Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Robin’s Inequality for Integers with at Most Eight Distinct Primes

Abstract

Robin’s inequality holds above 5040 for every integer with at most eight distinct prime factors, hence for every integer between 5040 and 223092870.

Application to the 5040 congruence cell: GoldenCell5040Shape.modEq_2241_shape takes 5040 dividing n and Nat.ModEq n (3^n) 2241 and gives n=2^a3^b57r, with 4<=a<=6, 2<=b<=3 and r coprime to 235783. The prime factors of this product are the disjoint union of {2,3,5,7} and those of r. Thus a residual with at most four distinct primes has total support at most eight, and the theorem below proves Robin for n>5040. These are applications of the existing shape and support theorems, rather than additional mathematical declarations. Support-cardinality partitions do not represent the FIB legal windows 000,100,010,101,001.

A further application of the existing positive-price and logarithm bounds excludes every weak Robin failure through log(n)=40. The preceding finite-floor theorem handles 5040<n<223092870. For the remaining integers with log(n)<=40, write A=log(n). The threshold criterion at price 1/60 has optimizer N=6983776800 and divisor ratio Z=249984/46189; at price 1/102 it has N=9316358251200 and Z=181416960/30808063. Occupied and next layers are checked at each prime in these integers. The first missing primes are 23 and 31; first-layer antitonicity and layer decrease extend those checks to every absent prime and further layer.

Optimality gives log(sigma(1,n)/n)<=log(Z)+(A-log(N))/d. Concavity of log(log(A)) makes the difference convex, so its endpoint bounds on [19,24] for d=60 and [24,40] for d=102 control the whole intervals. The existing rational logarithm bounds pay each endpoint strictly below 57/100, while log(89/50)>57/100. Together with log(223092870)>19, this yields sigma(1,n)/n<(89/50)*log(log(n)) on the whole remaining range. Since 89/50<exp(gamma), equality in the Robin comparison is excluded as well.

For arbitrary prime support of cardinality at most eleven, the saturation bound is sigma(1,n)/n<=product(p/(p-1))<=86822723/13271040<131/20. The sorted support’s i-th prime is at least the i-th prime overall: Mathlib’s finite order embedding, strictly increasing prime-count function, rank bound and nth/count inverse supply that comparison. The factor p/(p-1) reverses the order, and factors above one allow padding to eleven. This applies to arbitrary exponents and arbitrarily large support primes, not just smooth integers. Finally log(40)>92/25 and (89/50)*(92/25)>131/20 rule out such support after log(n)>40. Consequently every n>5040 with exp(gamma)*log(log(n))<=sigma(1,n)/n has log(n)>40 and at least twelve distinct prime factors. This is an application of existing results, not a new retained theorem, a claim of a new literature bound, or a proof of RH.

Theorem 1.1 (At most eight distinct prime factors).

Lean statement: D5/S3/Arith/Robin/FewDistinctPrimes.robin_of_card_primeFactors_le_eight

Proof. Machine-checked in Lean as D5/S3/Arith/Robin/FewDistinctPrimes.robin_of_card_primeFactors_le_eight (✓ std3). ∎

Citation. Emmanuel Elima (2022). On Robin’s inequality for positive integers and related bounds. URL: https://easychair.org/publications/preprint/xW8q.

Commentary.

For every natural number n > 5040 whose set of prime factors has at most eight elements, sigma(1,n)/n < e^gamma*log(log(n)). Exponents are unrestricted and no Riemann Hypothesis premise is used.

For n >= 6000 the frozen normal-form theorem gives m <= n with as many distinct primes, non-increasing exponents and sigma(1,n)/n <= sigma(1,m)/m. If m <= 5040, the frozen optimum of sigma(1,m)/m - m^(1/25) at 5040 gives sigma(1,m)/m <= 403/105, while e^gamma > 89/50 and log(log(6000)) > 54/25 give a capacity above 3.84. If m > 5040, at most eight distinct primes and decreasing exponents force m = 2^a3^b5^c7^d11^e13^f17^g*19^h with a >= b >= … >= h.

Below 10^12 a kernel-checked recursive enumeration visits all 4271 such products, carrying their divisor sums, and certifies 100sigma(1,m) <= Km on twelve intervals starting at 5040, 10080, 25200, 55440, 110880, 332640, 1441440, 7207200, 61261200, 367567200, 3491888400 and 41902660800, with K = 381, 395, 411, 425, 436, 452, 471, 490, 513, 530, 549, 568. A rational lower bound for log(log(t)) at each left end t, derived from the frozen dyadic logarithm brackets, pays each K/100. From 10^12 on, the product of p/(p-1) over the first eight primes, 323323/55296 < 5.85, is below 89/503317/1000 < e^gammalog(log(10^12)). Robin’s inequality at m then passes to n because n >= m.

For 5040 < n < 6000 the normal form could be 5040 itself, so this range is treated directly: an 11-smooth n is covered by the frozen eleven-smooth theorem, and otherwise a prime p >= 13 divides n, the factor sigma(1,p^a)/p^a is at most 13/12, the cofactor is below 462, and a kernel check gives sigma(1,r)/r <= 13/4 for r < 462.

Elima states the bound for at most twelve distinct primes by reduction to 21-full integers and calculation, and D5/L/ArithSums/mishra2026finiterobin states a stronger omega-graded bound for at most six distinct primes; neither is a machine verification, and this proof uses neither. This eight-prime theorem alone does not cover larger support, and the Riemann hypothesis remains open.

Theorem 1.2 (Every integer between 5040 and the ninth primorial).

Lean statement: D5/S3/Arith/Robin/FewDistinctPrimes.robin_lt_223092870

Proof. Machine-checked in Lean as D5/S3/Arith/Robin/FewDistinctPrimes.robin_lt_223092870 (✓ std3). ∎

Citation. Alexander Hertlein (2018). Robin’s inequality for new families of integers. URL: https://arxiv.org/pdf/1612.05186v2.

Commentary.

For every natural number n with 5040 < n < 223092870, where 223092870 = 23571113171923, sigma(1,n)/n < e^gamma*log(log(n)).

Nine distinct primes have product at least 223092870: removing the largest prime of a set of k <= 9 primes leaves k-1 primes, and the largest one is at least the k-th prime because fewer than k primes lie below it. So every n in the range has at most eight distinct prime factors and the theorem above applies.

The finite range is far inside the computationally verified range 5040 < n <= 10^(10^10) recorded by Hertlein; the point here is a kernel certificate independent of that computation.

References