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Denominators of the partial Euler product representation of Pi/4.
1

%I #27 Oct 09 2019 13:36:07

%S 4,16,128,512,6144,98304,393216,9437184,37748736,1207959552,

%T 43486543872,1739461754880,6957847019520,333976656936960,

%U 1335906627747840,80154397664870400,4809263859892224000,19237055439568896000,1385067991648960512000,99724895398725156864000

%N Denominators of the partial Euler product representation of Pi/4.

%C The Euler product representation follows from the classical Leibniz series representation of Pi/4 interpreted as a Dirichlet L-series using the unique non-principal Dirichlet characters modulo 4, whose (infinite) Euler product representation can be written as (3/4) * (5/4) * (7/8) * (11/12) * (13/12) * (17/16) * (19/20) * ... with each term in the product being the ratio of a prime number to its nearest multiple of 4. The sequence consists of the denominators of the partial products.

%H N. Elkies, <a href="http://www.math.harvard.edu/~elkies/M259.06/dirichlet.pdf">Introduction to Analytic Number Theory: Primes in Arithmetic Progression, Dirichlet Characters and L-Functions</a>

%H L. Euler, <a href="https://arxiv.org/abs/math/0506415">On the sums of series of reciprocals</a>, arXiv:math/0506415 [math.HO], 2005-2008.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Superparticular_ratio">Superparticular ratio</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wallis_product">Wallis Product</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gregory%27s_series">Gregory Series</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Madhava_series">Madhava Series</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Machin-like_formula">Machin-like Formula</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Inverse_trigonometric_functions">Inverse Trigonometric Functions</a>

%e a(3) = 128 = denominator((3/4) * (5/4) * (7/8)).

%o (PARI) a(n) = denominator(prod(k=2, n+1, my(p=prime(k)); if(p%4==1, p/(p-1), p/(p+1)))); \\ _Daniel Suteu_, Jan 22 2019

%Y Cf. A003881 (Decimal Expansion of Pi/4).

%Y Cf. A101455 (Dirichlet L-series of The Non-Principal Dirichlet Characters Mod 4).

%Y Cf. A323551 (Numerators of the Partial Euler Product Representation of Pi/4).

%K nonn,frac

%O 1,1

%A _Anthony Hernandez_, Jan 17 2019

%E More terms from _Daniel Suteu_, Jan 22 2019