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Theorem 1.2. Let n 2521, then σ(n) (eγ n log log n) 1+ c/(log log n)2 , where c> 0 ≥ ≤ is a small constant.

The same result appeared in [18], [17], and very recently in [1], but has a different constant. 10 The numerical data for n [5041, 1010 ] was compiled in [5]. On the other hand, there ∈ are several conditional criteria; some of these are listed below.

Theorem 1.3. ([18]) Let n N be an integer, and let σ(n)= d be the sum of divisors ∈ d|n function. Then P (i) If the Riemann Hypothesis is true, then, for each n 5041, ≥ σ(n) < eγ n log log n.

(ii) If the Riemann Hypothesis is false, then, there exists constants 0 <β < 1/2 and c> 0 such that cn log log n σ(n) eγ n log log n + ≥ (log n)β holds for infinitely many n.

The parameter 1 b<β< 1/2 arises from the possibility of a zero ρ = b + it C of the − ∈ analytic continuation of the zeta function

1 ( 1)n+1 ζ(s)= − (4) 1 21−s ns − nX≥1 on the half plane b = e(ρ) > 1/2 if the Riemann hypothesis is false. This in turns implies ℜ the existence of more or fewer primes than expected in some intervals. For example, under this condition, the number of primes would be

π(x) = li(x)+ O(xb+ε) > li(x)+ O(x1/2+ε) (5) infinitely often, some of this material is discussed in [13]. The effect of the zeros of the zeta function on the distribution of primes is readily revealed by the explicit formulas, consult the literature. All these inequalities involves the Euler constant γ. Another well known conditional result offers a constant free inequality, and it is written entirely in terms of the integer n 1. ≥ Theorem 1.4. ([13]) Let H = 1/m be the harmonic series. For each n 1, the n m≤n ≥ inequality P

Hn σ(n) < e log Hn + Hn (6) is equivalent to the Riemann Hypothesis.

The first few sections cover some background materials focusing on the sum of divisors function, and some associated finite sums and products over the prime numbers. The proof of Theorem 1.1 is given in Section 6. sum of divisors function inequality 3

2 Representations and Identities

The sum of divisors function d|n d is ubiquitous in number theory. It appears in the analysis many different problemsP in pure and applied mathematics. Its product repre- sentation in (8) unearths its intrinsic link to the distribution of the prime numbers. The totient function is defined by ϕ(n)=# k : gcd(k,n) = 1 . { } Lemma 2.1. Let n 1 be an integer, and let the symbol pv n denotes the maximum prime power divisor.≥ Then ||

(i) The Euler totient function has the product formula

1 ϕ(n)= n 1 . (7)  − p Yp|n

(ii) The sum of divisors function has the product formula

σ(n)= 1+ p + p2 + + pv . (8) · · · pYv||n

These representations of the divisor and totient functions are well known and/or easy to establish.

Lemma 2.2. Let n be an integer, and let the symbol pv n denotes the maximum prime power divisor.≥ Then, the sum of divisors function has th||e product formula

σ(n) n 1 = 1 . (9) n ϕ(n)  − pv+1  pYv||n

Proof. Use Lemma 2.1, and the geometric series formula r = (1 rx+1)/(1 r) 0≤k≤x − − to evaluate the sigma-phi identity P σ(n) ϕ(n) 1 1 1 1 = 1+ + + + 1 n n  p p2 · · · pv   − p pYv||n Yp|n 1 = 1 . (10)  − pv+1  pYv||n 

To illustrate the negligible effect of negligible multiplication by a prime power, a quanti- tative expression is computed in the next result.

Lemma 2.3. Let p 2 be a prime, and let t 0 be an integer. If pv n, then ≥ ≥ || σ(ptn) σ(n) 1 p−(t+v+1) = − . (11) ptn n · 1 p−(v+1) − Proof. By hypothesis pv n. Since σ(n) is multiplicative, it is sufficient to observe the || effect of extra prime power factor pt with t 0 on the value σ(pv)/pv. To achieve this ≥ sum of divisors function inequality 4 goal, modify the basic fact σ(pv) = (pv+1 1)/(p 1) to isolate the effect of multiplication − − by pa. That is,

σ(pt+v) pt+v+1 1 pv+1 1 pv(p 1) = − − − (12) pt+v pt+v(p 1) · pv(p 1) · pv+1 1 − − − σ(pv) pt+v+1 1 pv(p 1) = − − pv · pt+v(p 1) · pv+1 1 − − σ(pv) pt+v+1 1 1 = − pv · pt · pv+1 1 − σ(pv) pt+v+1 1 = − pv · pt+v+1 pt − σ(pv) 1 p−(t+v+1) = − . pv · 1 p−(v+1) − 

It is immediate that f(t) = σ(ptn)/ptn < 2 log log n is a slowly increasing function of t 0, but it is bounded above. ≥ 3 Extreme Values

The sum of divisors function is an oscillatory function, its values oscillate from its minimum σ(n)= n + 1 at the prime integers n to its maximum σ(n) < 2n log log n at the extremely abundant integers n.

3.1 Asymptotic Extrema The suprema of the sum of divisors function over various subsets of integers are known.

Theorem 3.1. ([10]) Let n 1 be an integer. Then ≥ (i) The limit supremum over the integers is

σ(n) lim sup = eγ . n→∞ n log log n

(ii) The limit supremum over the odd integers is

σ(n) eγ lim sup = . odd n→∞ n log log n 2

(iii) The limit supremum over the squarefree integers is

σ(n) 6eγ lim sup = . squarefree n→∞ n log log n π2

Refer to [12, p. 353], and similar references for proofs and other details on the maximal order of this function. sum of divisors function inequality 5

3.2 Conditional Lower and Upper Bounds Theorem 3.2. ([15]) Let n = 2 3 p be the product of the first k 1 primes. k · · · · k ≥ (i) If the Riemann Hypothesis is true, then, for each n 5041, k ≥ nk γ > e log log nk ϕ(nk) for all k 1. ≥ (ii) If the Riemann Hypothesis is false, then,

nk γ nk γ < e log log nk and > e log log nk ϕ(nk) ϕ(nk) occur for infinitely many k 1. ≥ Lemma 3.1. If the RH is false, then, the ratio σ(n)/n satisfies the upper bound

σ(n) log log n (13) n ≤ for infinitely many highly composite integers n 1. ≥ Proof. Apply Theorem 3.2 to the the phi-sigma inequality

σ(n) n 1 = 1 (14) n ϕ(n)  − pv+1  pYv||n n ≤ ϕ(n) log log n ≤ for infinitely many highly composite integers n. 

3.3 Unconditional Lower and Upper Bounds Several lower and upper estimates derived by several methods are computed in this sub- section.

Upper Bound I. An upper estimate derived from the sigma-phi identity, refer to Lemma 2.2, is computed below.

Lemma 3.2. Let n N and let p n, then the ratio σ(ptn)/ptn remains uniformly bounded and independent of∈ the | prime power pt. More precisely

(i) For any integer t 1, ≥ σ(ptn) n < . ptn ϕ(n)

(ii) For any integer r 1, ≥ σ(nr) n < . nr ϕ(n) sum of divisors function inequality 6

Proof. (i) Take the sigma-phi representation of the sum of divisors function, see Lemma 2.2, and use the fact that p n to simplify the ratio ptn/ϕ(ptn): | σ(ptn) ptn 1 = 1 (15) ptn ϕ(ptn)  − pv+1  pvY||ptn 1 −1 1 = 1 1  − p  − pv+1  pY|ptn pvY||ptn 1 −1 1 = 1 1  − p  − pv+1  Yp|n pvY||ptn 1 −1 < 1  − p Yp|n n = . ϕ(n) The product on the right side of third line, which is bounded by 1, absorbs the effect of multiplication by a prime power. (ii) The proof for this case is similar. 

Lower Bound II. An upper estimate derived from the prime divisors of an integer is computed in this subsection. Let n 1 be a highly composite number, and let p n be ≥ | the largest prime divisor of n. For the single prime p log n, the sum of divisors function ≈ has the unconditional upper bound

1 1 1 1 1 1 1+ 1+ (16) d ≥  2p d ≥  4 log n d ≥ d dX|pn Xd|n Xd|n Xd|n A more general estimate is given below. Lemma 3.3. Let n N be a highly composite integer, and let r 2. Then ∈ ≥ (i) 1 log log n + c 1 1 1+ 0 + O , d ≥  4 log n (log n)2  d dX|nr Xd|n (ii) 1 4 log log n + c 1 1 1+ 0 + O , d ≤  log n (log n)2  d dX|nr Xd|n

where c0 > 0 is a nonnegative constant. Proof. (i) Expand the sum into several subsums: 1 1 1 1 = + + d d pv+t d · · · dX|nr Xd|n 1≤Xt≤r pXv||n Xd|n 1 1 1 1 + ≥ d 2 pv+t d Xd|n 1≤Xt≤r pXv||n Xd|n

1 1 1 = 1+  . 2 pv+t d 1≤Xt≤r pXv||n Xd|n   (17) sum of divisors function inequality 7

The triple sum collects some of the divisors d nr not included in the basic sum of di- | v visors d|n 1/d. By Lemma 5.1, the prime power divisors p n have the upper bound v || p < 2P log n, including the largest prime divisor pk < 2 log n.

The first inner sum for t = 1 has the lower estimate 1 1 1 pv+1 ≥ 2 log n p pXv||n pXv||n 1 1 = 2 log n p p≤Xlog n log log n + γ + O(1/ log n) ≥ 2 log n log log n + γ 1 + O . (18) ≥ 2 log n (log n)2  The other inner double sum for the range 2 t r has the lower estimate ≤ ≤ 1 1 1 pv+t ≥ 2 log n pt pXv||n, 2≤Xt≤r pXv||n, 2≤Xt≤r 1 1 = 2 log n pt p≤Xlog n, 2≤Xt≤r 1 1 1 p−r−1 − ≥ 2 log n p2  1 p−1  p≤Xlog n − 1 1 ≥ 2 log n p2 p≤Xlog n 1 = O , (19) (log n)2  where π(log n) log n/2 log log n. Summing everything yields ≥

1 1 1 log log n + c0 1 1 1+  = 1+ + O , (20) 2 pv+t d  4 log n (log n)2  d 1≤Xt≤r pXv||n Xd|n Xd|n   where c0 = γ > 0 is a constant. The proof of (ii) is similar. 

4 Prime Harmonic Products

The prime harmonic products are sine qua non in number theory. It has a natural link to the totient function ϕ(n), the sieve of Eratosthenes, and other related concepts. Accurate estimates of this product and related products are essential in a variety of calculations in number theory.

Lemma 4.1. ([20, Theorems 7, 8]) Let x 2 be a large number. Then ≥ (i) 1 1 −1 1 1 1 < . eγ log x  − p − 2(log x)2 pY≤x

sum of divisors function inequality 8

(ii) 1 1 (eγ log x) 1 1 < .  − p − 2(log x)2 pY≤x

These are improved versions of the original works by Mertens , see [20], and [23] for more de- tails. The Euler constant is defined by γ = lim 1/n log x = .577215665 . . ., x→∞ n≤x − see [8, p. 28], [14, Theorem 2.2.1] for several definitionsP of this number and similar refer- ences. Lemma 4.2. ([21, Corollary 3]) Let x 14 be a large number. If the Riemann hypothesis holds, then ≥

(i) 1 1 −1 log x + 5 1 1 < . eγ log x  − p − 8π√x pY≤x

(ii) 1 3 log x + 5 (eγ log x) 1 1 < .  − p − 8π√x pY≤x

Theorem 4.1. (Mertens) Let n 1 be an integer. Then ≥ (i) The limit supremum over the prime is

1 1 −1 lim sup 1 = eγ . x→∞ log x  − p pY≤x

(ii) The limit supremum over the prime is

1 1 −1 6eγ lim sup 1+ = . x→∞ log x  p π2 pY≤x

Some references and other information on these limits are available in see [8, p. 31].

5 Highly Composite Numbers

Let p be the kth prime in increasing order, and let v = max m : pm n is the p-adic k p { | } valuation. Extremely abundant integers, and colossally abundant integers are related to the primorial integers n = 2v2 3v3 pvp , but the exponents have certain multiplicative · · · · structure 1 v v v . ≤ p ≤···≤ 3 ≤ 2 Definition 5.1. Let d(n) = 1. An integer n N is called highly composite if and d|n ∈ only if d(m)1 < d(n) for all integersP m 0 sum of divisors function inequality 9

Definition 5.3. An integer n 10080 is called extremely abundant if and only if ≥ σ(m) σ(n) < (22) m log log m n log log n for all integers m

These numbers are studied in [19], [2], [13], [5], [17], et alii.

Fixed a highly composite integer n 1, and let n = nr with a parameter r 1. The ≥ r ≥ sequence of functions σ(n ) σ(n ) σ(n ) r < r+1 < r+1 < (23) nr nr+1 nr+1 · · · is strictly monotonically increasing, but bounded above by eγ log log n, see equation (36).

Lemma 5.1. ([2, Theorem 2]) Let n be a large highly composite integer, then ≥ (i) Unconditionally, the largest prime divisor p n has the asymptotic | 1 p = (log n) 1+ O .  (log log n)2 

(ii) Modulo the Riemann hypothesis, the largest prime divisor p n has the asymptotic | log log n p = (log n) 1+ O .  (log n)1/2 

6 The Main Result

The verification of Theorem 1.1 involves the assumption that the inequality is false for infinitely many colossally abundant integers to derive a reductio ad absurdum.

Proof. (Theorem 1.1) Let n 5041 be a large colossally abundant integer, and suppose ≥ that σ(n) eγ n log log n (24) ≥ for infinitely many colossally abundant integers n 1. By Lemma ??, there is an uncon- ≥ ditional upper bound

σ(n2) log log n + c 1 σ(n) σ(n) 1+ 0 + O (25) n2 ≥  4 log n (log n)2  n ≥ n for all integers n 3, in particular, for all colossally abundant integers. By assumption ≥ (24), the Riemann hypothesis is false. Thus, by Theorem 1.3, there is the lower bound

σ(n) c (eγ log log n) 1+ , (26) n ≥  (log n)β  where c > 0 is a constant, and 0 <β < 1/2, for infinitely many colossally abundant integers n 1. Replacing the lower bound (26) into (25) produces ≥ log log n + c 1 σ(n) c 1+ 0 + O (eγ log log n) 1+ . (27)  4 log n (log n)2  n ≥  (log n)β  sum of divisors function inequality 10

Dividing everything in inequality (27) by eγ log log n reduces it to

log log n + c 1 σ(n) c 1+ 0 + O 1+ . (28)  4 log n (log n)2  eγ n log log n ≥ (log n)β

In addition, assuming that the RH is false, Lemma (3.1) implies that the ratio

σ(n) 1 (29) eγ n log log n ≤ holds for infinitely many colossally abundant integers. And replacing the ratio (29) into (28) returns log log n + c 1 c 1+ 0 + O 1+ . (30) 4 log n (log n)2  ≥ (log n)β Clearly, this contradicts the unconditional upper bound in (25) for any β (0, 1/2) and ∈ infinitely many sufficiently large colossally abundant integers n 5041. Ergo, ≥ σ(n) < eγ log log n (31) n as claimed. 

The upper bound in (31) is consistent with the well known conditional bound in Lemma 4.2, and the parameter β > 0 is within the correct range. This is easy to verify:

σ(n2) 1 −1 < 1 (32) n2  − p pY|n2 1 −1 = 1  − p Yp|n 1 −1 1 ≤  − p pY≤x 3 log x + 5 (eγ log x) 1+ ≤  x1/2  3 log log n + 5 < (eγ log log n) 1+  (log n)1/2  4 log log n < (eγ log log n) 1+ .  (log n)1/2 

The limit x = (log n)(1 + O(1/(log n)1/2)) in the third line follows from Lemma 5.1, and the fourth line in equation (32) follows from Lemma 4.2.

Comparing (27) and (32) returns,

4 log log n c (eγ log log n) 1+ > (eγ log log n) 1+ . (33)  (log n)1/2   (log n)β 

This immediately implies that β 1/2, and c < 4, as required. Accordingly, these ≥ parameters are consistent with the Riemann hypothesis. sum of divisors function inequality 11

7 Other Result

Another technique for proving the sum of divisors inequality for the sequence of integers n = nr, with r 2 is provided in this section. The verification involves the assumption r ≥ that the inequality is false for all colossally abundant integers to derive a reductio ad absurdum. Theorem 7.1. Let n 5041 be a sufficiently large integer, and let r 2. Then ≥ ≥ σ(nr) < eγ n log log n. (34) Proof. Let n 5041 be a large colossally abundant integer, and let r 2. Now, suppose ≥ ≥ that eγ nr log log nr σ(nr) (35) ≤ for all r 2 ≥ I. Applying Lemma 4.1 and Lemma 5.1 in succession reduce the right side to σ(nr) 1 −1 < 1 (36) nr  − p pY|nr 1 −1 = 1  − p Yp|n 1 −1 1 ≤  − p pY≤x 1 < eγ log x 1+  2(log x)2  1 1 = eγ log log n 1+ + O ,  2(log log n)2  (log log n)2  where x = (log n)(1 + O(1/(log log n)2)), this follows from Lemma 5.1, and routine calcu- lations.

II. By the hypothesis in equation (38), and elementary calculations, the left side reduces to σ(nr) eγ log log nr (37) nr ≥ = eγ log log n + eγ log log r. III. Combining equations (36) and (37) yield 1 1 σ(nr) eγ log log n 1+ + O > (38)  2(log log n)2  (log log n)2  nr eγ log log n + eγ log log r. ≥ Reexpressing (38) in the equivalent form 1 1 log r 1+ + O > 1+ (39) 2(log log n)2 (log log n)3  log log n shows that the left side converges to 1 at a faster rate than the right side by a factor of 1/ log log n. Clearly, there is a contradiction for all r 2. Ergo, σ(nr) < eγ n log log n for all sufficiently large integers n 1, and a fixed ≥ ≥ parameter r 2.  ≥ sum of divisors function inequality 12

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