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A positivity conjecture for a quotient of <i>q</i>-binomial coefficients. [PDF]
Ramanujan JGatzweiler M, Krattenthaler C.
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LRCS: Duality, LP bounds, and field size. [PDF]
Des Codes CryptogrGruica A, Jany B, Ravagnani A.
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The prime number theorem and fragments ofP A
Archive for Mathematical Logic, 1994Let \(I\Delta_ 0\) denote the subsystem of Peano arithmetic obtained by allowing induction for bounded formulas only. Let exp denote the axiom \(\forall x,y \exists z (z= x^ y)\), where \(z= x^ y\) is a \(\Delta_ 0\) formula defining the graph of the exponential function in \(\mathbb{N}\) and having, provably in \(I\Delta_ 0\), all the usual properties
Costas Dimitracopoulos
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A Heuristic for the Prime Number Theorem
The Mathematical Intelligencer, 2006The prime number theorem states that the number \(\pi(x)\) of primes less than \(x\) is asymptotic to \(x / \ln x\). As is well known, already Chebyshev proved that \(\pi(x)\) is bounded from below and above by functions of the type \(cx/\ln x\) for certain constants \(c\), and that if \(\pi(x) \sim cx/\ln x\), then \(c = 1\).
Montgomery, Hugh L., Wagon, Stan
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American Journal of Mathematics, 1942
A few years ago, Watson succeeded in proving an asymptotic congruence property formulated by Ramanujan, namely, the following theorem: If m and Ic are fixed positive integers, there are between n = 1 and n = N only o (N) integers n for which the sum of the (2m 1) -th powers of all divisors of n is not a multiple of k.
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A few years ago, Watson succeeded in proving an asymptotic congruence property formulated by Ramanujan, namely, the following theorem: If m and Ic are fixed positive integers, there are between n = 1 and n = N only o (N) integers n for which the sum of the (2m 1) -th powers of all divisors of n is not a multiple of k.
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2015
In the wake of Euclid’s proof of the infinitude of the primes, the question of how the primes were distributed among the integers became central — a question that has intrigued and challenged mathematicians ever since. The sieve of Eratosthenes provided a simple but very inefficient means of identifying which integers were prime, but attempts to find ...
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In the wake of Euclid’s proof of the infinitude of the primes, the question of how the primes were distributed among the integers became central — a question that has intrigued and challenged mathematicians ever since. The sieve of Eratosthenes provided a simple but very inefficient means of identifying which integers were prime, but attempts to find ...
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A Tauberian Theorem and Analogues of the Prime Number Theorem
Canadian Journal of Mathematics, 1968In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and1then ƒ(x) ∼ cx. His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s), on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).
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