The least prime with a given primitive root and a variant of the larger sieve

Daniel R. Johnston; Sunil Naik

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The least prime with a given primitive root and a variant of the larger sieve

Daniel R. Johnston, Sunil Naik

Number Theory · Submitted 6 Oct 2026 · 14 references

Pith's verdict: Solid conditional almost-all bound for the least prime primitive root, with a clean reusable larger-sieve variant; the proof holds up, and the one gap a referee must press is the asserted infimum A_1(g) ≥ 2/3 that pins the sharp constant.

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CitationDaniel R. Johnston, Sunil Naik. "The least prime with a given primitive root and a variant of the larger sieve." arXiv:2610.08569 (2026). https://pith.science/paper/ZSIV4UIM

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Editor's note

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Read it thoroughly. The paper delivers a real improvement: under GRH, p_g ≤ (log|g|)^{3.44} for almost all g, versus Fan–Pollack's uniform exponent 19, and the positive-proportion variant of Gallagher's larger sieve (Prop 2.2) is a genuinely reusable tool. I checked the new sieve proof, Lemmas 3.3–3.4, the alpha-interval arithmetic that forces B > 2 + 2e^{-1/3} ≈ 3.433, and the unconditional Theorem 1.3. Everything I checked holds. The reader's CONDITIONAL verdict is about right, but the specific worry is smaller than it looks.

The new things: Prop 2.2 with the weighted positive-proportion condition (2.4), which no existing variant (Croot–Elsholtz, Shao) contains; the almost-all exponent 3.44 from applying it inside the Hooley/Fan–Pollack machinery; and the elementary unconditional o(N)-exception result (Theorem 1.3), whose CRT proof is correct. The paper also flags where the c = 1 bound is weaker than Gallagher's original, and the GRH dependence is up front.

The soft spot: Section 3.3 asserts "directly from the definition (3.7), A_1(g) ≥ A_1(−15) = 2/3" with no derivation, and that single bound pins the sharp constant in (3.13). A referee should require the computation written out: the extremal case needs even omega(g1), the indicator 4 | g1 − 1 active, and the largest admissible product Pi(p−2) equal to 3, coming from {3,5}, attained at g1 = −15. The reader's sign worry is resolved by ordinary divisibility — 4 | −16 — so this is a presentation gap, not a correctness flaw; I verified the minimization goes through and the infimum really is 2/3. Minor: the Lemma 3.1 typo correction (log(Delta dx)) is quoted from Fan–Pollack and not independently checkable here, but it is benign since B > 2 makes that error o(#S_g). No other soft spots. The citation pattern is appropriate, and quoting the Fan–Pollack lemmas is standard practice.

The paper is for anyone working on Artin's conjecture, least primitive roots, or applications of the larger sieve, and the method section is short enough for a good reading-group talk.

Recommendation: send it to peer review. The only substantive request I would make is that the authors derive the A_1 infimum explicitly and state the extremal g1; that is a one-paragraph fix.

Referee report

Significance

If correct, the paper delivers a substantial quantitative improvement (uniform exponent 19 down to 3.44 for almost all g) and a clean, new unconditional o(N)-exception statement. Proposition 2.2 is a genuine methodological contribution: a positive-proportion larger sieve whose proof is self-contained and, in structure, machine-checkable, and which plausibly has further applications to Artin-type problems. I verified the Cauchy–Schwarz algebra in (2.6)–(2.7) and the rearrangement to (2.5), the existence of α in (3.2) exactly when B > 2 + 2e^{-1/3}, the exceptional-set exponent B(1-α) < B-1 in Lemma 3.4, and the CRT counting in Theorem 1.3; all are sound. The sharp constant in the headline theorem, however, rests entirely on a single assertion in §3.3, A_1(g) ≥ 2/3, which the manuscript states but does not derive; I independently verified that the claim is true (the minimum is attained at g1 = -15), so the gap is presentational rather than a correctness error, but the derivation must be added.

Major objections (2)

[§3.3, final step of Theorem 1.1] The assertion 'directly from the definition (3.7), we see that A_1(g) ≥ A_1(-15) = 2/3' is load-bearing: the admissible window for α in (3.2) and the threshold B > 2 + 2e^{-1/3} in (3.13) are pinned by A_1(g)/2 ≥ 1/3, and hence the headline exponent 3.44 in Corollary 1.2 depends on this single unproved extremal fact. The manuscript gives no computation. I verified the claim: the case analysis of (3.7) shows the minimum requires the indicator 1_{4|g1-1, g1|W} to be active with ω(g1) even, in which case A_1(g) = 1 - Π_{p|g1} 1/(p-2); the product Π(p-2) over distinct odd primes is minimized at {3,5}, giving Π = 1/3, and the congruence 4 | g1 - 1 forces g1 = -15 (since g1 = 15 fails 4 | 14), so A_1(-15) = 2/3; all other admissible g1 give a smaller Π or fail the congruence, so inf A_1 = 2/3. Because the claim is true, the result stands, but the paper must supply this derivation rather than asserting it, and it should state explicitly the sign convention used when g1 < 0 (namely that 4 | g1 - 1 means ordinary divisibility of the negative integer g1 - 1, and that ω(g1) counts the distinct prime divisors of |g1|).
[§3.3, completion of Theorem 1.1 (also §3.4)] The sentence 'Combining Lemmas 3.1-3.4, this is true provided (3.13)' omits the explicit verification that the accumulated error terms are dominated by the main term. Positivity in (3.4) requires A_0(x)li(x)(A_1(g)/2 - log(α/(1/2 - 1/B)) - ϵ) to exceed |O(x^{1/2+1/B+o(1)})| from (3.5), the o(#S_g) term from Lemma 3.2, and the O(li(x)/log x) term from Lemma 3.3. This does hold: B < 4 gives 1/2 + 1/B < 1, so x^{1/2+1/B+o(1)} = o(A_0 li(x)); li(x)/log x = x/(log x)^2 = o(x/(log x log log x)) = o(A_0 li(x)); and the bracket in (3.13) is bounded below by a fixed positive constant when α is chosen strictly inside the interval (3.2) and ϵ is taken sufficiently small (e.g., ϵ < 1/3 - log(αe^{1/3}/(1/2 - 1/B))). I recommend that the authors spell out this domination in a few lines, since (3.13) as written is only an inequality among leading coefficients.

Minor objections (3)

[§3.4, display (3.9)] The bound Σ_{ℓ|p-1, ℓ>x^α} 1 < 1/α is not valid: the number of divisors of p-1 exceeding x^α can grow like τ(p-1) = x^{o(1)}, which need not be below the constant 1/α. The estimate should instead read Σ_{ℓ|p-1, ℓ>x^α} 1 ≤ τ(p-1) = x^{o(1)}, giving ν(p) ≤ x^{1-α+o(1)}. This does not affect the conclusion of Lemma 3.4, since the incurred x^{o(1)} factor is absorbed by the fixed positive margin α - 1/B > 0, but the displayed inequality should be corrected.
[§1.2 and §4, Theorem 1.3 remark] In the remark following Theorem 1.3, the claimed bound N exp(-(c+o(1))Φ(N)/log Φ(N)) for the exceptional set when Φ(x) ≤ (1-ϵ)log x requires, in addition to the first term of (4.1), control of the second term P_N γ(N). It would be helpful to note explicitly that Lemma 4.2 gives P_N γ(N) = e^{(1+o(1))Φ} · exp(-(c+o(1))Φ/log Φ) = N^{1-ϵ+o(1)}·exp(-(c+o(1))Φ/log Φ) in this regime, so that P_N γ(N) is absorbed into the displayed bound. As written, the proof of Theorem 1.3 only covers the Φ = o(log x) regime (via the monotonicity reduction); a line clarifying how the remark's regime is handled would make the remark self-contained.
[Throughout] Minor typographical and notational issues: reference [1] reads 'Aumber Theory' for 'Number Theory'; the 'Möbius' function appears with a garbled combining character in the notation list of §1.3; the phrase 'least prime primitive root' is repeated in footnote 1; and it would help to state at the definition of S_g in (3.1) that the Legendre symbol (g/p) for negative g is the standard Kronecker-Jacobi symbol. None of these affect the mathematics.

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The paper

The paper claims

Under the Generalised Riemann Hypothesis, this paper proves that for almost every integer $g$ with $|g|\in[N,2N]$, the least prime $p_g$ for which $g$ is a primitive root satisfies $p_g\le(\log|g|)^{3.44}$, with at most $O((\log…

The reading

The paper studies $p_g$, the least prime modulo which a fixed integer $g$ is a primitive root, and asks how large $p_g$ must be relative to $|g|$. Working under the Generalised Riemann Hypothesis, it establishes that for each fixed $B>2+2e^{-1/3}=3.433\ldots$, all but $O((\log N)^{B-1})$ of the integers $g$ with $|g|\in[N,2N]$ satisfy $p_g\le(\log|g|)^B$; in the explicit form of Corollary 1.2 this reads $p_g\le(\log|g|)^{3.44}$. This improves the uniform bound $p_g\le(\log|g|)^{19}$ of Fan and Pollack by nearly an order of magnitude, at the price of excluding a tiny exceptional set rather than covering every $g$. The engine is a new 'positive proportion' variant of Gallagher's larger sieve, which bounds sets of integers that land in a fixed positive fraction of the surviving residue classes rather than in all of them, and the paper argues this sieve may have uses beyond primitive roots. It also proves an elementary unconditional analogue: for any $\Phi(N)\to\infty$, almost every $g$ in an interval of length $N$ has $p_g\le\Phi(N)$.

What carries the argument

The load-bearing object is the positive-proportion larger sieve (Proposition 2.2), an extension of Gallagher's larger sieve. Where the classical sieve counts integers passing every one of a family of residue-class filters, the variant counts integers passing at least a fraction $c\in(0,1]$ of the filters by $\Lambda$-weight, producing the bound $$\#A \le \left(\sum_{q\in Q}\Lambda(q)-\log N\right)\Big/\left(\frac{$c^{2}$(\sum_{q\in Q}\Lambda(q))^2}{\sum_{q\in Q}\nu(q)\Lambda(q)}-\log N\right),$$ obtained by running Gallagher's proof with one additional Cauchy–Schwarz application. This sieve feeds the $\ell$-test framework: a prime $p$ fails the $\ell$-test for $g$ if $p\equiv1\pmod{\ell}$ and $g^{(p-1)/\ell}\equiv1\pmod p$, and $g$ is a primitive root mod $p$ exactly when $p$ passes every $\ell$-test; the test primes are split into the three intervals $I_1,I_2,I_3$, with the large-prime interval $I_3$ being where the exceptional set of $g$ is created.

Extended reading notes

Core claim

The central claim is Theorem 1.1: under GRH, for every fixed $B>2+2e^{-1/3}$ and all sufficiently large $N$, every $g\in G$ (the integers that are neither $-1$ nor a perfect square) with $|g|\in[N,2N]$ satisfies $p_g\le(\log|g|)^B$, with at most $O((\log N)^{B-1})$ exceptions. The proof runs the Hooley/Fan–Pollack $\ell$-test machinery, splitting the test primes into three ranges $I_1,I_2,I_3$; the large-prime range is handled by the new larger-sieve variant, which leaves only a set of $g$ of the claimed size because each exceptional $g$ need only fall into a positive proportion of the forbidden residue classes rather than into all of them. The same framework yields tighter conditional exponents under restrictions: $3.35$ for positive $g$ and $3.22$ for squarefree $g$, both with correspondingly smaller exceptional sets. A separate elementary argument via the Chinese remainder theorem gives the unconditional statement that almost every $g\in[M,M+N]$ has $p_g\le\Phi(N)$ for any $\Phi(N)\to\infty$.

Load-bearing premise

The proof needs the uniform lower bound $A_1(g)\ge2/3$ for the constant defined in (3.7) across all $g\in G$ including negative $g$, asserted directly from the definition without derivation; if the true infimum were smaller, the sharp threshold $B>2+2e^{-1/3}$ in Theorem 1.1 would shift.

Editorial extensions

If this is right

Improves the uniform conditional bound $p_g\le(\log|g|)^{19}$ to $p_g\le(\log|g|)^{3.44}$ for all but $O((\log N)^{2.44})$ of the $g$ in a dyadic range.
Restricting to positive or squarefree $g$ sharpens the exponent to $3.35$ or $3.22$ respectively within the same framework.
Proposition 2.2 gives a general larger-sieve variant, plausibly reusable wherever residue-class sieving with a slack 'positive proportion' condition arises.
Unconditionally, almost every $g\in[M,M+N]$ admits a primitive-root prime $p_g\le\Phi(N)$ for any $\Phi(N)\to\infty$, with a stronger exceptional-set bound when $\Phi(N)\le(1-\epsilon)\log N$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The exact threshold $2+2e^{-1/3}$ rests on the uniform lower bound $A_1(g)\ge2/3$; for negative $g$ the indicator $4\mid g_1-1$ in (3.7) is sign-sensitive, and a complete derivation for that case would decide whether the minimum is truly $2/3$ or merely close to it, with any shortfall shifting the advertised $B$.
GRH enters at three distinct points (the $S_g$ asymptotic and the $I_1,I_2$ bounds), so pressing the constant further would likely require sharpening one of those estimates rather than refining the sieve alone.
The unconditional Section 4 result, though elementary, suggests a qualitatively different tradeoff shape (exponentially smaller exceptional set, weaker pointwise bound) that the conditional method does not reproduce.

Original abstract

For an integer $g$ not equal to $-1$ or a square, we study the least prime $p_g$ such that $g$ is a primitive root modulo $p_g$. We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $|g|\in[N,2N]$ have $p_g\leq(\log |g|)^{3.44}$ with at most $O((\log N)^{2.44})$ exceptions. This can be directly compared to the uniform bound $p_g\leq(\log |g|)^{19}$ proven recently by Fan and Pollack. To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest. In addition to our conditional result, we discuss other ``almost all" bounds for $p_g$ that hold unconditionally.

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For an integer $g$ not equal to $-1$ or a square, we study the least prime $p_g$ such that $g$ is a primitive root modulo $p_g$. We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $|g|\in[N,2N]$ have $p_g\leq(\log |g|)^{3.44}$ with at most $O((\log N)^{2.44})$ exceptions. This can be directly compared to the uniform bound $p_g\leq(\log |g|)^{19}$ proven recently by Fan and Pollack. To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest. In addition to our conditional result, we discuss other ``almost all" bounds for $p_g$ that hold unconditionally.

background contribution method result interpretation main claim

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s0	For an integer $g$ not equal to $-1$ or a square, we study the least prime $p_g$ such that $g$ is a primitive root modulo $p_g$.	background 0.71	0.61	0.05	asserted 0.58
s1	We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $\|g\|\in[N,2N]$ have $p_g\leq(\log \|g\|)^{3.44}$ with at most $O((\log N)^{2.44})$ exceptions.	result 1.00	0.96	0.90	asserted 1.00
s2	This can be directly compared to the uniform bound $p_g\leq(\log \|g\|)^{19}$ proven recently by Fan and Pollack.	interpretation 0.72	0.09	0.08	asserted 0.97
s3	To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest.	method 0.70	0.95	0.04	asserted 0.58
s4	In addition to our conditional result, we discuss other ``almost all" bounds for $p_g$ that hold unconditionally.	contribution 0.69	0.73	0.04	asserted 0.68

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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References

Works cited (14)

[1]

T. M. Apostol.Introduction to Analytic Aumber Theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976

View on Pith 1976
[2]

D. A. Burgess. “On character sums and primitive roots”.Proc. London Math. Soc. (3)12 (1962), pp. 179–192

View on Pith 1962
[3]

E. S. Croot III and C. Elsholtz. “On variants of the larger sieve”.Acta Math. Hungar. 103.3 (2004), pp. 243–254

View on Pith 2004
[4]

K. Fan and P. Pollack. “Counting primes with a given primitive root, uniformly”. Mathematika71.4 (2025), Paper No. e70055, 30

View on Pith 2025
[5]

P. X. Gallagher. “The large sieve”.Mathematika14 (1967), pp. 14–20

View on Pith 1967
[6]

P. X. Gallagher. “A larger sieve”.Acta Arith.18 (1971), pp. 77–81

View on Pith 1971
[7]

C. Hooley. “On Artin’s conjecture”.J. Reine Angew. Math.225 (1967), pp. 209–220

View on Pith 1967
[8]

G. Martin. “The least prime primitive root and the shifted sieve”.Acta Arith.80.3 (1997), pp. 277–288

View on Pith 1997
[9]

G. Martin. “Uniform bounds for the least almost-prime primitive root”.Mathe- matika45.1 (1998), pp. 191–207

View on Pith 1998
[10]

H. L. Montgomery and R. C. Vaughan.Multiplicative number theory. I. Classical theory. Vol. 97. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007

View on Pith 2007
[11]

P. Moree. “Artin’s primitive root conjecture—a survey”.Integers12.6 (2012), pp. 1305– 1416

View on Pith 2012
[12]

X. Shao. “On an inverse ternary Goldbach problem”.Amer. J. Math.138.5 (2016), pp. 1167–1191

View on Pith 2016
[13]

V. Shoup. “Searching for primitive roots in finite fields”.Math. Comp.58.197 (1992), pp. 369–380

View on Pith 1992
[14]

R. C. Vaughan. “Some applications of Montgomery’s sieve”.J. Number Theory5 (1973), pp. 64–79. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Email address:johnston@mpim-bonn.mpg.de Department of Mathematics, Indian Institute of Technology Palakkad, Kerala 678623, India Email address:sunilnaik@iitpkd.ac.in

View on Pith 1973

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

T. M. Apostol.Introduction to Analytic Aumber Theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976

View on Pith 1976

[2] [2]

D. A. Burgess. “On character sums and primitive roots”.Proc. London Math. Soc. (3)12 (1962), pp. 179–192

View on Pith 1962

[3] [3]

E. S. Croot III and C. Elsholtz. “On variants of the larger sieve”.Acta Math. Hungar. 103.3 (2004), pp. 243–254

View on Pith 2004

[4] [4]

K. Fan and P. Pollack. “Counting primes with a given primitive root, uniformly”. Mathematika71.4 (2025), Paper No. e70055, 30

View on Pith 2025

[5] [5]

P. X. Gallagher. “The large sieve”.Mathematika14 (1967), pp. 14–20

View on Pith 1967

[6] [6]

P. X. Gallagher. “A larger sieve”.Acta Arith.18 (1971), pp. 77–81

View on Pith 1971

[7] [7]

C. Hooley. “On Artin’s conjecture”.J. Reine Angew. Math.225 (1967), pp. 209–220

View on Pith 1967

[8] [8]

G. Martin. “The least prime primitive root and the shifted sieve”.Acta Arith.80.3 (1997), pp. 277–288

View on Pith 1997

[9] [9]

G. Martin. “Uniform bounds for the least almost-prime primitive root”.Mathe- matika45.1 (1998), pp. 191–207

View on Pith 1998

[10] [10]

H. L. Montgomery and R. C. Vaughan.Multiplicative number theory. I. Classical theory. Vol. 97. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2007

View on Pith 2007

[11] [11]

P. Moree. “Artin’s primitive root conjecture—a survey”.Integers12.6 (2012), pp. 1305– 1416

View on Pith 2012

[12] [12]

X. Shao. “On an inverse ternary Goldbach problem”.Amer. J. Math.138.5 (2016), pp. 1167–1191

View on Pith 2016

[13] [13]

V. Shoup. “Searching for primitive roots in finite fields”.Math. Comp.58.197 (1992), pp. 369–380

View on Pith 1992

[14] [14]

R. C. Vaughan. “Some applications of Montgomery’s sieve”.J. Number Theory5 (1973), pp. 64–79. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Email address:johnston@mpim-bonn.mpg.de Department of Mathematics, Indian Institute of Technology Palakkad, Kerala 678623, India Email address:sunilnaik@iitpkd.ac.in

View on Pith 1973