Oscillations in weighted arithmetic sums

Let $\omega(n)$ denote the number of distinct prime factors of $n$, and let $\Omega(n)$ denote its total number of prime factors, counting multiplicity. Questions involving the parity of these functions have a long history. For $\alpha\in[0,1]$, define

and

Pólya studied $L_{0}(x)$ in his 1919 article [23]. He noted that $L_{0}(x)$ was not positive after $L_{0}(1)=1$ up to approximately $x=1500$, and remarked that the Riemann hypothesis, as well as the simplicity of the zeros of the zeta function, would follow if this held true for all sufficiently large $x$. More generally, it is well known that both of these statements would hold if the normalized function $L_{0}(x)/\sqrt{x}$ were bounded by some constant, either above or below. In his influential 1942 paper [14], Ingham showed that considerably more would follow in this case: there would exist infinitely many linear dependencies over the rationals among the ordinates of the zeros of the zeta function on the critical line in the upper half plane. Stated another way, if the (positive) ordinates of the nontrivial zeros of $\zeta(s)$ are linearly independent, then arbitrarily large oscillations must exist in $L_{0}(x)/\sqrt{x}$. In fact, Ingham showed that this holds both for Pólya’s function $L_{0}(x)$ and for its close cousin, the Mertens function,

Since it seems there is no a priori reason to suspect such linear dependencies, and none has ever been detected among the zeros of the zeta function, it is often conjectured that these functions exhibit unbounded oscillations. Large oscillations have been shown to exist in the Mertens function [4, 13] and in the error term for the distribution of $k$-free numbers [19].

Turán studied $L_{1}(x)$ in 1948 [25], where he reported that this function is never negative over $2\leq x\leq 1000$, and connected its behavior to the zeros of the Riemann zeta function. Here, the Riemann hypothesis, as well as the simplicity of the zeros and the existence of linear dependencies among their ordinates, would similarly follow if $L_{1}(x)\sqrt{x}$ were bounded either above or below. Haselgrove [11] established that infinitely many sign changes in both $L_{0}(x)$ and $L_{1}(x)$ exist. Specific locations of sign changes in these functions were first determined respectively in [16] and in [6].

The broader family $L_{\alpha}(x)$ was investigated in [20], where the authors established how these functions are connected to the Riemann hypothesis, as well as the statement on the simplicity of the zeros of $\zeta(s)$ and the question of linear independence. In [21] it was shown that substantial oscillations must exist in the normalized functions $L_{\alpha}(x)x^{\frac{1}{2}-\alpha}$. For example, it was shown that each of the inequalities $L_{0}(x)/\sqrt{x}>1$ and $L_{1}(x)\sqrt{x}>2.37$ holds for infinitely many integers $x$ (see [21, Thms. 1.1 and 6.2]). The reader is referred to [6, 12, 20, 21] for additional information and background on this family of functions.

In [22], the authors established similar connections and statements for the function $H_{0}(x)$. Here, among other things, it was established that each of the inequalities $H_{0}(x)/\sqrt{x}>1.7$ and $H_{0}(x)/\sqrt{x}<-1.7$ holds for infinitely many positive integers $x$.

In [24, Conj. 1.1], Sun hypothesized that both $L_{1}(x)$ and $H_{1}(x)$ are $O_{\epsilon}(x^{-\frac{1}{2}+\epsilon})$ for any $\epsilon>0$. Both of these statements seem likely to be true, since either of these is equivalent to the Riemann hypothesis (see also [5, 20]). Sun also remarked there that $H_{1}(x)$ may be $O(x^{-\frac{1}{2}})$, but by the same argument employed in [21, §5] for $H_{0}(x)/\sqrt{x}$, bounding $H_{1}(x)\sqrt{x}$ would again imply the existence of infinitely many linear dependence relations among the ordinates of the zeros of the zeta function in the upper half plane, so a bound of this strength seems doubtful.

In this paper, we prove that large oscillations must exist in $H_{\alpha}(x)x^{\alpha-\frac{1}{2}}$ for every $\alpha\in[0,1]$. This presents an $\omega$-complement to the $\Omega$ work of [20] and [21], and generalizes the oscillation result concerning the case $\alpha=0$ from [22] in a natural way. For $\sigma>1$ and $s=\sigma+it$, let $h(s)$ denote the Dirichlet series

The series $h(1)$ was investigated by van de Lune and Dressler in 1975 [18], who proved that its value is $0$. Following [22, §5], we may write the Euler product above as

where

is an explicitly given function that converges in the half plane $\sigma>1/7$, and

so that $h(s)$ can be continued analytically to the left of the $\sigma=1$ line. (We remark as in [22] that there is no trouble in extending the form in (1.5) so that the corresponding function $F_{k}(s)$ converges on $\sigma>1/(k+1)$ for larger values of $k$.) For convenience we write $h(\alpha)$ in place of $F_{6}(\alpha)/Z_{6}(\alpha)$ for $\alpha\in(\frac{1}{2},1)$ throughout. For $0\leq\alpha\leq 1$, we define $\mathscr{H}_{a}(x)$ by

We establish the following theorem.

Sun made some additional conjectures regarding sums involving $\Omega(n)$ in [24], which we also address here. Let

In [24, Hypoth. 1.1], Sun made the following four hypotheses:

Certainly (1.10) and (1.11) are stronger than (1.8) and (1.9) respectively for sufficiently large $x$, and clearly (1.10) and (1.11) appear related. Sun reported that (1.10) was checked for $x\leq 10^{11}$ and (1.11) for $x\leq 2\cdot 10^{9}$.

We disprove all four of these statements in this article. In two of the conjectured inequalities above—the lower bounds in (1.10) and (1.11)—we also determine the minimal counterexample, after calculating these functions for $x\leq 7.5\cdot 10^{14}$. We summarize some of the results of our computations in the following theorem.

In addition, we determine lower bounds on the oscillations exhibited by $S_{0}(x)/\sqrt{x}$ and $S_{1}(x)\sqrt{x}$. These results show that each of the conjectured bounds in (1.8), (1.9), (1.10), and (1.11) is violated infinitely often. Our method in fact allows us to determine information on $S_{\alpha}(x)$ for any $\alpha\in[0,1]$. For this, we define

We determine bounds on oscillations for $\mathscr{S}_{\alpha}(x)$ for all $\alpha\in[0,1]$ in the following theorem.

In this article we investigate another conjecture made by Sun as well in [24]. Let

The similar function $W_{+}(x)=\sum_{n\leq x}2^{\Omega(n)}$ was considered by Grosswald [8] and by Bateman [2]: in particular, Grosswald proved that $\left\lvert W_{+}(x)\right\rvert\ll x\log^{2}x$. In [24, Conj. 1.1] Sun conjectured that a smaller and explicit bound holds for $W(x)$:

We investigate this function here, and verify that (1.17) holds for $x\leq 2.5\cdot 10^{14}$.

This article is organized in the following way. Section 2 describes the notion of weak independence of a set of real numbers, and its use in establishing lower bounds on the oscillation of sums of certain arithmetic functions. Section 3 presents the analysis of the functions $S_{\alpha}(x)$, and establishes Theorem 1.3. Following this, Section 4 considers the family $H_{\alpha}(x)$, and proves Theorem 1.1. Details on the computations that establish Theorem 1.2 and related facts, along with information on calculations for $W(x)$, are described in Section 5. Finally, in Section 6 we touch on some other conjectures of Sun from [24].

The work of Grosswald [9, 10], Bateman et al. [3], Diamond [7], Anderson and Stark [1], and others establishes connections between bounding oscillations in sums of certain arithmetic functions and the existence of integral linear relations among the ordinates of nontrivial zeros of $\zeta(s)$, where the linear relations have bounded coefficients. For example, in 1971 Bateman et al. [3] showed that if the Mertens function (1.3) satisfied $\left\lvert M(x)\right\rvert\leq\sqrt{x}$ for all $x>0$ then there would exist infinitely many linear relations among the nontrivial zeros of $\zeta(s)$ in the upper half plane with maximum coefficient $2$. They showed as well that the ordinates of the first $20$ zeros of $\zeta(s)$ on the critical line are $1$-independent, that is, there exist no nontrivial linear relations among these values using coefficients $\{-1,0,1\}$. In a more recent study of oscillations in the Mertens function [4], it was shown that the first $500$ zeros of $\zeta(s)$ are $10^{5}$-independent.

Here, in order to exhibit large oscillations in sums we require a particular form of weak linear independence for certain zeros of $\zeta(s)$. We assume the Riemann hypothesis and the simplicity of its zeros here and throughout this paper, as unbounded oscillations would follow anyway if either of these conditions did not hold. Let $\Gamma$ denote a set of positive real numbers, and for $T>1$ let $\Gamma^{\prime}=\Gamma^{\prime}(T)$ denote a subset of $\Gamma\cap[0,T]$. For each $\gamma\in\Gamma^{\prime}$, let $N_{\gamma}$ denote a positive integer. We say $\Gamma^{\prime}$ is $\{N_{\gamma}\}$-independent in $\Gamma\cap[0,T]$ if two conditions hold. First, whenever

then necessarily all $c_{\gamma}=0$. Second, for any $\gamma^{*}\in\Gamma\cap[0,T]$, if

then $\gamma^{*}\in\Gamma^{\prime}$, $c_{\gamma*}=1$, and all other $c_{\gamma}=0$. That is, the only linear relations with small coefficients here are the trivial ones. In addition, if $N$ is a positive integer with the property that each $N_{\gamma}\geq N$ with $\{N_{\gamma}\}$ as above, then we say that $\Gamma^{\prime}$ is $N$-independent in $\Gamma\cap[0,T]$.

Anderson and Stark [1] established a means for bounding oscillations by using weak independence. We summarize a particular result of theirs: suppose $g(u)$ is a piecewise continuous, real function, bounded on finite intervals, with Laplace transform $G(s)$,

Suppose also that $G(s)$ is absolutely convergent in $\sigma>\sigma_{0}$ for some $\sigma_{0}$, and can be analytically continued back to $\sigma\geq 0$, except for simple poles occurring at $\pm i\gamma$ for $\gamma\in\Gamma$, for a particular set $\Gamma$ of positive real numbers, and possibly a simple pole at $0$ as well. If $\Gamma^{\prime}\subseteq\Gamma$ is $\{N_{\gamma}\}$-independent in $\Gamma\cap[0,T]$, then

where $k_{T}(x)$ denotes an admissible weight function, which must be even, nonnegative, supported on $[-T,T]$, have the value $1$ at $x=0$, and be the Fourier transform of a nonnegative function. While the Fejér kernel ($k_{T}(x)=1-\left\lvert x\right\rvert/T$ for $\left\lvert x\right\rvert\leq T$ and $0$ otherwise) is admissible, we employ the kernel of Jurkat and Peyerimhoff [15] here, defined by

This kernel provides more weight to values of $x$ near the origin compared to the Fejér kernel (and correspondingly less weight to values farther away), and this is often advantageous.

The result (2.1) then provides a weak version of Ingham’s theorem for the case of weak independence. Ingham established a result similar to (2.1), where linear independence over the rationals for $\Gamma$ was required, the fractions $N_{\gamma}/(N_{\gamma}+1)$ were naturally omitted, and $\Gamma^{\prime}$ was replaced by $\Gamma\cap[0,T]$.

The LLL lattice reduction algorithm [17], combined with some additional linear algebra, may be employed to establish weak independence of a finite set $\Gamma^{\prime}$ of positive real numbers, relative to a parent set $\Gamma$ and a real parameter $T$. In this article, we are able to employ some weak independence results established in prior work, so we omit a detailed description of the algorithm, and refer the reader to the following sources. The article [20] provides a detailed explanation of Ingham’s method, extended to the family of functions $L_{\alpha}(x)$ from (1.1), including connections with the Riemann hypothesis, the simplicity of the zeros of the zeta function, and the linear independence question. The publications [4, 19, 21, 22] describe the method of weak independence, and computational strategies for establishing this property. The first of these treats the Mertens function $M(x)$, the third deals with the family $L_{\alpha}(x)$, and the last one considers $H_{0}(x)$ from (1.2). In particular, the paper [21] contains substantial exposition on this method, including for example a proof of (2.1).

Since $S_{\alpha}(x)$ from (1.7) is rather similar to Pólya’s function $L_{\alpha}(x)$, one might begin by employing some analysis similar to that of [20] and [21]. Indeed, when $\alpha=0$ there is a simple relationship connecting these two functions:

Thus, we might expect $S_{\alpha}(x)$ to exhibit a bias with sign opposite to that of $L_{\alpha}(x)$, and we might hope that methods that established large oscillations in $L_{\alpha}(x)$ might also produce similarly good results for $S_{\alpha}(x)$. Indeed, this turns out to be the case, as we show below.

Let $Y(s)$ denote the Dirichlet series

If $\sigma>1$, then it is straightforward to establish that

This was also given by Sun [24]. Standard results (see, e.g., [20]) show that if either the Riemann hypothesis is false or a zero of $\zeta(s)$ is not simple, then

Hence we may assume the Riemann hypothesis and the simplicity of the zeros in order to exhibit large oscillations. We follow the same procedure as in [20]. With $\mathscr{S}_{\alpha}(x)$ as in (1.12), we define

The Laplace transform of $B_{\alpha}(u)$ is

and as in [20] we compute

where

The adjustments made to $S_{\alpha}(x)$ when creating $\mathscr{S}_{\alpha}(x)$ in (1.12) are engineered so that the function $F_{\alpha}(s)$ in (3.1) is analytic in the open half-plane $\{s=\sigma+it\in\mathbb{C}:\,\sigma>0\}$ with only simple poles on the imaginary axis. In particular, the adjustment at $\alpha=1/2$ removes the pole of order $2$ at $s=0$ in (3.2), and the adjustment for $1/2<\alpha<1$ removes the simple pole on the real axis at $s=\alpha-1/2$. This prepares us to employ the result of Anderson and Stark and the machinery of weak independence.

For $s=0$, we compute

where $\gamma_{0}=0.57721\ldots$ denotes Euler’s constant. For $s=i\gamma_{n}$, where $\rho_{n}=\frac{1}{2}+i\gamma_{n}$ is the $n$th zero of the zeta function on the critical line in the upper half plane, we have

If $\Gamma^{\prime}\subseteq\Gamma\cap[0,T]$ and $\Gamma^{\prime}$ is $N$-independent in $\Gamma\cap[0,T]$, then using the result of Anderson and Stark (2.1) (where $\sigma_{0}=1/2$) we conclude that

In particular, when $\alpha=0$, this produces (writing $\rho=\frac{1}{2}+i\gamma$)

for an infinite sequence of $u\to\infty$, and likewise

for another infinite sequence of $u\to\infty$. Note that $-(1+\sqrt{2})/\zeta(1/2)=1.653\ldots$ , so that, in a loose sense, one expects $S_{0}(x)$ to be biased towards positive values. Similarly, $S_{1}(e^{u})e^{u/2}$ exhibits the same oscillation bounds about $(1+\sqrt{2})/\zeta(1/2)=-1.653\ldots$ , since $\left\lvert\rho_{n}-1\right\rvert=\left\lvert\rho_{n}\right\rvert$. We note that the conjectures (1.10) and (1.11) are the same as stating that the magnitude of the oscillations here never exceeds approximately $0.653$, since $1.653-0.653=1$ and $1.653+0.653\approx 2.3$.

We may now complete the proof of Theorem 1.3 by employing a result on weak independence.

We remark that it is possible to improve the bounds in Theorem 1.3 with a new computation, using zeros selected to maximize contributions according to the residues (3.4) in this problem, rather than employing the set of residues that were selected for the functions $L_{\alpha}(x)$. However, we would not witness a substantial gain with a calculation of approximately the same size: we estimate gaining $0.015$ in the oscillation bound in each direction from a computation with approximately $n=240$ zeros. This computation would require several months of core time, so with a high cost relative to a small benefit we opted against a new calculation here.

Figures 3 and 4 illustrate the actual oscillations exhibited by $S_{0}(e^{u})e^{-u/2}$ and $S_{1}(e^{u})e^{u/2}$ respectively over $24\leq u\leq 34.25\ldots$ , along with their respective center line at $u=\pm(1+\sqrt{2})/\zeta(1/2)$. Also, note that when $\alpha=1/2$ the oscillations of $S_{1/2}(e^{u})$ center on the line

Figure 5 displays the sampled values of $S_{1/2}(u)$ over $24\leq u\leq 30$, along with this line.

In order to prove Theorem 1.1, we generalize our analysis in [22], in a manner analogous to the prior section and our study of the behavior $L_{\alpha}(x)$ from [20] and [21]. In [22], for the case $\alpha=0$ we computed the Laplace transform of $H_{0}(e^{u})/e^{u/2}$: under the Riemann hypothesis, this function is analytic in the half plane $\sigma>0$. For the general case, we proceed in a similar manner, except that when $\alpha\in(1/2,1)$ one must incorporate an adjustment to prevent the appearance of a pole on the positive real axis in the corresponding Laplace transform. In essence one must account for the value of $h(\alpha)$ for $1/2<\alpha<1$, with $h(\alpha)$ defined in this interval by (1.5). We made precisely the analogous adjustment in the analysis of $L_{\alpha}(x)$ in [20] and [21]. Using the function $\mathscr{H}_{\alpha}$ from (1.6), for $u\geq 0$ we set $A_{\alpha}(u)$ to be a suitable scaling of $\mathscr{H}_{\alpha}(e^{u})$:

Next, we set

and

Using an argument very similar to that employed in [20] for $L_{\alpha}(x)$, we find that for all $\alpha\in[0,1]$, the function $G_{\alpha}(s)$ is the Laplace transform of $A_{\alpha}(s)$:

and, under the Riemann hypothesis, the integral in (4.2) converges for all $\sigma>0$. One slight difference arises here: when $\alpha=1/2$ there is a pole of $h_{\alpha}(s)$ in (4.1) when $s=0$. However, this is canceled by the zero of $h(1/2)$, which itself comes from the pole of $\zeta(2s)$ in (1.5). As such, unlike the case of $L_{1/2}(x)$ or $S_{1/2}(x)$, there is no dramatic bias in $H_{1/2}(x)$.

It is now straightforward to calculate the residue of $G_{\alpha}(s)$ at each pole along $\sigma=0$. We have

From (4.3) and the fact that $h(1)=0$, we expect no bias in the sum $H_{\alpha}(x)$ for $0\leq\alpha\leq 1/2$ and for $\alpha=1$. Over $1/2<\alpha<1$, we expect a bias in this function whenever $h(\alpha)\neq 0$. Figure 1 exhibits $h(\alpha)$ over $1/2\leq\alpha\leq 1$. It is interesting that the sign of the bias changes over this range. For $1/2<\alpha<\alpha^{*}\approx 0.63093$, our plot indicates that we expect $H_{\alpha}(x)$ to exhibit a negative bias, with maximal negative bias occurring near $\alpha_{1}=0.55336$, where $h(\alpha_{1})=-0.0950579\ldots$ . That is, the values of $H_{\alpha_{1}}(x)$ will oscillate relative to the base curve $h(\alpha_{1})e^{(\alpha_{1}-\frac{1}{2})u}$. For $\alpha^{*}<\alpha<1$, we expect the sum $H_{\alpha}(x)$ to display a positive bias, with maximal bias occurring near $\alpha_{2}=0.73587$, where $h(\alpha_{2})=0.0804324\ldots$ . Figure 7 displays $H_{\alpha}(e^{u})e^{\alpha-1/2}$ for two values of $\alpha$ near these points of maximal bias. In these plots, the bias specified by $h(\alpha)$ was not removed as in (1.6), so in each of these graphs the oscillations center on an exponential function. In part (b) of this figure, where $\alpha=3/4$, the bias is quite evident, since the centering curve is $e^{u/4}$. However, in part (a) it is not so obvious, since the guiding exponential $e^{u/20}$ remains rather small over the plotted interval.

We may now establish Theorem 1.1 by computing the relevant residues and employing a particular set of zeros of the zeta function known to possess a weak independence property.