1. Weierstrass products 2. Poisson-Jensen formula 3. Hadamard products Apart from factorization of polynomials, after Euler's sin πz πz = ∞ n=1 1 − z 2 n 2 there is Euler's product for Γ(z), which he used as the definition of the Gamma function: ∞ 0 e −t t z dt t = Γ(z) = 1 z e γz ∞ n=1 1 + z n e −z/n where the Euler-Mascheroni constant γ is essentially defined by this relation. The integral (E...

For an infinite sequence of independent coin tosses with P(Heads)=pE(O, l), the longest run of consecutive heads in the first n tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum of n(1 p ) i.i.d. exponential random variables. These results are ext...

In the fastest-performing integer factoring algorithms, one creates a sequence of integers (in a pseudo-random way) and wishes to rapidly determine a subsequence whose product is a square. In 1994 Pomerance stated the following problem which encapsulates all of the key issues: Select integers a1, a2, . . . , at random from the interval [1, x], until some (non-empty) subsequence has product equa...

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ(n) := ∑ d|n d < en log log n is satisfied for n ≥ 5041, where γ denotes the Euler(Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreo...

Robin’s criterion states that the Riemann hypothesis is true if and only inequality $$\sigma (n) < e^{\gamma } \times n \log n$$ holds for all natural numbers $$n > 5040$$ , where (n)$$ sum-of-divisors function of $$\gamma \approx 0.57721$$ Euler–Mascheroni constant. We show Robin are not divisible by some prime between 2 1771559. prove when $$\frac{\pi ^{2}}{6} n' \le $$n>5040$$ $$n'$$ square ...

We generalize well-known Catalan-type integrals for Euler’s constant to values of the generalized Euler constant function and its derivatives. Using generating functions appearing in these integral representations, we give new Vacca and Ramanujan-type series for values of the generalized Euler constant function and Addison-type series for values of the generalized Euler constant function and it...

Here is an immediate proof in the following cases: 1. a(n) = n: 2. a(n) = d(n) =number of divisors of n; 3. a(n) = w(n) =number of distinct prime divisors of n: 4. a(n) = D(n) =number of total prime divisors of n (that is. counted with repetitions): 5. a(n) = dJ(n) =the Euler function of n: 6. a( n) = cr( n) =the sum of the divisors of n; 7. a(n) = Pn =the nth prime: 8. a(n) = 71(n) =the number...