Steinhaus Random Multiplicative Function

• Steinhaus random multiplicative functions are defined via independent, uniformly distributed prime phases and extended multiplicatively to all integers.
• They exhibit exact orthogonality in second moments while higher moments capture intricate dependencies leading to non-Gaussian behaviors.
• The theory connects Euler products, cancellation in partial sums, and multiplicative chaos, underpinning advanced Gaussian and random variance limit results.

A Steinhaus random multiplicative function is a completely multiplicative random function $f:\mathbb N\to \mathbb C$ obtained by choosing the prime values $f(p)$ independently and uniformly on the unit circle and then extending by multiplicativity to all integers. It is a central model in probabilistic number theory because it combines exact multiplicative structure with primewise independence, producing a sequence that is orthogonal in second moment but strongly dependent globally. The resulting theory links partial sums, Euler products, multiplicative energy, martingale central limit theorems, and Gaussian multiplicative chaos, and it now includes both Gaussian limit theorems for suitable restrictions and genuinely non-Gaussian limits for full or heavily correlated sums (Xu, 20 May 2026).

1. Foundational model

For each prime $p$, a Steinhaus random multiplicative function is specified by an independent random variable $f(p)$ distributed uniformly on

$\{z\in\mathbb C:|z|=1\}.$

Complete multiplicativity is then imposed: $f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$ equivalently,

$f(n)=\prod_p f(p)^{v_p(n)}$

when $n=\prod_p p^{v_p(n)}$ (Xu, 20 May 2026).

This model differs sharply from the Rademacher random multiplicative function, whose prime values are $\pm1$ and whose standard extension is supported on squarefree integers. In the Steinhaus case every prime power contributes, and the function remains unit-modulus on all integers. That complete multiplicativity is responsible both for the algebraic tractability of the model and for the long-range dependencies that distinguish it from sums of independent random phases (Heap et al., 2015).

The basic objects of study are partial sums and weighted partial sums such as

$$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$

together with twisted sums and polynomial subsequences. In the Gaussian limit statements, the reference law is the standard complex Gaussian $f(p)$0, meaning a rotationally invariant complex normal random variable with mean $f(p)$1 and variance $f(p)$2 (Soundararajan et al., 2022).

2. Orthogonality, moments, and Euler products

The primewise independence gives an exact orthogonality relation: $f(p)$3 Hence for deterministic coefficients $f(p)$4,

$f(p)$5

This identity explains the natural variance normalizations in the subject, but it does not imply approximate independence of the values $f(p)$6, because complete multiplicativity creates many higher-order product relations (Xu, 20 May 2026).

The higher moments are governed by exact multiplicative matching conditions. For fixed $f(p)$7 and $f(p)$8,

$f(p)$9

where

$p$0

At $p$1,

$p$2

and the geometric factor is related to the Birkhoff polytope by

$p$3

(Heap et al., 2015).

For larger real moments, the asymptotic shape remains rigid. Uniformly for

$p$4

one has

$p$5

Thus the $p$6-th moment is not Gaussian even after the natural $p$7 scaling; the excess logarithmic factor $p$8 records the cumulative effect of multiplicative correlations (Harper, 2018).

The corresponding Euler product is a persistent structural device. For the weighted model,

$p$9

and the partial sum

$f(p)$0

is closely approximated by

$f(p)$1

This relation converts multiplicative fluctuations into primewise additive fluctuations and is the main reason random Euler products dominate the fine asymptotic theory (Hardy, 2023).

3. Full partial sums, cancellation, and extremal behavior

For the full sum

$f(p)$2

the low moments exhibit “better than square-root cancellation.” Uniformly for $f(p)$3,

$f(p)$4

In particular,

$f(p)$5

This proves Helson’s conjecture that the first moment is $f(p)$6 and shows that the $f(p)$7-scale suggested by the second moment is not the correct typical scale (Harper, 2017).

Before this sharp first-moment theorem, Bondarenko and Seip proved the lower bounds

$f(p)$8

for all $f(p)$9, and in particular

$\{z\in\mathbb C:|z|=1\}.$0

Their argument used a decomposition of $\{z\in\mathbb C:|z|=1\}.$1 into homogeneous chaoses indexed by $\{z\in\mathbb C:|z|=1\}.$2, the number of prime factors counted with multiplicity, and identified $\{z\in\mathbb C:|z|=1\}.$3 as the critical degree where $\{z\in\mathbb C:|z|=1\}.$4 and $\{z\in\mathbb C:|z|=1\}.$5 comparability breaks down (Bondarenko et al., 2014).

The failure of a naive central limit theorem is now explicit. Harper’s work showed that $\{z\in\mathbb C:|z|=1\}.$6 exhibits more than square-root cancellation, and in particular

$\{z\in\mathbb C:|z|=1\}.$7

does not have a complex Gaussian limiting distribution (Soundararajan et al., 2022). The full-sum regime is therefore qualitatively different from genuinely Gaussian sums of weakly dependent phases.

At the almost sure level, the partial sums fluctuate far above $\{z\in\mathbb C:|z|=1\}.$8 along subsequences. Almost surely there exist arbitrarily large $\{z\in\mathbb C:|z|=1\}.$9 such that

$f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$0

This answers a question of Halász and proves a conjecture of Erdős in the random multiplicative setting (Harper, 2020).

The same arithmetic dependence produces unusually large maxima for Dirichlet polynomials with Steinhaus coefficients. For

$f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$1

the supremum over polynomially long $f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$2-ranges satisfies, with high probability,

$f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$3

for fixed $f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$4. This is dramatically larger than the corresponding independent-coefficient scale and reflects the same multiplicative reinforcement already visible in the moments of $f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$5 (Benatar et al., 2022).

4. Gaussian regimes on sparse, structured, and moving sets

Although the full sum is non-Gaussian, many restricted sums satisfy genuine complex central limit theorems. A general criterion is phrased in terms of multiplicative energy

$f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$6

If a large set $f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$7 contains a density-one subset $f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$8 with asymptotically minimal multiplicative energy,

$f(mn)=f(m)f(n)\qquad \text{for all }m,n\ge 1,$9

then

$f(n)=\prod_p f(p)^{v_p(n)}$0

converges in distribution to $f(n)=\prod_p f(p)^{v_p(n)}$1. The proof uses a quantitative martingale central limit theorem based on decomposing the sum by largest prime factor (Soundararajan et al., 2022).

This mechanism yields several concrete Gaussian families. For short intervals

$f(n)=\prod_p f(p)^{v_p(n)}$2

one has a CLT when

$f(n)=\prod_p f(p)^{v_p(n)}$3

The same paper proves complex Gaussian limits for sums over shifted primes, for sums over integers representable as sums of two squares in suitable short intervals, and for additive twists

$f(n)=\prod_p f(p)^{v_p(n)}$4

whenever $f(n)=\prod_p f(p)^{v_p(n)}$5 is irrational and satisfies

$f(n)=\prod_p f(p)^{v_p(n)}$6

(Soundararajan et al., 2022).

A different regime is obtained by averaging over moving intervals. If

$f(n)=\prod_p f(p)^{v_p(n)}$7

then for each fixed $f(n)=\prod_p f(p)^{v_p(n)}$8,

$f(n)=\prod_p f(p)^{v_p(n)}$9

whenever

$n=\prod_p p^{v_p(n)}$0

Moreover, if $n=\prod_p p^{v_p(n)}$1 is chosen uniformly from $n=\prod_p p^{v_p(n)}$2 and

$n=\prod_p p^{v_p(n)}$3

then for almost all realizations $n=\prod_p p^{v_p(n)}$4,

$n=\prod_p p^{v_p(n)}$5

Here “almost all” is with respect to the Steinhaus law on the prime values (Pandey et al., 2022).

Polynomial subsequences provide another large Gaussian class. If $n=\prod_p p^{v_p(n)}$6 has degree at least $n=\prod_p p^{v_p(n)}$7 and is not of the form $n=\prod_p p^{v_p(n)}$8, then

$n=\prod_p p^{v_p(n)}$9

For polynomials not expressible as products of linear factors over $\pm1$0, there almost surely exist arbitrarily large $\pm1$1 such that

$\pm1$2

which matches the law-of-the-iterated-logarithm scale expected for independent Steinhaus variables and sharply contrasts with the non-Gaussian linear case $\pm1$3 (Klurman et al., 2022).

5. Density thresholds and nonstandard normalization

A major refinement of the theory concerns what happens when the averaging set is not sparse. Let $\pm1$4. If

$\pm1$5

then necessarily

$\pm1$6

Thus the naive normalization $\pm1$7 can produce a standard complex Gaussian limit only in the zero-density regime (Xu, 20 May 2026).

This obstruction is sharp. If $\pm1$8 has density $\pm1$9 satisfying

$$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$0

then for most sets $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$1 with density $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$2,

$$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$3

The factor $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$4 is essential whenever $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$5 does not tend to $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$6. In the Bernoulli model $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$7 with $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$8, centering gives $$\sum_{n\le x} f(n),\qquad \sum_{n\le x}\frac{f(n)}{\sqrt n},\qquad \sum_{n\in A}f(n),$$9 with

$f(p)$00

so the fluctuation term naturally has variance $f(p)$01, not $f(p)$02 (Xu, 20 May 2026).

An analogous correction appears for short intervals very close to full length. For

$f(p)$03

there exists a deterministic scaling factor $f(p)$04 such that

$f(p)$05

with

$f(p)$06

Equivalently,

$f(p)$07

If

$f(p)$08

then $f(p)$09, so the classical $f(p)$10-normalization is recovered. When $f(p)$11 is very close to $f(p)$12, however, the correct Gaussian scale is smaller than $f(p)$13. In contrast, when $f(p)$14, there is no deterministic normalization for which the limiting distribution is a non-degenerate Gaussian (Harper et al., 27 Jun 2026).

Taken together, these results identify density and interval length as structural parameters: sparse sets and genuinely short intervals behave in an i.i.d.-like manner, whereas dense subsets and near-full intervals retain a visible trace of the multiplicative-chaos structure of the full sum.

6. Multiplicative chaos, mixed Gaussian limits, and weighted extensions

The most refined distributional results no longer produce fixed Gaussian limits but Gaussian mixtures with random variance. For the large-prime-factor subsum

$f(p)$15

one has

$f(p)$16

Here $f(p)$17 is a nonnegative random variable defined through a critical Gaussian multiplicative chaos measure, and the limit is conditionally Gaussian with random variance $f(p)$18 (Hardy, 8 Mar 2025).

A general version of this phenomenon concerns sums

$f(p)$19

where $f(p)$20 is Steinhaus and $f(p)$21 is deterministic multiplicative. If $f(p)$22 with $f(p)$23 and suitable summability conditions hold, then

$f(p)$24

stably, where $f(p)$25 is independent of

$f(p)$26

The random measure $f(p)$27 is obtained from the normalized modulus square of the Euler product associated with $f(p)$28, so the limiting law is genuinely non-Gaussian unless $f(p)$29 is almost surely constant (Gorodetsky et al., 2024).

That construction has since been extended to the full $f(p)$30-regime $f(p)$31. For a broad class of multiplicative $f(p)$32, the measures

$f(p)$33

converge, for both prime truncation and critical-line approximation schemes, to the same nontrivial random Radon measure $f(p)$34. As an application, the same generalized central limit theorem with random variance holds for normalized sums of $f(p)$35 throughout this full subcritical regime (Gorodetsky et al., 13 Mar 2025).

Weighted and twisted models reveal further regimes. For the critical weight $f(p)$36,

$f(p)$37

the almost sure upper and lower envelopes satisfy, for every $f(p)$38,

$f(p)$39

and

$f(p)$40

This confirms the scale predicted by exponentiating a law of the iterated logarithm for the associated prime-indexed random walk (Hardy, 2023).

Beyond the critical case, divisor twists lead into a supercritical multiplicative-chaos phase. For $f(p)$41, if $f(p)$42 is the generalized divisor function defined by $f(p)$43, then uniformly for $f(p)$44,

$f(p)$45

This extends the critical-chaos picture of the untwisted Steinhaus sum into the supercritical range and matches the predictions of supercritical Gaussian multiplicative chaos (Hamdan, 7 Apr 2026).

In aggregate, the modern theory shows that the Steinhaus random multiplicative function does not admit a single universal asymptotic law. Sparse and low-energy restrictions can restore ordinary complex Gaussian behavior; dense subsets and near-full intervals require corrected deterministic variance; full or near-full sums exhibit multiplicative-chaos effects; and in several weighted settings the limiting object is a Gaussian mixture whose variance is itself random.