1 and quotes the conjecture at which he and Rényi independently arrived, namely that

$$\displaystyle{\lim _{n\rightarrow \infty }Q(n) = \infty.}$$

A good value of the constant c has been given by W. Verdenius [12] and the above conjecture has been proved by the writer [7]. The result of [7] has been improved in [9] to the form

$$\displaystyle{\mathbb{Q}(n)> 2 + \frac{\log (n - 1)} {\log 8} \quad \ (n> 1).}$$

The subject is still alive as shown in Coppersmith and Davenport’s paper [2] and Zannier’s paper [14].

The papers [E4] and [E12] concern the greatest prime factor P(f, x) of \(\prod _{k=1}^{x}f(k)\), where f is an irreducible polynomial of degree d > 1. The first estimate for P(f, x) in the case \(f = {x}^{2} + 1\) was given by Chebyshev and his result was extended to all relevant polynomials by T. Nagell in 1921 in the following form

$$\displaystyle{P(f,x)> c(f,\epsilon )x{(\log x)}^{1-\epsilon }}$$

for all ε > 0 and a suitable c(f, ε) > 0.

In [E4] Erdős proved that

$$\displaystyle{P(f,x)> x{(\log x)}^{c(f)\log \log \log x}\mbox{ for }c(f)> 0,\ x> x_{ 0}(f)}$$

and asserted that by a much more complicated argument one can show that

$$\displaystyle{ P(f,x)> x\exp ({(\log x)}^{\delta })\mbox{ for }\delta =\delta (f)> 0,\ x> x_{l}(f). }$$
(*)

An attempt made in [E12] to reconstruct the proof of ( ∗ ) led only to a weaker estimate

$$\displaystyle{P(f,x)> x\exp \exp (c{(\log \log x)}^{1/3})\mbox{ for }x> x_{ 2}(f)}$$

where \(c\,>\,0\) is an absolute constant. However G. Tenenbaum [10] has succeeded in proving (*) with any δ less than 2 − log4.

In the paper [E5] Erdős considers the sum

$$\displaystyle{S(x) =\sum _{k\leq x}d(f(k)),}$$

where d(n) is the divisor function and \(f \in \mathbb{Z}[x]\) is irreducible. He proves that

$$\displaystyle{c_{1}x\log x \leq S(x) \leq c_{1}x\log x,}$$

where c 1, c 2 are positive constants depending upon f.

The upper estimate, which is much deeper has been considerably generalized by D. Wolke [13] He has replaced d(n) by a multiplicative function and the polynomial f by a quickly growing integer valued function, both functions subject only to mild restrictions. The subject is still alive as shown by Nair’s paper [6].

In the paper [E6] Erdős considers values of a polynomial free from k-th powers. It was proved by G. Ricci in 1933 that a primitive irreducible polynomial of degree d > 1 represents infinitely many integers free from d-th powers. Erdős improves this as follows: every irreducible polynomial of degree d > 2 without a fixed (d − 1)-th power divisor greater than 1 represents infinitely many integers free from (d − 1)-th powers. This result has been improved by C. Hooley [3], who has given an asymptotic formula for the number of integers in question and also by M. Nair [5] who has replaced d − 1 by [λd] + 1, where \(\lambda = \sqrt{2} -\frac{1} {2}\). Hooley gives a tribute to Erdős by saying “It is to the perspicacity of Erdős that we owe our present appreciation of the manifold uses to which sieve methods can be put” (l.c., p. xi).

[E7] is a survey of problems and results, in which in particular [E4], [E5] and [E6] are discussed.

The remaining paper [E11] contains the following theorem. Let \(F \in \mathbb{Z}[x]\) be a polynomial of degree d ≥ 2 such that F(n) ≥ 1 for all n ≥ 1. Let \(O_{F} =\{ F(n)\}_{n=1}^{\infty }\). Then F(N) is called prime in O F , if F(n) is not the product of strictly smaller terms in O F . If F(x) is not of the form a(bx + c)d, then almost all terms of O F are prime in O F . The “almost all” is indeed, quantified. For a later, related work, see [8].