Department of Mathematics and Statistics

Boston University

Boston, MA 02215

(617) 353-9545

rohrlich@math.bu.edu.

1. Self-dual Artin representations

2. Artin representations of Q of dihedral type

3. Quaternionic Artin representations of Q

4. Appendix to: A. Karnataki, Self-dual
Artin representations of dimension three

5. A Taniyama product for the Riemann zeta function

6. Almost abelian Artin representations of Q

*A Taniyama product for the Riemann zeta function*. In:
*Exploring the Riemann Zeta Function: 190 Years from Riemann's
Birth*,
H. Montgomery, A. Nikeghbali, and M. Rassias eds.,
Springer
(2017).

I deeply apologize for being completely unaware of, and hence not
acknowledging, the following prior literature:

[1] K. Joshi, R. Ragunathan, * Infinite product identities for
L-functions*, Illinois J. Math. 49 (2005), 885-891.

[2] K. Joshi, C. S. Yogananda, * A remark on products of Dirichlet
L-functions*, Acta Arith. 91 (1999), 325-327.

[3] J. Kaczorowski, A. Perelli, * Some remarks on infinite products
of L-functions*, J. Math. Anal. and Appl. 406 (2013), 293-298.

In particular, Kaczorowski and Perelli consider product identities
for a vastly more general class of L-functions,
the identity for $1/\zeta(s)$ being merely a special case.
I am grateful to Alberto Perelli for drawing my attention to the
above literature and for showing me a proof of Theorem 2 valid
for the subclass of Euler products in [3] which are of "polynomial
type."

* Quaternionic Artin representations of Q *,
Math. Proc. Cambridge Phil. Soc. 163 (2017), 95-114.

In the first paragraph of the introduction, the
asymptotic relation proved by Klueners is misstated:
It should be `$Z(k,G;x)\sim cx^a$`

with a positive constant `$c$`

.

*Self-dual Artin representations*. In:
*Automorphic Representation and L-functions*,
D. Prasad, C.S. Rajan, A. Sankaranarayanan, J. Sengupta, eds.,
Tata Institute of Fundamental Research Studies in Mathematics Vol. 22
(2013), 455 -- 499.

The following errors have been corrected in the above pdf file
but disfigure the published version:

(i) The sentence following (4.1) on p. 465 should be corrected as follows:
"Indeed let `$m$`

be a positive integer not divisible by 4 such
that the greatest common divisor of `$m$`

and the discriminant of
`$F$`

divides 2, and let ..."

(ii) In the second line from the bottom of p. 472, `$\mathfrak q$`

should be `$q$`

.

(iii) The constant in the asymptotic
relation (9.3) on p. 487 is incorrect, both because of double-counting and
because the formula for `$q(\rho)$`

holds in general only if `$q(\psi)$`

is
relatively
prime to `$d_K$`

. However, the bound
`$\vartheta^{\ab, K}_{\Q,2}(x) \ll x$`

still holds,
and if (9.3) is replaced by this bound then the rest of the argument goes
through as before.

*A deformation of the Tate module*,
J. of Algebra 229 (2000), 280 -- 313.

The formula for ` $\theta(c)$ `

on p. 294 is in error:
The coefficient of ` $X^2$ `

should be
` $(c^4-c^2)/12$, `

not ` $(c^4+2c^2-3c)/24$. `

Although very embarrassing,
the error is inconsequential in the sense that there
was no need to display the coefficient of ` $X^2$ `

in the first place.

*Galois theory, elliptic curves, and root numbers*,
Compositio Math. (1996), 311 -- 349.

In the first line of the paragraph containing displayed formula
(3.6) on p. 331, the equation ` $K=F(\Delta^{1/e})$`

should be ` $K=F(\varpi^{1/e})$`

.