Breusch's theorem on primes between x and 9x/8

The modulus \(Q_9 = 210\) to The grid function reaches \(2\) at \(657\)

• The modulus \(Q_9 = 210\)
• The Möbius divisor table for \(210\)
• The divisor table for \(210\) has 16 entries
• The divisor table for \(210\) lists every divisor
• Divisors in the table for \(210\) are positive
• Entries of the divisor table for \(210\) divide \(210\)
• Summand of the Breusch grid function
• Breusch grid function for \(k = 9\)
• Grid bound \(G_9 \le 3\) for \(0 \le j < 2700\)
• Grid bound \(G_9 \le 3\) for \(2700 \le j < 5400\)
• Grid bound \(G_9 \le 3\) for \(5400 \le j < 8100\)
• Grid bound \(G_9 \le 3\) for \(8100 \le j < 10800\)
• Grid bound \(G_9 \le 3\) for \(10800 \le j < 13500\)
• Grid bound \(G_9 \le 3\) for \(13500 \le j < 16200\)
• Grid bound \(G_9 \le 3\) for \(16200 \le j < 18900\)
• Grid bound \(G_9 \le 3\)
• The grid function reaches \(1\) at \(279\)
• The grid function stays below \(1\) on \((10, 279)\)
• First index from \(10\) where the grid function reaches \(1\)
• The grid function reaches \(2\) at \(657\)

The grid function stays below \(2\) on \((10, 657)\) to Quotients of Q₉ by its divisors are positive

• The grid function stays below \(2\) on \((10, 657)\)
• First index from \(10\) where the grid function reaches \(2\)
• The grid function reaches \(3\) at \(963\)
• The grid function stays below \(3\) on \((10, 963)\)
• First index from 10 on with grid value at least 3
• Signs in the divisor table for \(210\)
• The divisor table for \(210\) has no repeated entries
• Sum over a duplicate-free list
• Exponential bound by the geometric series
• Logarithm is at least 1 − 1/t
• Weak Stirling lower bound for factorials
• Weak Stirling upper bound for factorials
• Scaled factorial argument \(N_{10}\)
• Scaled factorial argument N₉
• Scaled factorial argument \(N_1\)
• Binomial factor ratio₉
• Logarithm of Breusch's product \(F_9(n)\)
• Q₉ is positive
• Value of Q₉
• Quotients of Q₉ by its divisors are positive

Positivity of the scaled factorial argument \(N_{10}\) to Exact formula for the scaled factorial argument \(N_{10}\)

• Positivity of the scaled factorial argument \(N_{10}\)
• Positivity of the scaled argument N₉
• Positivity of the scaled factorial argument \(N_1\)
• Upper bound for the scaled factorial argument \(N_{10}\)
• Upper bound for the scaled factorial argument \(N_9\)
• Upper bound for the scaled factorial argument \(N_1\)
• N₁₀ splits as N₉ + N₁
• Factorial divisibility behind ratio₉
• Positivity of ratio₉
• Logarithm of the binomial factor
• Möbius reciprocal sum over the divisors of 210, in the rationals
• Möbius reciprocal sum over the divisors of 210, in the reals
• Lower bound for \(\log N!\)
• Upper bound for \(\log N!\)
• Main Stirling term
• Termwise Stirling lower bound in Breusch's argument
• Divisors of 210 divide multiples of 210
• The constant \(K_9\)
• Closed form of the Möbius-weighted main-term sum
• Exact formula for the scaled factorial argument \(N_{10}\)

Exact formula for the scaled factorial argument \(N_9\) to Power bound for the multinomial coefficient \(M(t)\), small \(t\)

• Exact formula for the scaled factorial argument \(N_9\)
• Exact formula for the scaled factorial argument \(N_1\)
• Combined main term of the Möbius-weighted Stirling sums
• Lower bound for \(\log F_9(n)\)
• Extending the primorial bound leftward across an interval
• The multinomial coefficient \(M(t)\) for the 3.4 primorial bound
• Factorial divisibility behind the multinomial coefficient
• Positivity of the multinomial coefficient \(M(t)\)
• Cleared-denominator identity for the multinomial coefficient \(M(t)\)
• The multinomial coefficient \(M(t)\) as a product of binomial coefficients
• Half-power bound for the central binomial coefficient
• Multinomial bound for \((3t)!\)
• Binomial factorization of the factorial of a multiple of three
• Four-power lower bound for the central binomial coefficient
• Key estimate for the multinomial coefficient \(M(t)\)
• Base case of the threshold inequality
• Induction step of the threshold inequality
• Threshold inequality for the multinomial bound
• Power bound for the multinomial coefficient \(M(t)\), large \(t\)
• Power bound for the multinomial coefficient \(M(t)\), small \(t\)

Power bound for the multinomial coefficient \(M(t)\) to Upper bound for log 3

• Power bound for the multinomial coefficient \(M(t)\)
• Superadditivity of floor division for three terms
• Floor-difference bound on the range 3t < p ≤ 6t
• Floor-difference bound on the range 2t < p ≤ 3t
• Floor-difference bound on the range 3t/2 < p ≤ 2t
• Floor-difference bound on the range 6t/5 < p ≤ 3t/2
• Floor-difference bound on the range t < p ≤ 6t/5
• Floor-difference bound on the range t < p ≤ 6t
• Primes in \((t, 6t)\) occur in \(M(t)\)
• Splitting the primorial with a multinomial coefficient
• Primorial bound \(3.4^m\) below \(133\)
• Primorial bound by powers of 3.4, integer form
• Primorial bound by powers of 3.4, real form
• Bounding a logarithm above by the exponential
• Logarithm lower bound from an exponential bound
• Exponential lower bound from a power comparison
• Power comparison above e bounds an exponential
• Lower bound for log 2
• Lower bound for log 5
• Upper bound for log 3

Upper bound for log 17 to One-period shift of \(G_9\)

• Upper bound for log 17
• Logarithm of 10
• Logarithm of 9
• Logarithm of 3.4
• Evaluation of a three-term rational sum
• Numerical margin of the main term
• Prime-counting bound \(\pi(m) \le m/3\) for \(36 \le m < 156\)
• Prime-counting bound \(\pi(m) \le m/3\) from \(156\) on
• Prime-counting bound \(\pi(m) \le m/3\) for natural \(m \ge 36\)
• Prime-counting bound \(\pi(x) \le x/3\) for real \(x \ge 36\)
• Upper bound for log 10
• Strict decay of log over square root beyond e squared
• Logarithm bounded by a multiple of the square root
• Positivity of the gap in Breusch's prime interval argument
• Continuous Breusch function
• Floor of a quotient of natural numbers
• Remainder bound for common multiples
• Real floors at divisors of 210 agree with grid floors
• Shifting a floor quotient by one period
• One-period shift of \(G_9\)

Periodicity of \(G_9\) in multiples of \(18900\) to The continuous Breusch function vanishes at small ratios

• Periodicity of \(G_9\) in multiples of \(18900\)
• \(G_9\) depends only on the residue modulo \(18900\)
• The continuous Breusch function agrees with the grid function
• The Breusch grid function vanishes below 9
• Value of \(G_9\) at \(9\)
• Bound \(G_9 \le 1\) below \(10\)
• The continuous Breusch function is at most 3
• The continuous Breusch function vanishes near zero
• Level-\(1\) threshold for the continuous Breusch function
• Level-\(2\) threshold for the continuous Breusch function
• Level-\(3\) threshold for the continuous Breusch function
• Integer cast of the floor of a quotient of natural numbers
• Legendre terms of \(N_{10}\) as real floors
• Legendre terms of \(N_9\) as real floors
• Legendre terms of \(N_1\) as real floors
• Legendre's formula with integer summands
• Valuation of a factorial ratio as a difference of valuations
• One divisor's contribution to the valuation
• The \(p\)-adic valuation of the weighted product
• The continuous Breusch function vanishes at small ratios

Valuation identity for the weighted product to The primorial as a product of primes

• Valuation identity for the weighted product
• Single-term valuation identity for large primes
• Value of \(10Qn\)
• Logarithm of a natural number from its factorization
• Bound on the prime factors of the ratio factors
• Valuation term of one binomial factor
• Valuation terms vanish off the prime support
• A weighted log-factor as a sum over primes
• Prime factorization of \(\log F_9(n)\)
• Logarithms of primes are nonnegative
• \(G\) is at most its positive part
• The truncated continuous Breusch function is at most 3
• Positive lower bounds on a truncation lift to the integer
• Layer-cake decomposition with three levels
• A level bound turns into a bound on the prime
• Primes up to \(1890n\) have \(n/p \ge 1/1890\)
• Upper bound for primes where a Möbius-weighted floor sum is at least 1
• Upper bound for primes where a Möbius-weighted floor sum is at least 2
• Upper bound for primes where a Möbius-weighted floor sum is at least 3
• The primorial as a product of primes

Chebyshev-type bound for the sum of prime logarithms to Primes in \((X, 9X/8)\) for large \(X\), assuming the upper bound

• Chebyshev-type bound for the sum of prime logarithms
• Prime-logarithm bound for one level of \(G\)
• Contribution of primes with \(p^2 > 2100n\)
• Bound for one small-square prime term
• Counting small-square primes
• Contribution of the small-square primes
• Upper bound for \(\log F_9(n)\) when \((1890n, 2100n)\) has no prime
• Breusch's endgame upper-bound inequality
• Breusch's prime chain
• Entries of Breusch's prime chain are prime
• Breusch's prime chain is strictly increasing
• Consecutive entries of Breusch's prime chain are within a factor \(9/8\)
• Last entry of Breusch's prime chain
• Bracketing a real number in an increasing list
• Consecutive pairs survive prepending
• Adjacent entries form a consecutive pair
• Primes in \((X, 9X/8)\) for \(53 \le X < 9820997\)
• Every large real lies in a window \((5600n/3, 1890n)\)
• Endgame contradiction in Breusch's theorem
• Primes in \((X, 9X/8)\) for large \(X\), assuming the upper bound

Breusch's theorem from the upper bound to Breusch's theorem on primes between \(x\) and \(9x/8\)

• Breusch's theorem from the upper bound
• Breusch's theorem on primes between \(x\) and \(9x/8\)

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