Erdős's conjecture on the non-vanishing of $L(1,f)$ for $\pm1$-valued periodic functions (Erdősian functions)

Erdős, in a 1960s communication recorded by Livingston, conjectured that if a periodic arithmetic function $f$ of period $q$ takes only the values $\pm1$ away from multiples of $q$ (and vanishes on them), then the associated Dirichlet series at $s=1$, namely $L(1,f)=\sum_{n\ge1}f(n)/n$, is never zero. The conjecture sits at the interface of analytic number theory and transcendence theory, and its resolution is the main stated obstruction in Murty and Saradha's study of the arithmetic of Euler–Lehmer constants $\gamma(a,q)$, the generalized Euler constants of arithmetic progressions introduced by D. H. Lehmer. All cases are settled except a set of moduli $q\equiv1\pmod 4$: the conjecture is known for $q$ even, $q$ prime, prime powers, products of two odd primes, several classes of multiplicative $f$, odd $f$, all $q\equiv3\pmod 4$, and at least 82% of the moduli $q\equiv1\pmod 4$ by lower natural density (an alternative method in the same work gives 80%). The problem is to prove or refute the conjecture in full generality, the open core being the surviving $q\equiv1\pmod 4$ cases. The problem remains open as of 2026.

Background

For a function $f:\mathbb{Z}\to\mathbb{Q}$ that is periodic with period $q$, the associated Dirichlet series is $L(s,f)=\sum_{n\ge1}f(n)/n^s$. The series at $s=1$ converges exactly when $f$ has mean zero over one period, $\sum_{n=1}^{q}f(n)=0$, and then $L(1,f)$ is a finite $\mathbb{Q}$-linear combination of logarithms of algebraic numbers. This follows from the digamma function $\psi(x)=\Gamma'(x)/\Gamma$ (the logarithmic derivative of the gamma function): partial sums of $1/n$ along residue classes are controlled by $\psi(a/q)$, and Gauss's digamma theorem expresses each $\psi(a/q)$ in terms of Euler's constant $\gamma$, $\pi$ times cotangents of rational multiples of $\pi$, and logarithms of sines $\log\sin(\pi j/q)$, all of which live in the $q$-th cyclotomic field (the field obtained by adjoining a primitive $q$-th root of unity to the rationals; the cyclotomic polynomial $\Phi_q$ is the minimal polynomial of such a root over the rationals).

The natural constants attached to residue classes are the Euler–Lehmer constants introduced by D. H. Lehmer (1975): for $q\ge2$ and $1\le a\le q$ with $(a,q)=1$, $\gamma(a,q)=\lim_{x\to\infty}\Bigl(\sum_{\substack{n\le x\\ n\equiv a\pmod q}}\frac1n-\frac{\log x}{q}\Bigr),$ a generalization of Euler's constant $\gamma=\gamma(1,1)$, about whose irrationality nothing is known. Murty and Saradha (2010) proved that for fixed $q$ at most one of the $\gamma(a,q)$ is algebraic, and that the vanishing of $L(1,f)$ is precisely the kind of obstruction that blocks pinning down individual constants; they also left open whether two distinct $\gamma(a,q)$ can be equal and whether $\gamma$ itself is transcendental.

The central tool is the theorem of Baker, Birch, and Wirsing (1973), based on Baker's theory of linear forms in logarithms of algebraic numbers (which gives lower bounds for nonzero such forms): if $f$ is rational-valued and periodic with period $q$, has mean zero, and the cyclotomic polynomial $\Phi_q$ is irreducible over the field $\mathbb{Q}(f(1),\dots,f(q))$, then $L(1,f)\ne0$ (indeed it is transcendental unless identically of a special form). Okada (1982) turned this into a concrete numerical criterion: $L(1,f)=0$ forces an explicit system of congruences on divisor sums of the values of $f$, weighted by the Möbius function $\mu(d)$ (the function on positive integers equal to $(-1)^k$ on a product of $k$ distinct primes and $0$ whenever a square divides the argument). Call $f$ Erdősian mod $q$ if $f(n)=\pm1$ when $q\nmid n$ and $f(n)=0$ when $q\mid n$. Erdős conjectured (in writing to Livingston, who recorded it in 1965) that $L(1,f)\ne0$ for every Erdősian $f$. Known cases: $q$ even (the mean cannot vanish, so the series diverges and there is nothing to prove); $q$ prime (Baker–Birch–Wirsing 1973); $2\varphi(q)+1>q$, where $\varphi$ is Euler's totient function, which covers prime powers and products of two distinct odd primes (Okada 1982); $f$ completely multiplicative, i.e. $f(mn)=f(m)f(n)$ for all $m,n$ (Tijdeman 2002); $f$ multiplicative with $|f(p^k)|<p-1$ on prime divisors $p$ of $q$ (Saradha–Tijdeman 2003); $f$ odd, i.e. $f(q-a)=-f(a)$ for $1\le a<q$ (Murty–Saradha 2007); and all moduli $q\equiv3\pmod4$ (Murty–Saradha 2010, Theorem 7 of that paper). Murty–Saradha (2010) also proved, by the same circle of ideas, that among the Euler–Lehmer constants $\gamma(a,q)$ (with $(a,q)=1$, $1\le a\le q$) at most one is algebraic, and that as $x$ ranges over rational numbers in $(0,1]$ at least one of $\Gamma(x)$, $\Gamma'(x)$ is transcendental with at most one exceptional $x$; whether every individual $\gamma(a,q)$ is transcendental, whether two can be equal, and removing the exceptional $x$ remain open side questions of that paper, distinct from the conjecture stated here. A refutation of Erdős's conjecture would exhibit an exact linear relation among logarithms of cyclotomic algebraic numbers, which is widely believed impossible.

Problem Statement

Erdős's conjecture (recorded by Livingston, 1965): let $q\ge2$ and let $f:\mathbb{Z}\to\mathbb{Z}$ be periodic with period $q$, with $f(a)=\pm1$ for all integers $a$ such that $q\nmid a$ and $f(a)=0$ when $q\mid a$ (an Erdősian function), and suppose $\sum_{n=1}^{q}f(n)=0$, so that $L(1,f)=\sum_{n\ge1}f(n)/n$ converges. Prove that $L(1,f)\ne0$ for every modulus $q$ and every such $f$, or exhibit an explicit pair $(q,f)$ with $L(1,f)=0$. The conjecture is already proved for $q$ even, $q$ prime, all $q$ with $2\varphi(q)+1>q$ ($\varphi$ denoting Euler's totient function), suitable multiplicative $f$, odd $f$, all $q\equiv3\pmod4$, and at least 82% of the moduli $q\equiv1\pmod4$ by lower natural density (an alternative method in the same work gives 80%), so the open core is the remaining Erdősian functions with $q\equiv1\pmod4$.

References

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